Spaces of Algebra Structures and Cohomology of

Operads

by

Charles W. Rezk

Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 1996

\begin{matrix}c \\ ◯ \end{matrix} Massachusetts Institute of Technology 1996. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Department of Mathematics May 3, 1996

Certified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Michael J. Hopkins Professor of Mathematics

Thesis Supervisor

Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

David A. Vogan

Chairman, Departmental Committee on Graduate Students Spaces of Algebra Structures and Cohomology of Operads

by Charles W. Rezk

Submitted to the Department of Mathematics on May 3, 1996, in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract

The aim of this paper is two-fold. First, we compare two notions of a “space” of algebra structures over an operad A:

1. the classification space, which is the nerve of the category of weak equivalences of A-algebras, and

2. the moduli space A{X}, which is the space of maps from A to the endomorphism operad of an object X.

We show that under certain hypotheses the moduli space of A-algebra structures on X is the homotopy fiber of a map between classification spaces.

Second, we address the problem of computing the homotopy type of the moduli space A{X}. Because this is a mapping space, there is a spectral sequence computing its homotopy groups with E_{2}-term described by the Quillen cohomology of the operad A in coefficients which depend on X. We show that this Quillen cohomology is essentially the same, up to a dimension shift, as the Hochschild cohomology of A, and that the Hochschild cohomology may be computed using a “bar construction”.

Thesis Supervisor: Michael J. Hopkins Title: Professor of Mathematics

Contents

1 Introduction and statement of results 4 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Moduli spaces of algebra structures . . . . . . . . . . . . . . . . . . . 9 1.3 Cohomology of operads . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Operads and related categories 16 2.1 Monoidal categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Operads and related categories . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Constructions involving Σ-objects . . . . . . . . . . . . . . . . . . . . 29 2.4 Monoidal functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Homotopy theory of operads 38 3.1 Closed model categories . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Model categories over simplicial sets . . . . . . . . . . . . . . . . . . . 44 3.3 Model category structures over simplicial modules . . . . . . . . . . . 47 3.4 Compatibility of actions with model category structure . . . . . . . . 50 3.5 Cofibrant objects and forgetful functors . . . . . . . . . . . . . . . . . 54 3.6 Relations between homotopy categories . . . . . . . . . . . . . . . . . 55 3.7 Standard resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Moduli spaces of algebra structures 61 4.1 Moduli spaces of algebra structures . . . . . . . . . . . . . . . . . . . 61 4.2 Relation between moduli space and classification spaces . . . . . . . . 65

5 Cohomology of operads 71 5.1 Abelian group objects in simplicial R-module operads . . . . . . . . . 71 5.2 The fundamental exact sequence . . . . . . . . . . . . . . . . . . . . . 73 5.3 Model category structures for abelian group objects . . . . . . . . . . 78 5.4 Equivalence of Quillen and Hochschild cohomologies . . . . . . . . . . 79

A Free monoids 81

Chapter 1

Introduction and statement of results

1.1 Introduction

1.1.1 Operads

Operads and algebras over operads may be viewed as generalizations of the familiar concepts of rings and modules. Thus, let 𝒜 denote the category of abelian groups, and let ⊗ denote tensor product of abelian groups. Any abelian group A induces a functor A⊗−: 𝒜 → 𝒜 sending an abelian group X to A⊗X. We see that composition of such functors A ⊗ − and B ⊗ − corresponds to tensoring with the abelian group A ⊗ B. An associative ring A is simply a monoid-object with respect to this tensor product. An A-module is an algebra over the induced triple A ⊗ − on 𝒜.

We can proceed in an analogous fashion when we replace 𝒜 with a category of Σ-objects. Let (C, ⊗) be a symmetric monoidal category, such as (𝒮, ×), the category of of simplicial sets with monoidal product given by cartesian product, or (ℳ_{R}, ⊗_{R}), the category of simplicial R-modules with monoidal product given by tensor product over R. A Σ-object on C is essentially a collection of objects {A[n] ∈ C} where each A[n] is equipped with an action of the symmetric group Σ_{n}; we let ΣC denote the category of Σ-objects on C. Each Σ-object A induces a functor A(−): C → C defined by the formula

A(X) ≃ ∐_{n≥0} A[n] ⊗_{Σ_{n}} X^{⊗n}.

There is a Σ-object I which induces the identity functor on C.

It turns out that for A, B ∈ ΣC the composite functor A(B(−)): C → C is itself induced by a Σ-object, which we denote by A∘B. The operation −∘−: ΣC×ΣC → ΣC makes ΣC into a (non-symmetric) monoidal category with unit I.

An operad A on C, is an object of ΣC equipped with maps A ∘ A → A and I → A making the appropriate associativity and unit diagrams commute; i.e., an operad is simply a “monoid” in ΣC. The category of operads on C is denoted by operC. Thus any operad A induces a corresponding triple (also called a monad) on

An A-algebra is an object X ∈ C equipped with a map A(X) → X making the appropriate associativity and unit diagrams commute; equivalently, an A-algebra is an algebra over the triple A(−). We refer to such a map A(X) → X as an A-algebra structure on X. The category of A-algebras is denoted by C^{A}.

1.1.2 Classifying objects for the category of algebras

The introduction of the concept of operads allows us to note that the structure of the category of A-algebras can be recovered entirely in terms of the operad A. We illustrate this first with the example of rings and modules. Given an abelian group M, the group End_{M} = hom_{𝒜} (M, M) is naturally a ring, and given a ring R there is a natural bijective correspondence

{R-module structures on M} ⟷ {ring maps R → End_{M}}.

Likewise, given R-modules M and N, the set of abelian group homomorphisms hom_{𝒜} (M, N) has a natural structure as an R-bimodule, and there is a natural bijective correspondence

{R-module maps M → N} ⟷ {R-bimodule maps R → hom_{𝒜} (M, N)}.

These observations have analogies when we replace 𝒜 with a category of Σ-objects. Let C be a symmetric monoidal category as before. Given an object X ∈ C, there is an endomorphism operad E_{X} defined by E_{X}[n] = map_{C}(X^{⊗n}, X). It has the property that there is a natural bijective correspondence

{A-algebra structures on X} ⟷ {operad maps A → E_{X}},

(see 2.1.11 and 2.2.13).

Likewise, given two A algebras X, Y there is a Σ-object ℌ𝔬𝔪(X, Y) defined by ℌ𝔬𝔪(X, Y)[n] = map_{C}(X^{⊗n}, Y). The A-algebra structures on X and Y induce on ℌ𝔬𝔪(X, Y) the structure of an A-biobject; i.e., there are maps A ∘ ℌ𝔬𝔪(X, Y) → ℌ𝔬𝔪(X, Y) and ℌ𝔬𝔪(X, Y) ∘ A → ℌ𝔬𝔪(X, Y) satisfying obvious properties. It has the property that there is a natural bijective correspondence

{A-algebra maps X → Y } ⟷ {A-biobject maps A → ℌ𝔬𝔪(X, Y)}

(see 2.1.19 and 2.2.15).

The above shows that the category of A-algebras can be understood by looking at appropriate maps from A into certain “classifying objects”; in particular, we can get derived functors of these by resolving the operad A. In the case of maps of operads A → E_{X}, the relevant derived functor is the moduli space of A-algebra structures on X.

1.1.3 Moduli space of algebra structures

Let C denote either 𝒮, the category of simplicial sets, or ℳ_{R}, the category of simplicial R-modules. These are categories “enriched over simplicial sets”; in particular, for any pair of objects X, Y ∈ C there is a simplicial set map_{C}(X, Y) having as 0-simplices the maps X → Y (see 2.3.19). Thus given an operad A over C we may consider the simplicial set of maps

A{X} = map_{C}(A, E_{X}).

We call this space the moduli space of A-algebra structures.

1.1.4 Classification space

This is another kind of “moduli space” defined entirely in terms of a category C along with a subcategory of weak equivalences.

Let C be a category, and let wC ⊂ C denote a subcategory containing all the objects of C, the maps of which we will call “weak equivalences”. One defines the classification space (1.2.4) to be the simplicial nerve of wC. The classification space may be viewed as a “moduli space of objects of C”; in particular, the components of wC are in one-to-one correspondence with weak equivalence classes of objects in C. If C is in fact a simplicial closed model category (see 3.1.1), then the component of wC which contains an object X ∈ C has the homotopy type of the classifying space of the monoid of self-equivalences of X (see Proposition 1.2.6).

Clearly, if π: C → D is a functor between such categories which preserves all weak equivalences, then there is an induced map wC → wD of classification spaces.

We will introduce simplicial model category structures on the categories of Aalgebras and the category of operads over simplicial sets or simplicial R-modules, allowing us to state and prove the following theorem.

1.1.5. Theorem. Let C denote the category 𝒮 or the category ℳ_{R}. Let A be a cofibrant operad, and let X be a fibrant and cofibrant object of C. Then the moduli space A{X} is naturally weakly equivalent to the homotopy fiber over X of the map of classification spaces N(wC^{A}) → N(wC).

This result is discussed in more detail in Section 1.2.

1.1.6 Quillen cohomology of operads

Recall that the moduli space A{X} of algebra structures is a space of maps between operads. If A is a cofibrant operad and X ∈ C is a fibrant and cofibrant object (whence E_{X} is a fibrant operad), and f: A → E_{X} is a given map, then there is a second quadrant spectral sequence of the form

E_{2}^{st} ≃ H_{oper}^{s}(A, π_{t}(E_{X}; f)) ⇒ π_{t−s}(A{X}; f),

where H_{oper}^{s}(A, U) denotes the Quillen cohomology of the operad A with coefficients in an abelian group object U over A, and π_{t}(E_{X}; f) is the abelian group object whose fiber over a point a ∈ A[n] is given by

π_{t}(E_{X}; f)_{a} ≃ π_{t}(map_{C}(X^{n}, X); f(a)).

An instance of this kind of spectral sequence is described by Robinson [16].

Quillen cohomology of an operad A is defined by resolving A by a cofibrant operad. Such cofibrant resolutions of operads are difficult to compute with, and so it is useful to have another description of Quillen cohomology.

1.1.7 Hochschild cohomology of operads

Because operads are “monoids” in the monoidal category ΣC, we define an Abiobject to be a Σ-object M with a two-sided action by the monoid A. We may consider the Quillen cohomology of an A-biobject M. A special example of an A-biobject is A itself. Thus we may define Hoch^{s}(A, K), the Hochschild cohomology of an operad A with coefficients in an abelian group object U over A, to be H_{Abiobj}^{s}(A, U), the Quillen cohomology of A as an A-biobject.

1.1.8. Theorem. Let C denote the category ℳ_{R} (and hopefully the category 𝒮). Let A ∈ operC be an operad with cofibrant underlying Σ-object, and let U be an abelian group object over A. Then there is a natural exact sequence

0 → Hoch^{0}(A, U) → Hoch^{0}(I, η^{∗}U) → H_{oper}^{0}(A, U) → Hoch^{1}(A, U) → 0

and for n ≥ 1 natural isomorphisms

Hoch^{n+1}(A, U) ≃ H_{oper}^{n}(A, U).

(Here I denotes the trivial operad, and η^{∗}U is the pull-back of the coefficient system U along the unit map η: I → A.)

Furthermore, the cohomology of a Σ-cofibrant operad may be computed using a “bar complex”; the bar complex B(A, A, A) of an operad A is a simplicial object whose n-th degree term is the A-biobject A^{∘(n+2)}; see 3.7.1. (This is not the same as the Ginzburg-Kapranov bar complex of an operad of [8].)

Finally, we note that the argument which proves Theorem 1.1.8 is largely formal. We call a monoidal category M right-closed (2.1.9) if the monoidal product − ∘ − has a “right adjoint on one side”; i.e., if for each A ∈ M the functor B ↦ B ∘ A has a right adjoint. Given a monoid A in M and mild conditions on the category M, we will show (Proposition 5.2.14) that there is a short exact sequence of abelian group objects

0 → D(A) → A_{b}(A ∘ A) → A_{b}(A) → 0,

where D(A) denotes the “abelianization” of the operad A and A_{b}(M) denotes the “abelianization” of an A-biobject M. This short exact sequence is a key step in the proof of Theorem 1.1.8.

1.1.9 Notation

We will use 𝒮 to denote the category of simplicial sets, and will use ℳ_{R} to denote the category of simplicial modules over a commutative ring R.

For a category C and objects X, Y ∈ C we write C[X, Y] to denote the set of maps from X to Y in C.

Given categories C, D, we write C^{D} for the category of functors D → C.

Given a functor X: D → C from a small category D we write colim^{D} X for the colimit of this functor. If D = G is a group, we write X^{G} ∈ C for the limit and X_{G} ∈ C for the colimit.

Given a functor π: D → C and an object X ∈ C we let (π ↓ X) denote the over category, having as objects pairs (Y ∈ D, f: πY → X ∈ C) and as maps (Y, f) → (Y′, f′) morphisms g: Y → Y′ ∈ D such that f′g = f. If C = D and π = 1_{C} we write (C ↓ X).

1.1.10 Looking ahead

In this paper we consider only operads acting on simplicial sets or on simplicial modules. It is desirable to extend this theory to a category of spectra; however, it seems some care is needed in order to carry this out. For example, the category of operads on a universe, as described by Lewis-May-Steinberger [11], does not admit an action by a category of Σ-objects; rather, it is Σ-objects over a certain fixed operad which act on these spectra. Once these differences are worked out, the theory proceeds similarly.

1.1.11 Organization of the paper

In Section 1.2 we describe Theorem 1.1.5 and some of its consequences. In Section 1.3 we do the same for Theorem 1.1.8.

In Sections 2.1 through 2.4 we establish the categories in which we work. In Section 2.1 we define the notion of a right-closed monoidal category. In Section 2.2 we define the notion of an operad, along with the related notions of algebras over an operad, Σ-objects, and biobjects over an operad; we also define certain enriched homobjects. In Section 2.3 we show that various categories are complete and cocomplete. In Section 2.4 we discuss monoidal functors.

In Sections 3.1 through 3.7 we study the homotopy theory of the categories in question. In Section 3.1 we recall the notion of a closed model category. In Sections 3.2 and 3.3 we show that in the contexts of simplicial sets or simplicial modules, all of these categories can be given the structure of a Quillen simplicial closed model category. In Section 3.4 we show that monoidal structures and enriched hom-objects are compatible with the closed model category structures as best they can be. In Section 3.5 we examine when objects such as operads or A-algebras are cofibrant in some underlying category. In Section 3.6 we show that weakly equivalent operads which are cofibrant in an underlying category generate categories of algebras with equivalent homotopy theories. In Section 3.7 we show that the “standard resolutions” of an A-algebra, or of A as an A-biobject, are cofibrant.

In Sections 4.1 and 4.2 we describe the relationship between the moduli space of algebra structures and classification spaces. In Section 4.1 we define the moduli space of algebra structures of a diagram, and in Section 4.2 we prove Theorem 1.1.5.

In Sections 5.1 through 5.4 we define and compare two notions of cohomology of operads. In Section 5.1 we describe categories of abelian group objects and show that the relevant abelianization functors exist. In Section 5.2 we demonstrate an exact sequence relating the abelianizations of operads and biobjects. In Section 5.3 we show that these categories of abelian group objects admit a Quillen simplicial closed model category structure. In Section 5.4 we use the results of the previous sections to prove the equivalence of Quillen and Hochschild cohomologies of operads.

In Appendix A we discuss the construction of the free operad.

1.2 Moduli spaces of algebra structures

In this section we define for each simplicial set X and each operad A over simplicial sets a structure space. We then state a theorem relating this structure space, under certain hypotheses, to the classification spaces of the categories of simplicial sets and of A-algebras. We then state analogous results concerning operads over simplicial R-modules.

1.2.1 Operads over simplicial sets and their algebras

The category 𝒮 of simplicial sets is a symmetric monoidal category via cartesian product (2.2.1). As described in Section 2.2, we can therefore define in 𝒮 the notion of an operad. In brief, an operad over 𝒮 is a collection of simplicial sets {A[n]} such that A[n] is equipped with a Σ_{n}-action, and such that certain other structure maps are present which make the functor

X ↦ A(X) = ∐_{n=0} A[n] ×_{Σ_{n}} X^{n}

into a triple on 𝒮 (for the full definition, see 2.2.9). Any operad over 𝒮 thus determines a category of algebras (2.2.13), denoted by 𝒮^{A}.

Each such category 𝒮^{A} will be given the structure of a Quillen closed model category, in which the weak equivalences are maps f: X → Y of algebras which are weak equivalences on the underlying simplicial sets (3.2.4). Thus we obtain a homotopy category Ho 𝒮^{A} of A-algebras by formally inverting the weak equivalences.

Any map of operads f: A → B determines a pair of adjoint functors

f_{∗} : 𝒮^{A} ⇄ 𝒮^{B} : f^{∗}

where the right adjoint forgets structure, but preserves the underlying simplicial set (2.3.10). These functors induce by Corollary 3.6.3 a pair of adjoint functors

Ho 𝒮^{A} ⇄ Ho 𝒮^{B}

between the corresponding homotopy categories.

It seems appropriate to define a notion of weak equivalence of operads for which weakly equivalent operads would have equivalent homotopy categories of algebras. Thus we say a map of operads f: A → B is a weak equivalence if for each n ≥ 0 the geometric realization |f[n]|: |A[n]| → |B[n]| is a Σ_{n}-equivariant homotopy equivalence (i.e., there is a Σ_{n}-equivariant map g: |B[n]| → |A[n]| and Σ_{n}-equivariant homotopies |f[n]|g ∼ 1 and g|f[n]| ∼ 1.) Then we obtain the following.

1.2.2. Proposition. A weak equivalence A → B of operads over 𝒮 induces an equivalence Ho 𝒮^{A} ≃ Ho 𝒮^{B}.

Proof. This follows from Corollary 3.6.5 using Proposition 3.5.1.

1.2.3 The structure space

The category of operads on 𝒮 is enriched over simplicial sets; thus, for any A, B ∈ oper𝒮 there is a simplicial set map_{oper𝒮}(A, B), in which the 0-simplices correspond to the usual maps from A to B.

The set of A-algebra structures on a simplicial set X is in bijective correspondence with the set of operad maps A → E_{X}, where E_{X} denotes the endomorphism operad of X. Thus we define the moduli space of algebra structures by

A{X} = map_{oper𝒮}(A, E_{X}).

Note that this construction defines a functor oper𝒮^{op} → 𝒮 sending A ↦ A{X}; note that A{X} is not a functor of X.

We note that an n-simplex of A{X} corresponds precisely to a map of operads Δ[n] ⊗ A → E_{X}, where K ⊗ A is the “tensor product” of a simplicial set K with an operad A (see 2.3.19), and Δ[n] is the standard n-simplex. The map Δ[n] → Δ[0] of simplicial sets is a simplicial homotopy equivalence, and it follows that the map Δ[n] ⊗ A → Δ[0] ⊗ A ≃ A is a weak equivalence of operads.

1.2.4 The classification space

Let C be a simplicial model category (3.1.1). Both 𝒮 and 𝒮^{A} are simplicial model categories, by Propositions 3.1.7 and 3.2.5. Let wC ⊂ C denote the subcategory consisting of all objects and all weak-equivalences between them. We call the simplicial nerve of wC the classification space of C. By abuse of notation we will write wC for its simplicial nerve; a t-simplex in (wC)_{t} is a chain of t composable maps. 1.2.5. Remark. Note that since C is not a small category, wC is not a simplicial set, but merely a simplicial class. We take the position that C is a category defined in some Grothendieck universe of sets (see [10]) U, and that we can regard wC as a simplicial set in some higher universe U′.

Dwyer-Kan prove the following results:

1.2.6. Proposition. [3, 4.6], [4, 2.3]. Given C as above we have that:

1. The components of wC are in one-to-one correspondence with the weak equivalence classes of objects in C.

2. Each component of wC is homotopically small; i.e., has the homotopy type of a small simplicial set.

3. If X ∈ C is a fibrant and cofibrant object, then the component of wC containing X has the weak homotopy type of the classifying space B haut(X) of the simplicial monoid haut(X) ⊆ map_{C}(X, X) of self-homotopy equivalences of X.

More succinctly, we may write

wC ≃ ∐_{ho. types [X]} B haut_{C}(X).

If π: C → D is a functor which preserves weak equivalences, then it induces a map wπ: wC → wD of classification spaces. If π is part of a Quillen equivalence, then the induced map wπ is a weak equivalence. Thus, we have the following:

1.2.7. Proposition. If f: A → B is a map of operads, there is an induced map w𝒮^{B} → w𝒮^{A} of classification spaces; furthermore, if f is a weak equivalence of operads, then the induced map on classification spaces is one as well.

1.2.8 Comparison between structure space and classification space

We can relate A{X} and w𝒮^{A} as follows. Let w𝒮^{Δ[−]⊗A} denote the bisimplicial set having (w𝒮^{Δ[t]⊗A})_{s} as its (s, t)-simplices. There are maps w𝒮^{A} → w𝒮^{Δ[t]⊗A} induced by the map Δ[t] ⊗ A → Δ[0] ⊗ A ≃ A of operads, and maps A{X} → (w𝒮^{Δ[−]⊗A})_{0} which send an n-simplex of of A{X}, considered as a map f: Δ[t] ⊗ A → E_{X}, to the corresponding object (X, f) in 𝒮^{Δ[t]⊗A}.

Thus we obtain a diagram of simplicial sets

[figure: A{X}; i; diag(w𝒮^{Δ[−]⊗A}); j; w𝒮^{A}; k; ∼; π; pt ≃ {X}; w𝒮; _{w𝒮,}]

where k is a weak equivalence by Proposition 1.2.7. Furthermore, i is exactly inclusion of the fiber of j over the vertex in w𝒮 corresponding to the object X.

Since k is a weak equivalence the top row gives a map A{X} → w𝒮^{A} in the homotopy category of simplicial sets.

1.2.9 Main theorem

To state our theorem, we need another definition. An operad A over 𝒮 is said to be cofibrant if it is a retract of one for which there exists a collection S_{t} ⊆ A_{t} of Σ-objects over sets which are closed under degeneracy maps, and isomorphisms F S_{t} ⟶^{∼} A_{t}, where F S_{t} denotes the free operad on S_{t} (see 2.3.6).

1.2.10. Theorem. Let A ∈ oper𝒮 be a cofibrant operad over 𝒮, and let X ∈ 𝒮 be a Kan complex. Then the square

\xymatrix{A\{X\} \ar[d] \ar[r] & w𝒮^{A} \ar[d] \\ pt \ar[r] & w𝒮}

is a homotopy pull-back square of simplicial sets.

1.2.11. Remark. 1. By Proposition 1.2.6, if we restrict the maps in this fiber square to the component of w𝒮 containing X, we obtain a homotopy pull-back square

\xymatrix{A\{X\} \ar[r] \ar[d] & ∐_{[Y],πY ≃X} B haut_{𝒮}^{A}(X) \ar[d] \\ pt \ar[r] & B haut_{𝒮}(X),}

where the coproduct in the upper-right corner is indexed over representatives of weak-equivalence-types of A-algebras whose underlying simplicial sets are weakly equivalent to X. We can think of this square as defining a haut_{𝒮}(X)- action on A{X} up to homotopy, and we can think of ∐ B haut_{𝒮}^{A}(X) as the homotopy quotient by this action.

2. By Proposition 3.2.11 operads over 𝒮 are a closed model category having cofibrant objects as above, and weak equivalences as described in 1.2.1. Thus for any A ∈ oper𝒮 there exists a weak equivalence B → A ∈ oper𝒮 from a cofibrant operad, whence Ho 𝒮^{B} ≃ Ho 𝒮^{A}. Thus, even if A is not cofibrant, there is a homotopy pull-back square

\xymatrix{B\{X\} \ar[r] \ar[d] & w𝒮^{A} \ar[d] \\ pt \ar[r] & w𝒮.}

There is a “relative” version of Theorem 1.2.10:

1.2.12. Corollary. Let A → B ∈ oper𝒮 be a map between cofibrant of operads over 𝒮, and let X ∈ 𝒮 be a Kan complex. Then the square

\xymatrix{B\{X\} \ar[r] \ar[d] & w𝒮^{B} \ar[d] \\ A\{X\} \ar[r] & w𝒮^{A}}

is a homotopy pull-back square in the homotopy category of simplicial sets.

1.2.13 Corresponding results over simplicial R-modules

Let R be a commutative ring, and let ℳ_{R} denote the category of simplicial R-modules. As before, we can define the category of operads over ℳ_{R}, denoted by operℳ_{R}, and their categories of algebras, denoted by ℳ_{R}^{A} for A ∈ operℳ_{R}. We can obtain similar results in this setting, but we need to be more careful.

We say an A-algebra X over ℳ_{R} is ℳ_{R}-cofibrant if the underlying simplicial R-module of X is projective in each dimension. We say an operad A over ℳ_{R} is Σ-cofibrant if in each degree A[n] is a retract of some RS, a free R-module on a Σ_{n}-set S. Clearly, if R is a field, then every A-algebra is ℳ_{R}-cofibrant, and if R is a field of characteristic zero, then every operad A is Σ-cofibrant.

The statements of Sections 1.2.1, 1.2.3, 1.2.8, and 1.2.9 hold with 𝒮 replaced by ℳ_{R}, with the following exceptions:

We say that a map A → B ∈ operℳ_{R} is a weak equivalence if for each subgroup H ⊆ Σ_{n} the map of fixed point sets A[n]^{H} → B[n]^{H} is a weak equivalence of simplicial sets. Then we obtain

1.2.14. Proposition. A weak equivalence A → B ∈ operℳ_{R} of operads over ℳ_{R} which are Σ-cofibrant induces an equivalence Ho ℳ_{R}^{A} ≃ Ho ℳ_{R}^{B}.

Proof. This follows from Corollary 3.6.5 when we take C = ℳ_{R}.

Thus the corresponding statement of the main theorem is

1.2.15. Theorem. Let A ∈ operℳ_{R} be a cofibrant operad over ℳ_{R}, and let X ∈ ℳ_{R} be a degree-wise projective object. Then the square

\xymatrix{A\{X\} \ar[d] \ar[r] & w𝒮^{A} \ar[d] \\ pt \ar[r] & w𝒮}

is a homotopy pull-back square in the homotopy category of simplicial sets.

1.3 Cohomology of operads

In this section we define the notions of Quillen cohomology of operads and Hochschild cohomology of operads in the category of simplicial R-modules. We then state a theorem which states that, under mild hypotheses, these cohomology theories are essentially the same up to a dimension shift.

1.3.1 Coefficient systems

Let A be an operad in the category of R-modules. A coefficient system over A is an abelian group object in the category of operads over A. Such an object consists of maps of operads π: B → A, 0: A → B, −: B → B and +: B ×_{A} B → B satisfying an obvious set of relations. In particular π0 = 1_{A}, whence B has the form B[n] = A[n] ⊕ U[n] for some Σ-object U; this U is a square-zero ideal of the operad B. We shall use the symbol U to denote the coefficient system.

Given such a coefficient system U, the t-th Eilenberg-MacLane object K(U, t) is an operad in the category of simplicial R-modules whose n-th space is defined by

K(U, t)[n] = A[n] ⊕ W̄^{t}U[n],

where W̄ is the construction of Eilenberg-Mac Lane (see [12, IV]).

More generally, if A is an operad in the category of simplicial R-modules, then given a coefficient system U over π_{0}(A) we obtain Eilenberg-MacLane objects K(U, t) over A by pulling back.

1.3.2 Quillen cohomology of operads

The t-th Quillen cohomology group H_{oper}^{t}(A, U) of an operad A with coefficients in a coefficient system is defined to be the abelian group of maps A → K(U, t) in the homotopy category Ho (operℳ_{R} ↓ A) of operads over A. This is just a special case of the general formulation of Quillen [13].

1.3.3 Biobjects and a bar complex

Recall that an operad in ℳ_{R} is a monoid in the monoidal category Σℳ_{R} (2.2.2). A biobject M of A is a Σ-object equipped with a two-sided action by A (2.2.15). A map of operads f: A → B gives B the structure of an A-biobject. Thus in particular, A is itself an A-biobject, and furthermore via the “zero-section” A → K(U, t) every Eilenberg-MacLane object becomes an A-biobject.

The bar complex B(A, A, A) of an operad A is a simplicial object in the category of A-biobjects, with

B(A, A, A)_{n} = A^{∘n}.

The simplicial structure maps are the obvious ones induced by the unit I → A and multiplication A ∘ A → A maps.

1.3.4 Quillen cohomology of biobjects

Let A be an operad in ℳ_{R}. The t-th Quillen cohomology group H_{A-biobj}^{t}(M, U) of an A-biobject M with coefficients in U is defined to be the abelian group of maps M → K(U, t) in the homotopy category Ho (A-biobjℳ_{R} ↓ M) of A-biobjects over M. Here K(U, t) is an Eilenberg-MacLane object, i.e., an A-biobject in the category of simplicial R-modules whose n-th space is defined by

K(U, t)[n] = M[n] ⊕ W̄^{t}U[n],

where W̄ is the construction of Eilenberg-Mac Lane (see [12, IV]).

1.3.5 Hochschild cohomology of operads

Let A be an operad, and U a coefficient system over A as in 1.3.1. As noted in 1.3.3, A has the structure of an A-bimodule, as do the Eilenberg-MacLane objects K(U, t). Thus we define the t-th Hochschild cohomology group Hoch^{t}(A, U) of an operad A with coefficients in U by

Hoch^{t}(A, U) = H_{A-biobj}^{t}(A, U).

If A is a Σ-cofibrant operad, then its Hochschild cohomology can be computed by means of a bar construction.

1.3.6. Proposition. Let A be a Σ-cofibrant operad over ℳ_{R}. Then

Hoch^{t}(A, U) ≃ H^{t}(A-biobjℳ_{R}(A^{∘∗+2}, A ⊕ U), ∂),

the cohomology of a cochain complex with differential

∂: A-biobjℳ_{R}(A^{∘s}, A ⊕ U) → A-biobjℳ_{R}(A^{∘s+1}, A ⊕ U)

induced by the alternating sum of the face maps of B(A, A, A).

Proof. This is an immediate consequence of the definition of Hochschild cohomology and Proposition 5.4.3.

1.3.7 Equivalence of Quillen and Hochschild cohomology

We will prove the following theorem in Section 1.3.

1.3.8. Theorem. Let A be an operad in the category of R-modules which is Σcofibrant. Let U be a coefficient system on A. Then there are natural isomorphisms

Hoch^{t+1}(A, U) ≃ H_{oper}^{t}(A, U)

for t ≥ 1, and a natural exact sequence

0 → Hoch^{0}(A, U) → U[1] → H_{oper}^{0}(A, U) → Hoch^{1}(A, U) → 0.

Operads and related categories

2.1 Monoidal categories

In this section we discuss the notion of a right-closed monoidal category, and the related notion of a right-closed action. These notions provide a convenient language for our discussion of Σ-objects and operads in Sections 2.2 and 2.3.

2.1.1 Monoidal categories

A monoidal category M, as defined in [10, VII], is a category equipped with functors

η: 1 → M

and

μ: M × M → M

along with natural isomorphisms

a: μ(1_{M} × μ) ⟶^{∼} μ(μ × 1_{M}): M × M × M → M,

ℓ: μ(η × 1_{M}) ⟶^{∼} 1_{M} : M = 1 × M → M,

and

r: μ(1_{M} × η) ⟶^{∼} 1_{M} : M = M × 1 → M,

such that the following diagrams commute,

[figure: μ(1 × μ(1 × μ)); μ(1×a); a(1×1×μ); μ(1 × μ(μ × 1)); a(1×μ×1); μ(μ × μ); a(μ×1×1); μ(μ(μ × 1) × 1); μ(μ(1 × μ) × 1),; μ(a×1)]

[figure: μ(1 × μ(η × 1)); a(1×η×1); μ(1×ℓ); μ(μ(1 × η) × 1); μ(r×1); μ,]

and such that

ℓη = rη: 1 = 1 × 1 → M.

We will generally write I ∈ M for the object which is the image of the functor η, and will write A ∘ B for μ(A, B). We usually suppress mention of the structural isomorphisms a, ℓ, and r; thus we write A ∘ B ∘ C for the object naturally isomorphic to A ∘ (B ∘ C).

2.1.2 Actions by monoidal categories

An action of a monoidal category M on a category C is a functor

ψ: M × C → C

along with natural isomorphisms

b: ψ(μ × 1_{C}) ⟶^{∼} ψ(1_{M} × ψ): M × M × C → C

and

u: ψ(η × 1_{C}) ⟶^{∼} 1_{C} : C = 1 × C → C

making the following diagrams commute:

[figure: ψ(1 × ψ(1 × ψ)); μ(1×b); b(1×1×ψ); ψ(1 × ψ(μ × 1)); b(1×μ×1); ψ(μ × ψ); b(μ×1×1); ψ(μ(μ × 1) × 1); ψ(μ(1 × μ) × 1),; ψ(a×1)]

[figure: ψ(1 × ψ(η × 1)); b(1×η×1); ψ(1×u); ψ(μ(1 × η) × 1); ψ(r×1); ψ.]

We will generally write A ∘ (X) or even A(X) for ψ(A, X). We usually suppress mention of the structure isomorphisms b and u; thus we write A ∘ B ∘ (X) or AB(X) for the object naturally isomorphic to A ∘ (B ∘ (X)).

2.1.3. Example. Let C be a category, and let M = C^{C} the category of endofunctors on C. Then M is a monoidal category under composition of functors, and there is an action of M on C induced by evaluation of functors. In particular, given A: C → C and X ∈ C, we see that A ∘ X is just evaluation of the functor A at X. 2.1.4. Remark. It is not hard to show that an action of M on C is equivalent to giving a monoidal functor (see Section 2.4) M → C^{C} where C^{C} denotes the category of functors from C to C.

2.1.5 Right-closed action

A right-closed action of a monoidal category M on a category C is an action such that for each X ∈ C the functor −∘X: M → C has a right adjoint ℌ𝔬𝔪(X, −): C → M. These right adjoints fit together to give a functor

ℌ𝔬𝔪(−, −): C^{op} × C → M

inducing for X, Y ∈ C and A ∈ M a natural isomorphism

C[A ∘ X, Y] ≃ M[A, ℌ𝔬𝔪(X, Y)]. (2.1.6)

Furthermore, for X, Y, Z ∈ C there is a natural “unit” map

I → ℌ𝔬𝔪(X, X)

which is adjoint to the identity on X, and a natural “composition” map

ℌ𝔬𝔪(Y, Z) ∘ ℌ𝔬𝔪(X, Y) → ℌ𝔬𝔪(X, Z)

which is induced by “evaluation” maps ℌ𝔬𝔪(X, Y) ∘ X → Y and ℌ𝔬𝔪(Y, Z) ∘ Y → Z which are adjoint to the identity on ℌ𝔬𝔪(X, Y) and ℌ𝔬𝔪(Y, Z) respectively.

2.1.7. Proposition. Given a right-closed action of M on C and an object X ∈ C, the functor − ∘ X: M → C preserves colimits whenever they exist, and the functor ℌ𝔬𝔪(X, −): C → M preserves limits whenever they exist.

Proof. Straightforward.

2.1.8. Remark. Note, however, that for X ∈ C the functor X ∘ − need not preserve colimits, and the functor ℌ𝔬𝔪(−, X) need not take colimits to limits.

2.1.9 Right-closed monoidal categories

A right-closed monoidal category is a monoidal category M such that the action of M on itself induced by the monoidal structure is right-closed. This means that there is a functor

F(−, −): M^{op} × M → M

inducing for A, B, C ∈ M a natural isomorphism

M[A ∘ B, C] ≃ M[A, F(B, C)].

2.1.10. Remark. There are corresponding notions of left-closed actions and leftclosed monoidal categories.

2.1.11 Monoids and algebras

A monoid in a monoidal category M is an object A ∈ M along with maps μ: A∘A → A ∈ M and η: I → A ∈ M making the following diagrams commute.

[figure: A ∘ A ∘ A; μ∘1; 1∘μ; A ∘ A; μ; I ∘ A; η∘1; A ∘ A; μ; A ∘ I; 1∘η; A ∘ A; μ; A; A.]

Let monM denote the category of monoids in M.

Suppose M acts on C. Then an algebra over the monoid A is an object X ∈ C equipped with a map ψ: A ∘ X → X ∈ C making the following diagrams commute.

[figure: A ∘ A ∘ X; μ∘1; 1∘ψ; A ∘ X; ψ; I ∘ X; η∘1; A ∘ X; ψ; X.]

Such a map ψ is called an A-algebra structure on X. The category of A-algebras is denoted by C^{A}. We can characterize the set of A-algebra structures on X as the equalizer of the pair of maps

C[A(X), X] ⇉ C[A^{2}(X), X] × C[X, X] (2.1.12)

where the top map takes ψ: A(X) → X to the pair (ψμ, ψη) and the bottom map takes ψ to the pair (ψA(ψ), 1_{X}).

Now suppose that the action of M on C is right-closed. Then for any object X ∈ C there is a monoid E_{X} in M defined by E_{X} = ℌ𝔬𝔪(X, X) with structure maps given by the unit and composition maps e: I → E_{X} and m: E_{X} ∘ E_{X} → E_{X} as described in 2.1.9. This E_{X} is called the endomorphism monoid of X. It classifies A-algebra structures on X as described in the following proposition.

2.1.13. Proposition. If A ∈ monM and X ∈ C, then monM[A, E_{X}] is naturally isomorphic to the the equalizer of the pair of maps in 2.1.12.

Proof. The set of monoid maps from A to E_{X} is exactly the equalizer of the pair of maps

M[A, E_{X}] ⇉ M[A ∘ A, E_{X}] × M[I, E_{X}], (2.1.14)

where the top map sends α to (αμ, αη) and the bottom map sends α to (m(α ∘ α), e). Thus it suffices to show that the adjunction isomorphism of 2.1.6 carries the diagram 2.1.12 to the diagram 2.1.14, which is a routine verification.

2.1.15. Example. Let M denote any closed symmetric monoidal category. Then M is clearly a right-closed monoidal category. Examples of such M include the category of simplicial sets 𝒮, in which case monM is the category of simplicial monoids, and the category of simplicial R-modules ℳ_{R}, in which case monM is the category of associative R-algebras.

2.1.16. Example (O-categories). Let O be a set, and let C = Set^{O} be the category of sets indexed by O; thus, an object X ∈ Set^{O} is a collection of sets {X(x)}_{x∈O}. The category M = Set^{O×O} is equivalent to the category of directed graphs with vertex set O. There is a right-closed monoidal structure on M given by

(A ∘ B)(x, y) = ∐_{z∈O} A(z, y) × B(x, z)

and

F(B, C)(y, z) = ∏_{x∈O} Set[B(x, y), C(x, z)],

and a right-closed action of M on C given by

(A ∘ X)(x) = ∐_{z∈O} A(z, x) × X(z)

and

ℌ𝔬𝔪(X, Y)(x, y) = Set[X(x), Y (y)].

Then a monoid in M is an O-category; i.e., a category with a fixed set of objects O, and maps between such monoids are functors which are the identity on the set of objects. The category of algebras over an O-category C is equivalent to the category of C-diagrams.

It is not hard to see that there is an analogous left-closed action of M on C.

2.1.17. Example (Triples). Let C be a complete category, and let M = C^{C} the category of endofunctors on C. Then C^{C} is a monoidal category under composition of functors, and there is a right-closed action of C^{C} on C induced by evaluation of functors. In fact, objects X, Y ∈ C can be identified with functors X, Y : 1 → C; then ℌ𝔬𝔪(X, Y): C → C is the “right Kan extension” (see [10, X]) of X along Y. Explicitly, this is the functor defined at C ∈ C by

ℌ𝔬𝔪(X, Y)(C) = ∏_{C[C,X]} Y.

A monoid in C^{C} is exactly a triple on C (also known as a monad), and an algebra over a monoid is just an algebra over the triple. The triple E_{X}: C → C may be called the endomorphism triple; it classifies triple algebra structures on X ∈ C.

This category C^{C} does not seem in general to be a right-closed monoidal category: F(X, Y) would necessarily be defined as the right Kan extension of X: C → C along Y : C → C; one expects this to exist only if C is a small complete category, which is rare (see [10, V.2]).

2.1.18. Example (Operads). Another example of a right-closed monoidal category and right-closed action is that of Σ-objects acting on a closed symmetric monoidal category. This example is described in detail in Section 2.2.

2.1.19 Biobjects

Let M be a monoidal category. Suppose A ∈ monM, and let X ∈ M. A left-A-action of A on X is a map ψ: A ∘ X → X making the diagram

[figure: A ∘ A ∘ X; ψ∘X; A∘ψ; A ∘ X; ψ; X; η∘X; A ∘ X; ψ; X]

commute; in other words, an algebra over the triple M → M defined by X ↦ A ∘ X. The category of algebras over this triple is denoted by _{A}M. Likewise given B ∈ monM, a right-B-action of B on X is a map ϕ: X ∘ B → X making the diagram

[figure: X ∘ B ∘ B; X∘ϕ; ϕ∘B; X ∘ B; ϕ; X; X∘η; X ∘ B; ϕ; X]

commute; in other words, an algebra over the triple M → M defined by X ↦ X ∘ B. The category of algebras over this triple is denoted by M_{B}.

More generally given A, B ∈ monM and X ∈ M an A, B-action on X consists of a left-A-action and a right-B-action which commute; in other words, an algebra over the triple M → M defined by X ↦ A ∘ X ∘ B. The category of algebras over this triple is denoted by _{A}M_{B}, which we refer to as the category of A, B-biobjects.

We are particular interested in the case when A and B are the same monoid. We shall refer to an X ∈ M with an A, A-action as an A-biobject; the category of A-biobjects is thus denoted by _{A}M_{A}.

Suppose M is a right-closed monoidal category with a right closed action on C. Then if X ∈ C^{A} and Y ∈ C^{B} then ℌ𝔬𝔪(X, Y) has a natural B, A-action, with action map given by

B ∘ ℌ𝔬𝔪(X, Y) ∘ A → ℌ𝔬𝔪(Y, Y) ∘ ℌ𝔬𝔪(X, Y) ∘ ℌ𝔬𝔪(X, X) → ℌ𝔬𝔪(X, Y),

where A → ℌ𝔬𝔪(X, X) and B → ℌ𝔬𝔪(Y, Y) are maps of monoids classifying the algebra structures on X and Y respectively. Furthermore, note that A^{∘n} = A ∘ ⋯ ∘ A for n ≥ 1 has a natural structure as an A-biobject. The following proposition says that maps between A-algebras are represented by an A-biobject.

2.1.20. Proposition. If A ∈ monM and X, Y ∈ C^{A}, then there is a natural isomorphism

AM_{A}[A, ℌ𝔬𝔪(X, Y)] ≃ C^{A}[X, Y].

Proof. A map of A-biobjects is exactly a map which is both a map of left-A-objects and a map of right-A-objects. Thus _{A}M_{A}[A, ℌ𝔬𝔪(X, Y)] is the equalizer of

M_{A}[A, ℌ𝔬𝔪(X, Y)] ⇉ M_{A}[A ∘ A, ℌ𝔬𝔪(X, Y)],

where the top arrow sends f to fμ and the bottom arrow sends f to ℓ(A ∘ f), where ℓ: A ∘ ℌ𝔬𝔪(X, Y) → ℌ𝔬𝔪(X, Y) is the left-A-action. A straightforward argument shows that this diagram is isomorphic to

M[I, ℌ𝔬𝔪(X, Y)] ⇉ M[A, ℌ𝔬𝔪(X, Y)],

where the top arrow sends g to r(g ∘ A) and the bottom arrow sends g to ℓ(A ∘ g). Here r: ℌ𝔬𝔪(X, Y) ∘ A → ℌ𝔬𝔪(X, Y) denotes the right-A-action. A straightforward argument shows that by the adjunction isomorphism of 2.1.6 this diagram is the same as

C[X, Y] ⇉ C[A(X), Y],

where the top arrow sends h to hψ_{X} and the bottom arrow sends h to ψ_{Y} h, where ψ_{X}: A(X) → X and ψ_{Y} : A(Y) → Y are the algebra structures. But the equalizer of this diagram is exactly C^{A}[X, Y]

2.1.21 Monoidal structures for over-categories

In this section, we note that right-closed monoidal structures and right-closed actions naturally induce similar structures on over-categories. We will make use of this fact in Section 5.2.

2.1.22. Proposition. Let M be a monoidal category, and let A be a monoid in M. Then there is a natural monoidal structure on (M ↓ A).

Proof. The monoidal structure is defined as follows. Given objects (B ⟶^{b} A), (C ⟶^{c} A) ∈ (M ↓ A), we define

(B ⟶^{b}A) ∘ (C ⟶^{c}A) = B ∘ C ⟶^{b∘c} A ∘ A ⟶^{μ} A.

The unit of this monoidal structure is the unit map η: I → A of A.

2.1.23. Proposition. Let M be a complete category with a right-closed action on C. Let A be a monoid in M and let X be an A-algebra. Then there is a canonical right-closed action

(M ↓ A) × (C ↓ X) → (C ↓ X).

Proof. The right-closed action of (M ↓ A) on (C ↓ X) is defined as follows. Given objects (B ⟶^{b}A) ∈ (M ↓ A), and (Y ⟶^{y} X), (Z ⟶^{z}X) ∈ (C ↓ X), we define

(B ⟶^{b}A) ∘ (Y ⟶^{y} X) = B ∘ Y ⟶^{b∘y} A ∘ X ⟶^{ψ} X

and

ℌ𝔬𝔪_{A,X}(Y ⟶^{y} X, Z ⟶^{z}X) = ℌ𝔬𝔪(Y, Z) ×_{ℌ𝔬𝔪(Y,X)} A → A.

2.1.24. Corollary. Let M be a complete right-closed monoidal category. Let A be a monoid in M. Then there is a canonical right-closed monoidal structure on (M ↓ A).

2.1.25 Enriched categories

In this section, we note that all the constructions described above can be carried out when C and M are replaced with categories enriched over some symmetric closed monoidal category V [9], with the definitions suitably modified to take into account the enriched structure. If C is enriched over V we write map_{C}(X, Y) ∈ V for the enriched hom-object.

Thus, if C and M are categories enriched over V, and if M is is right-closed monoidal category with a right-closed action on C for which the structure functors η: 1 → M, μ: M×M → M, and ψ: M×C are V-functors, then all the constructions of this section apply with hom sets C[−, −] and M[−, −] replaced by hom objects map_{C}(−, −) and map_{M}(−, −).

In particular, the discussion of 2.1.11 implies that for X ∈ C and A ∈ monM there exists a V-object

map_{monM}(A, E_{X}) ∈ V

of A-algebra structures on X. In the sequel we will consider the case when V = 𝒮, the category of simplicial sets.

2.2 Operads and related categories

In this section we define (following Getzler-Jones [7]) the notion of a Σ-object in a symmetric monoidal category (Getzler and Jones call them S-objects). A Σ-object in a category C induces a functor from C to itself. We will define a monoidal structure on Σ-objects corresponding to composition of the induced functors; this monoidal structure will turn out to be right-closed, and have a right-closed action on C. Using this construction we will be able to define operads, algebras over an operad, and biobjects over an operad over the category C.

2.2.1 Closed symmetric monoidal categories

In this section, we let C denote a closed symmetric monoidal category [10, VII, 7] with monoidal structure − ⊗ − and unit object k, which has internal hom-objects hom(−, −), so that

C[X ⊗ Y, Z] ≃ C[X, hom(Y, Z)].

Furthermore, we assume that C has all small limits and colimits. Note that the functors X ⊗ − and − ⊗ Y must commute with all colimits.

The key examples of symmetric monoidal categories we need are:

1. The category Set (resp. 𝒮) of (simplicial) sets, with cartesian product as the monoidal product,

2. The category 𝒯 of compactly generated weak-Hausdorff topological spaces, with the categorical product as the monoidal product,

3. The category ℳ_{R} (resp. ℳ_{R}) of (simplicial) modules over a commutative ring R, with tensor product over R as the monoidal product. (This monoidal structure will be symmetric via the natural isomorphism X ⊗ Y ⟶^{∼} Y ⊗ X sending x ⊗ y ↦ y ⊗ x.)

4. The category ℳ_{R} (resp. ℳ_{R}) of graded (simplicial) modules over a graded commutative ring R, with tensor product over R as the monoidal product. (This monoidal structure will be symmetric via the natural isomorphism X ⊗ Y ⟶^{∼} Y ⊗ X sending x ⊗ y ↦ (−1)^{pq}y ⊗ x, where x and y are homogeneous elements of degree p and q respectively.)

Each of the categories 𝒮, 𝒯, and ℳ_{R} is enriched over simplicial sets, and the monoidal structure is likewise enriched, in the sense that the monoidal product and the hom functor are simplicial functors.

2.2.2 Σ-objects

Let Π denote the category whose objects are finite totally ordered sets, and whose maps are all functions between sets, which need not preserve the orderings; in other words, the objects of Π are finite sets which are “secretly” equipped with an ordering. Note that if S is an object of Π then any subset of T of S with the induced ordering is also an object of Π. Let Σ denote the subcategory of Π consisting of the objects and all isomorphisms (again, these maps need not preserve the ordering). Let Σ_{S} denote the group of automorphisms of the set S.

If S = {s_{1}, …, s_{n}} ∈ Σ with s_{1} < ⋯ < s_{n}, and if we have a collection of objects {X_{s} ∈ C}_{s∈S} in some symmetric monoidal category C, then we define

⨂_{s∈S} X_{s} = X_{s_{1}} ⊗ ⋯ ⊗ X_{s_{n}}.

Thus given a map f: T → S ∈ Σ there is an obvious induced map

⨂_{t∈T} X_{f(t)} → ⨂_{s∈S} X_{s}.

In particular, if Y ∈ C, then the notation Y^{⊗S} for the tensor product of copies of Y indexed by S is non-ambiguous, and has a natural Σ_{S}-action.

Let ΣC denote the category of functors from Σ^{op} to C. We refer to ΣC as the category of Σ-objects in C. Let k = {1, …, k} denote a distinguished ordered set with k elements, and let Σ′ denote the full subcategory of Σ consisting of the objects 0, 1, 2, . . .. Then the inclusion Σ′ → Σ is clearly an equivalence of categories. Thus there is an equivalence of the categories ΣC and the functor category C^{Σ′^{op}}. Since an object in ΣC is thus determined by its restriction to the subcategory Σ′^{op}, we will often regard an X ∈ ΣC as a sequence of objects X[0], X[1], X[2], … ∈ C where each X[n] is equipped with a right action by Σ_{n}.

2.2.3 Monoidal structure in the category of Σ-objects

To any Σ-object A is associated an endofunctor A(−) on the category C. It is defined to be the coequalizer in

∐_{T →T′∈Σ} A[T′] ⊗ X^{⊗T} ⇉ ∐_{T ∈Σ} A[T] ⊗ X^{⊗T} → A(X).

This defines a functor ΣC × C → C. Using the equivalence Σ′ → Σ, we see that

A(X) ≃ ∐_{s≥0} A[s] ⊗_{Σ_{s}} X^{⊗s}.

(This construction may also be described as the coend [10] of the functor Σ → C defined by T ↦ X^{⊗T} and the functor A: Σ^{op} → C.)

We will define a monoidal structure −∘− on ΣC such that (A∘B)(X) ≃ A(B(X)). For this purpose we need to describe the functor X ↦ A(X)^{T} in terms of a Σ-object, to be denoted by A[−, T].

To make this construction, we note that one can define a symmetric monoidal category structure on ΣC as follows. For A, B ∈ ΣC define

(A ⊗ B)[n] = ∐_{r+s=n} A[r] ⊗ B[s] ⊗_{Σ_{r}×Σ_{s}} Σ_{n}.

Thus A ⊗ B ∈ ΣC, and − ⊗ −: ΣC × ΣC → ΣC is a functor. It is not hard to show that this defines a symmetric monoidal product on ΣC with the following property.

2.2.4. Lemma. There are natural and coherent isomorphisms

(A ⊗ B)(X) ≃ A(X) ⊗ B(X)

for A, B ∈ ΣC and X ∈ C.

Proof. The isomorphism is defined by

(A ⊗ B)(X) ≃ ∐_{n≥0}(A ⊗ B)[n] ⊗_{Σ_{n}} X^{⊗n}

≃ ∐_{n} ∐_{r+s=n}(A[r] ⊗ B[s]) ⊗_{Σ_{r}×Σ_{s}} X^{⊗n}

≃ ∐_{r≥0,s≥0}(A[r] ⊗_{Σ_{r}} X^{⊗r}) ⊗ (B[s] ⊗_{Σ_{s}} X^{⊗s})

≃ A(X) ⊗ B(X).

It immediately follows that A(X)^{⊗T} ≃ A^{⊗T} (X). We define a functor

A[−, −]: Σ^{op} × Σ → C

by A[S, T] = A^{⊗T} [S]. It is not hard to show that

A[S, T] ≃ ∐_{π∈Π(S,T)} ⨂_{t∈T} A[π^{−1}(t)],

Given A and B in ΣC, define A ∘ B ∈ ΣC by the following coequalizer diagram in C

∐_{T →T′∈Σ} A[T′] ⊗ B[S, T] ⇉ ∐_{T ∈Σ} A[T] ⊗ B[S, T] → (A ∘ B)[S].

Using the equivalence of categories Σ′ → Σ, we obtain the following description of this construction:

(A ∘ B)[s] ≃ ∐_{t≥0} A[t] ⊗_{Σ_{t}} B[s, t] ≃ ∐_{t≥0} A[t] ⊗_{Σ_{t}} B^{⊗t}.

(Equivalently, (A ∘ B)[S] is the coend of functors B[S, −]: Σ → C and A: Σ^{op} → C.) Also define an object I in ΣC by

I[S] = \begin{cases}k & if |S| = 1, \\ ∅ & otherwise, \end{cases}

where ∅ is the initial object in C.

2.2.5. Lemma. There are natural and coherent isomorphisms

(A ⊗ A′) ∘ B ≃ (A ∘ B) ⊗ (A′ ∘ B)

for A, A′, B ∈ ΣC.

Proof. Similar to proof of Lemma 2.2.4.

2.2.6. Proposition. With the given structure ΣC becomes a monoidal category with an action on C. In particular, there are natural isomorphisms

(A ∘ B)(X) ≃ A(B(X)),
(X ∘ Y) ∘ Z ≃ X ∘ (Y ∘ Z),
X ∘ I ≃ X,  I ∘ X ≃ X,

and

I(X) ≃ X.

Proof. Straightforward, using Lemmas 2.2.4 and 2.2.5. For example the first equation follows from

A(B(X)) ≃ ∐_{n≥0} A[n] ⊗_{Σ_{n}} B(X)^{⊗n}

≃ ∐_{n≥0}(A[n] ⊗_{Σ_{n}} B^{⊗n})(X)

≃ (A ∘ B)(X).

Given two objects X and Y in C, we define an object ℌ𝔬𝔪(X, Y) in ΣC by

ℌ𝔬𝔪(X, Y)[S] = hom[X^{⊗S}, Y].

This gives a functor ℌ𝔬𝔪: C^{op} × C → ΣC.

2.2.7. Proposition. This structure defines a right-closed action of ΣC on C. In particular, there is a natural isomorphism

C[A(X), Y] ≃ ΣC[A, ℌ𝔬𝔪(X, Y)].

Proof. Straightforward.

Given two objects B and C in ΣC, we define F(B, C) ∈ ΣC by the following equalizer in C

F(B, C)[T] → ∏_{S∈Σ} hom(B[S, T], C[S]) ⇉ ∏_{S′→S∈Σ} hom(B[S, T], C[S′]).

Using the equivalence Σ′ → Σ we obtain the description

F(Y, Z)[t] ≃ ∏_{s≥0} hom(Y [s, t], Z[s])^{Σ_{s}}.

2.2.8. Proposition. With the given structure, ΣC becomes a right-closed monoidal category. In particular, there is a natural isomorphism

ΣC[X ∘ Y, Z] ≃ ΣC[X, F(Y, Z)].

Proof. Straightforward.

2.2.9 Operads

An operad is defined to be a monoid in the monoidal category ΣC. We will refer to the category of operads by operC.

2.2.10. Example. Let C = 𝒮 be the category of simplicial sets. The Σ-object P defined by

P [S] = \begin{cases}pt & if |S| = 0, 1, \\ ∅ & otherwise, \end{cases}

admits a unique structure as an operad. We have that

P (X) = pt ⨿ X.

2.2.11. Example. Again, let C be the category of simplicial sets. The Σ-object N defined by

N[S] = pt

for all S admits a unique structure as an operad, and is referred to as the symmetric operad. We have that

N(X) = ∐_{s≥0} (X^{×s})_{Σ_{s}}.

2.2.12. Example. Again, let C be the category of simplicial sets. The Σ-object M defined by

M[S] = Σ_{S},

with the obvious Σ action admits the structure of an operad, and is referred to as the associative operad. We have that

M(X) = ∐_{s≥0} X^{×s}.

2.2.13 Algebras over an operad

Let A be an operad. The category of A-algebras is the category of algebras over the triple A(−) on C, and is denoted by C^{A}.

2.2.14. Example. The category of algebras over P is equivalent to the category of pointed simplicial sets. The category of algebras over N is equivalent to the category of simplicial commutative monoids. The category of algebras over M is equivalent to the category of simplicial associative monoids.

Given X in C, the endomorphism operad is the operad E_{X} = ℌ𝔬𝔪(X, X). By Proposition 2.1.13 it has the property that the set of operad maps A → E_{X} is in natural one-to-one correspondence with the set of A-algebra structures on X.

2.2.15 Biobjects over an operad

Let A and B be operads. An A, B-biobject is an object X in ΣC equipped with a map A∘X∘B → X which satisfies the obvious unital and associativity properties; i.e., it is an object with a left-A-action and a right-B-action as in 2.1.19. We denote the category of A, B-biobjects by (A, B)-biobjC. It is not hard to see that (A, B)-biobjC is equivalent to the category of algebras over a triple on ΣC which takes X to A∘X∘B.

As noted in 2.1.19, if X is an A-algebra and Y is a B-algebras, then ℌ𝔬𝔪(X, Y) has a natural structure of a B, A-biobject, defined by

B ∘ ℌ𝔬𝔪(X, Y) ∘ A → ℌ𝔬𝔪(Y, Y) ∘ ℌ𝔬𝔪(X, Y) ∘ ℌ𝔬𝔪(X, X) → ℌ𝔬𝔪(X, Y),

where A → E_{X} ≃ ℌ𝔬𝔪(X, X) and B → E_{Y} ≃ ℌ𝔬𝔪(Y, Y) are maps of operads which classify the algebra structures on X and Y respectively. 2.2.16. Example. An operad A has the structure of an A-biobject in a natural way; the structure map A ∘ A ∘ A → A is just multiplication in the operad. As shown in 2.1.19, there is a natural isomorphism

C^{A}[X, Y] ≃ A-biobjC[A, ℌ𝔬𝔪(X, Y)].

2.3 Constructions involving Σ-objects

In this section we show the existence of limits and colimits in Σ-objects, operads, and biobjects, and describe a “relative” circle-product construction for biobjects, which will lead in particular to adjoint functor pairs

f_{∗} : C^{A} ⇄ C^{B} : f^{∗}

for any map f: A → B of operads.

2.3.1 Limits and colimits

In this section we show the existence of limits and colimits in the categories of operads, and of algebras and biobjects over an operad. We shall assume in this section that (C, ⊗) denotes a symmetric monoidal category with all small limits and colimits.

A reflexive pair in any category C is a pair of maps f, g: X ⇉ Y along with a section s: Y → X such that fs = 1_{Y} = gs. The coequalizer of such a reflexive pair is a map k: Y → Z ∈ C which is initial among all maps such that kf = kg. The significance of reflexive coequalizers comes from the following lemma; the usefulness of reflexive coequalizers in these sorts of arguments was noticed by Mike Hopkins.

2.3.2. Lemma. Let C be as in 2.3.1.

1. Let f_{i}, g_{i}: X_{i} ⇉ Y_{i} ⟶^{k_{i}} Z_{i}, s_{i}: Y_{i} → X_{i} for i = 1, …, n be reflexive coequalizers in C. Then the induced diagram

X_{1} ⊗ ⋯ ⊗ X_{n} ⇉ Y_{1} ⊗ ⋯ ⊗ Y_{n} → Z_{1} ⊗ ⋯ ⊗ Z_{n}

is a reflexive coequalizer in C.

2. Let D be a filtered category, and let X_{i}: D → C for i = 1, …, n be diagrams. Then there is a canonical isomorphism

colim^{D}(X_{1} ⊗ ⋯ ⊗ X_{n}) ≃ (colim^{D} X_{1}) ⊗ ⋯ ⊗ (colim^{D} X_{n}).

Proof. The lemma follows from the fact that ⊗ distributes over colimits and the easily checked fact that in both cases (1) and (2) the functor D → D × ⋯ × D is final in the sense of [10, IX.3]. (For case (1) take D to be the category which indexes a reflexive pair of maps.)

This leads immediately to the following statements for the action of ΣC on C, and for the monoidal structure on ΣC.

2.3.3. Lemma. Let C be as in 2.3.1.

1. Let f, g: X ⇉ Y ⟶^{k} Z, s: Y → X be a reflexive coequalizer in C. Then for A ∈ ΣC the induced diagram

A(X) ⇉ A(Y) → A(Z)

is a reflexive coequalizer in C.

2. Let D be a filtered category, and let X: D → C be a diagram. Then for A ∈ ΣC there is a canonical isomorphism

A(colim^{D} X) ≃ colim^{D} A(X).

Proof. This follows immediately from Lemma 2.3.2 and the fact that A(X) is constructed from A and X using only colimits and the monoidal product in C.

2.3.4. Lemma. Let C be as in 2.3.1.

1. Let f_{i}, g_{i}: A_{i} ⇉ B_{i} ⟶^{k_{i}} C_{i}, s_{i}: B_{i} → A_{i} for i = 1, …, n be reflexive coequalizers in ΣC. Then the induced diagram

A_{1} ∘ ⋯ ∘ A_{n} ⇉ B_{1} ∘ ⋯ ∘ B_{n} → C_{1} ∘ ⋯ ∘ C_{n}

is a reflexive coequalizer in C.

2. Let D be a filtered category, and let A_{i}: D → ΣC for i = 1, …, n be diagrams. Then there is a canonical isomorphism

colim^{D}(A_{1} ∘ ⋯ ∘ A_{n}) ≃ (colim^{D} A_{1}) ∘ ⋯ ∘ (colim^{D} A_{n}).

Proof. This follows immediately from Lemma 2.3.2 and the fact that A ∘ B is constructed from A and B using only colimits and the monoidal product in C.

Lemmas 2.3.3 and 2.3.4 allow us to show that categories of algebras over an operad (respectively, categories of biobjects over an operad) are complete and cocomplete.

2.3.5. Proposition. Let A and B be operads over C.

1. The category C^{A} of A-algebras is complete and cocomplete; furthermore, the forgetful functor C^{A} → C preserves limits, reflexive coequalizers, and filtered colimits.

2. The category (A, B)-biobjC of A, B-biobjects is complete and cocomplete; furthermore, the forgetful functor (A, B)-biobjC → ΣC preserves limits, reflexive coequalizers, and filtered colimits.

Proof. We prove part 1; the proof of part 2 is similar.

It is straightforward to show that C^{A} has all limits, and that these limits are preserved by the forgetful functor.

We now show that C^{A} has all reflexive coequalizers, and that these are preserved by the forgetful functor. Let f, g: X ⇉ Y, s: Y → X ∈ C^{A} be a reflexive pair. Let k: Y → Z ∈ C be the coequalizer of f and g in the underlying category C. We claim that Z admits a unique A-algebra structure making k the coequalizer in C^{A} of f and g. Consider the following diagram.

[figure: A^{2}X; A^{2}Y; A^{2}Z; AX; ψ_{X}; AY; ψ_{Y}; AZ; ψ_{Z}; X; f; g; η_{X}; Y; k; η_{Y}; Z.]

The rows are coequalizers in C by Proposition 2.3.3. Thus there exist unique dotted arrows in C making the appropriate squares commute. Let ψ_{Z}: A(Z) → Z be one of the induced arrows. It is then easy to check that, by uniqueness, the other dotted arrows must be μ, A(ψ_{Z}): A^{2}Z → AZ and η_{Z}: Z → AZ, and thus that ψ_{Z} induces an A-algebra structure on Z. It is now easy to check that Z is the coequalizer of f and g in the category of A-algebras as desired.

A similar argument shows that one may also construct filtered colimits in C^{A} by first constructing them in C.

We can now construct arbitrary small colimits in C^{A}. Let X: D → C^{A} be a functor from a small category. We consider a pair of parallel maps

f, g: A(colim^{C} A(X)) ⇉ A(colim^{C} X),

where colim^{C} denotes the colimit of a diagram in the underlying category C, defined as follows. Applying the functor C^{A}[−, W] for any object W ∈ C to the parallel maps and using standard adjunctions, we get a pair of parallel maps

lim C[A(X), W] ⇉ lim C[X, W]

where in each factor of the limit the top arrow sends f: A(X) → W to ψ_{W} A(f), and the bottom arrow sends f to fψ_{X}. It is easy to see that if the coequalizer of f and g in C^{A} exists then it is exactly the colimit of X in C^{A}. The natural transformation η: I → A induces a map t: colim^{C} X → colim^{C} A(X); let s = A(t). It is not hard to see that s is a section of f and g; thus, by the first part of the proof the coequalizer of f and g exists.

2.3.6 Free operads

Given a Σ-object A, the free operad on A is an operad F A together with a map of Σ-objects A → F A inducing a bijection

operC[F A, B] ⟶^{∼} ΣC[A, B]

for all operads B. If the free operad exists for every A ∈ ΣC, we call the resulting functor F: ΣC → operC the free operad functor; it is left adjoint to the forgetful functor operC → ΣC.

2.3.7. Proposition. Let C be as in 2.3.1. Then the free operad functor F: ΣC → operC exists.

Proof. A construction of the free operad is given in Appendix A.

An argument similar to those given above shows that the category of operads on C is complete and cocomplete.

2.3.8. Lemma. Let C be as in 2.3.1.

1. Let f, g: A ⇉ B ⟶^{k} C, s: Y → X be a reflexive coequalizer in ΣC. Then the

induced diagram

F A ⇉ F B → F C

is a reflexive coequalizer in C.

2. Let D be a filtered category, and let A: D → C be a diagram. Then there is a canonical isomorphism

F (colim^{D} A) ≃ colim^{D} F A.

Proof. This follows from Lemma 2.3.3 and from the construction of F A in Appendix A: there F A is defined as a colimit F A = colim_{n} F_{n}A in ΣC where F_{0}A = I and F_{n}A = I ⨿ A ∘ F_{n−1}A for n > 0. Thus F A is constructed from A using only colimits and the circle product in ΣC.

2.3.9. Proposition. The category operC of operads on C is complete and cocomplete; furthermore, the forgetful functor operC → ΣC preserves limits, reflexive coequalizers, and filtered colimits.

Proof. The proof is similar to that of Proposition 2.3.5.

2.3.10 The “circle-over” construction

As before, let (C, ⊗) be a complete and cocomplete symmetric monoidal category. Given an operad A over C, a right A-object M, and an A-algebra X, we define M ∘_{A} (X) ∈ C to be the coequalizer in the diagram

(M ∘ A)(X) ⇉ M(X) → M ∘_{A} (X),

where the two left hand maps are induced by the action maps of M and X respectively. This is a reflexive coequalizer, with reflection M(X) → (M ∘ A)(X) induced by the unit map I → A of the operad.

Similarly, if M is a right A-object and N a left A-object we may define M ∘_{A} N ∈ ΣC to be the coequalizer in

M ∘ A ∘ N ⇉ M ∘ N → M ∘_{A} N

where the two left hand maps are the action maps of M and N respectively. This is a reflexive coequalizer, with reflection induced by the unit map of the operad.

2.3.11. Proposition. Let A be an operad over C, and let M be a right A-object, N a left A-object, and X an A-algebra. Then there are natural isomorphisms

M ∘_{A} A ≃ M,  A ∘_{A} N ≃ N,  A ∘_{A} (X) ≃ X.

2.3.12. Proposition. Let M be an A, B-biobject, let N be a B, C-biobject, and let X be a B-algebra.

1. The object M ∘_{B} (X) ∈ C has a natural structure as an A-algebra.

2. The object M ∘_{B} N ∈ ΣC has a natural structure as an A, C-biobject.

Proof. This follows immediately from Lemmas 2.3.3 and 2.3.4.

2.3.13. Proposition. Let M be a right A-object, let N be an A, B-biobject, let P be a left B-object, and let X be a B-algebra.

1. There is a unique isomorphism

(M ∘_{A} N) ∘_{B} (X) ≃ M ∘_{A} (N ∘_{B} (X))

commuting with the obvious maps (M ∘ N)(X) → (M ∘_{A} N) ∘_{B} (X) and M ∘ (N(X)) → M ∘_{A} (N ∘_{B} (X)).

2. There is a unique isomorphism

(M ∘_{A} N) ∘_{B} P ≃ M ∘_{A} (N ∘_{B} P)

commuting with the obvious maps (M ∘ N) ∘ P → (M ∘_{A} N) ∘_{B} P and M ∘ (N ∘ P) → M ∘_{A} (N ∘_{B} P).

Proof. This follows immediately from Lemmas 2.3.3 and 2.3.4.

2.3.14. Corollary. The category of A-biobjects is a monoidal category with monoidal product

− ∘_{A} −: A-biobjC × A-biobjC → A-biobjC

and unit object A ∈ A-biobjC, and there is an action

− ∘_{A} (−): A-biobjC × C^{A} → C^{A}.

2.3.15. Example. Let C = 𝒮 and let P and M be operads as in Example 2.2.14. There is a map of operads P → M. Given a P -algebra X, i.e., a pointed simplicial set, the object M ∘_{P} (X) is the free associative monoid on X such that the basepoint is the identity element. In other words, M ∘_{P} (X) is the James construction on X.

2.3.16. Remark. Although we will not need it, one can show that with the structure of Corollary 2.3.14 the category A-biobjC is a right-closed monoidal category, and its action on C^{A} is a right-closed action. In fact, for X ∈ C^{B} there is a pair of adjoint functors

− ∘_{B} X : (A, B)-biobjC ⇄ C^{A} : ℌ𝔬𝔪(X, −),

where ℌ𝔬𝔪 coincides with the one defined on C, and for N ∈ (B, C)-biobjC there is a pair of adjoint functors

− ∘_{B} N : (A, B)-biobjC ⇄ (A, C)-biobjC : F^{C}(N, −),

where F^{C}(N, P) is defined to be the equalizer of

F^{C}(N, P) → F(N, P) ⇉ F(N ∘ C, P),

where the top arrow on the right is induced by N ∘ C → N, and the bottom arrow on the right is adjoint to the map F(N, P) ∘ N ∘ C → P ∘ C → P.

2.3.17 Adjoint functors

Finally, we note that associated to maps of operads f: A → B and g: A′ → B′ there are obvious forgetful functors f^{∗}: C^{B} → C^{A} and (f, g)^{∗}: (B, B′)-biobjC → (A, A′)-biobjC.

2.3.18. Proposition. The functors

f_{∗} = B ∘_{A} −: C^{A} → C^{B}

and

(f, g)_{∗} = B ∘_{A} − ∘_{A′} B′: (A, A′)-biobjC → (B, B′)-biobjC

are left adjoint to f^{∗} and (f, g)^{∗} respectively.

2.3.19 Simplicial constructions

If C is enriched over simplicial sets (2.1.25), and if the structure functor − ⊗ −: C × C → C is a simplicial functor, then it is not hard to see that ΣC, monC, C^{A}, and (A, B)-biobjC are enriched over simplicial sets, and that structure functors of the right-closed monoidal structure on ΣC and the right-closed action of ΣC on C are simplicial functors.

Examples of such categories C are simplicial sets and simplicial R-modules. Thus, the functors ×: 𝒮 × 𝒮 → 𝒮 and ⊗_{R}: ℳ_{R} × ℳ_{R} → ℳ_{R} extend to simplicial functors.

Moreover, for these categories there exist functors

− ⊗ −: 𝒮 × C → C

(not to be confused with the monoidal structure ⊗ in C) and

(−)^{(−)} : 𝒮^{op} × C → C

such that for X, Y ∈ C and K ∈ 𝒮 there are natural isomorphisms

C[K ⊗ X, Y] ≃ 𝒮[K, map_{C}(X, Y)] ≃ C[X, Y^{K}].

These remarks in turn imply that the categories of sigma objects, operads, algebras over an operad, and biobjects over an operad, over either 𝒮 or ℳ_{R} are enriched over simplicial sets.

2.4 Monoidal functors

In this section we define monoidal functors and show how they induce corresponding functors on categories of operads and biobjects.

2.4.1 Monoidal functors

Let (S, ⊗, I_{S}) and (T, ⊗, I_{T}) be monoidal categories. A monoidal functor L: S → T is a functor equipped with natural isomorphisms

b: LX ⊗ LY ⟶^{∼} L(X ⊗ Y)

and

u: I_{T} ⟶^{∼} LI_{S}

making the following diagrams commute:

[figure: L(X ⊗ (Y ⊗ Z)); La; b; L((X ⊗ Y) ⊗ Z); b; LX ⊗ L(Y ⊗ Z); 1⊗b; L(X ⊗ Y) ⊗ LZ; b⊗1; LX ⊗ (LY ⊗ LZ); a; (LX ⊗ LY) ⊗ LZ,]

and

[figure: I_{T} ∘ LX; ℓ; u∘1; LX; Lℓ; LI_{S} ∘ LX; b; L(I_{S} ∘ X),; LX ∘ I_{T}; r; 1∘u; LX; Lr; LX ∘ LI_{S}; b; L(X ∘ I_{S}).]

If L: S → T is a monoidal functor between symmetric monoidal categories such that the diagram

[figure: LX ∘ LY; b; τ; L(X ∘ Y); τ; LY ∘ LX; b; L(Y ∘ X)]

commutes, where τ: X ∘ Y ⟶^{∼} Y ∘ X denotes the symmetry isomorphism in a symmetric monoidal category, then we say that L is a symmetric monoidal functor.

2.4.2. Proposition. If L: S → T is a monoidal functor which has a right adjoint R: T → S, then there is a natural map

c: RX ⊗ RY → R(X ⊗ Y).

Furthermore, if S and T are right-closed, then there is a natural isomorphism

d: hom_{T}(Y, RZ) ≃ R hom_{S}(LY, Z).

If furthermore L is a symmetric monoidal functor between symmetric monoidal categories then the natural map c commutes with the symmetry operators τ.

The main examples of symmetric monoidal functors which we will be interested in are

|−| : 𝒮 ⇄ 𝒯 : Sing

where |−| is geometric realization and Sing is the singular complex, and

R(−) : 𝒮 ⇄ ℳ_{R} : U

where R(−) is the free R-module functor and U is the underlying set functor. All of these are examples of simplicial functors.

2.4.3. Proposition. If L : S ⇄ T : R are an adjoint pair of functors between closed symmetric monoidal categories, and L is a monoidal functor, then there are naturally induced functors

L′ : ΣS ⇄ ΣT : R′.

Furthermore L′ is a monoidal functor, and there are natural isomorphisms

L(A(X)) ≃ (L′A)(LX),

for X ∈ S and A ∈ ΣS.

Proof. The functors L′ and R′ are defined “object-wise”, so that for A ∈ ΣS and B ∈ ΣT we have that (L′A)[S] = L(A[S]) and (R′B)[S] = R(B[S]). The adjunction is clear. To show that L′ is monoidal, it suffices to note that (L′A)[S, T] ≃ L(A[S, T]) and that L is monoidal and preserves colimits. The proof of the last part of the proposition is similar.

2.4.4. Proposition. There are induced adjoint functors

L′ : operS ⇄ operT : R′

which coincide with L′ : ΣS ⇄ ΣT : R′ on underlying Σ-objects, and which induce a natural equivalence

F (L′A) ≃ L′(F A),

where A ∈ ΣS and F denotes the free operad functor.

Proof. The existence of the induced functors follows from the fact that L′ is monoidal, and from Proposition 2.4.2. The rest of the argument is straightforward from the construction of the free operad in Appendix A.

2.4.5. Corollary. Given A ∈ operS, there are induced adjoint functors

L : S^{A} ⇄ T^{L′A} : R,

which coincide with L : S ⇄ T : R on the underlying categories.

2.4.6. Corollary. Given A, B ∈ operS, there are induced adjoint functors

L′ : (A, B)-biobjS ⇄ (L′A, L′B)-biobjT : R′,

which coincide with L′ : ΣS ⇄ ΣT : R′ on underlying Σ-objects.

Homotopy theory of operads

3.1 Closed model categories

In this section we recall the notion of a simplicial closed model category and give a principle for recognizing such structures in the cases that interest us. We also recall certain examples of simplicial closed model categories which we will need.

3.1.1 Simplicial closed model categories

A closed model category M is a category equipped with three distinguished classes of maps, called fibrations, cofibrations, and weak equivalences, satisfying axioms (1)-(5) of [14]. The properties of closed model categories are elaborated in [13]; in particular, we note that for any closed model category M the homotopy category Ho M, obtained by formally inverting the weak equivalences of M, exists.

A simplicial closed model category is a closed model category M which is enriched over simplicial sets and has enriched limits and colimits over simplicial sets (2.3.19), and which satisfies one of the following three equivalent conditions:

1. Let i: K → L ∈ 𝒮 be a cofibration and j: X → Y ∈ M be a cofibration. Then the induced map

f: K ⊗ Y ⨿_{K⊗X} L ⊗ X → L ⊗ Y ∈ M

is a cofibration. If furthermore either i or j is a weak equivalence, then so is f.

2. Let i: K → L ∈ 𝒮 be a cofibration and p: Z → W ∈ M be a fibration. Then

the induced map

g: Z^{L} → Z^{K} ×_{W^{K}} W^{L} ∈ M

is a fibration. If furthermore either i or p is a weak equivalence, then so is g.

3. Let j: X → Y ∈ M be a cofibration and p: Z → W ∈ M be a fibration. Then the induced map

h: map_{M}[Y, Z] → map_{M}[X, Z] ×_{map_{M}[X,W]} map_{M}[Y, W] ∈ 𝒮

is a fibration. If furthermore either j or p is a weak equivalence, then so is h.

3.1.2 Cofibrantly generated model categories

If M is a closed model category, we say that M is cofibrantly generated if there exist sets of maps I = {i_{α}: S_{α} → D_{α}} and J = {j_{β}: Λ_{β} → Δ_{β}} in M such that

1. a map in M is a trivial fibration if and only if it has the right lifting property with respect to every i_{α} ∈ I,

2. a map in M is a fibration if and only if it has the right lifting property with respect to every j_{β} ∈ J,

3. the domains S_{α} and Λ_{β} of the maps in I and J are small; an object S ∈ M is said to be small if for any countable directed sequence X_{i} in M the natural map

colim_{i} M[S, X_{i}] → M[S, colim_{i} X_{i}]

is an isomorphism.

It follows from this definition that the maps in I are cofibrations, and the maps in J are trivial cofibrations. We say that the set I (resp. J) is a set of generators of the class of cofibrations (resp. trivial cofibrations).

3.1.3. Remark. The formulation of the notion of cofibrantly generated closed model category given in 3.1.2 is not the most general form possible; our definition is in fact a very restricted case of a definition of Dwyer and Kan [2].

Note that if M is a cofibrantly generated closed model category, and X ∈ M is any object, then (M ↓ X) is also a cofibrantly generated closed model category, with generators

I′ = {S_{α} → D_{α} → X}

and

J′ = {Λ_{β} → Δ_{β} → X}.

3.1.4 Recognizing model categories

The following proposition will be useful in showing that various categories are simplicial closed model categories.

3.1.5. Proposition. Let M and N be categories enriched over simplicial sets, having all enriched limits and colimits, and let

L : M ⇄ N : R

be a pair of adjoint simplicial functors. Suppose that M is a simplicial closed model category with sets of maps I = {S_{α} → D_{α} ∈ M} and J = {Λ_{β} → Δ_{β} ∈ M} which generate cofibrations and trivial cofibrations respectively. If 1. the objects LS_{α} and LΛ_{β} are small in N, and

2. there is a simplicial functor

E: N → N

and a natural transformation

ϵ: 1 → E: N → N

such that for each object X ∈ N the object REX ∈ M is fibrant and the map Rϵ_{X}: RX → REX ∈ M is a weak equivalence,

then N is a simplicial closed model category with the following structure.

1. A map f: X → Y ∈ N is a weak equivalence (resp. a fibration) if and only

if the map

Rf: RX → RY ∈ M

is a weak equivalence (resp. a fibration) in M.

2. The sets of maps

I′ = {LS_{α} → LD_{α}}

and

J′ = {LΛ_{β} → LΔ_{β}}

are generators for the cofibrations and trivial cofibrations of N respectively.

3. There exist functorial factorizations of maps in N into a cofibration followed by a trivial fibration, or respectively a trivial cofibration followed by a fibration.

Proof. The proof is a variation of an argument of Quillen. To show that N is a simplicial closed model category, we carry out the small object argument [13, Ch.II,4], using the sets of maps I′ and J′ to construct the factorizations. The only tricky part of the argument is to show that if a map f: X → Y in N has the left lifting property with respect to maps p ∈ N for which Rp is a fibration, then Rf is a weak equivalence. To show this, we use the map ϵ_{X}: X → EX. Thus there is a lifting g in

[figure: X; ϵ_{X}; f; EX; Y .; g]

Since Rϵ_{X} is a weak equivalence, it suffices to show that Rg is also. To prove this, we show that Ef · g is simplicially homotopic to ϵ_{Y}; this follows from the lifting in the diagram

[figure: X; ϵ_{Y} f=Ef·ϵ_{X}; f; EY; EY^{Δ[1]}; Y; (ϵ_{Y} ,Ef·g); EY × EY .]

3.1.6 Model category structure for simplicial sets

We recall that the category 𝒮 of simplicial sets forms a simplicial closed model category. 3.1.7. Proposition. [13] The category 𝒮 of simplicial sets is a simplicial closed model category with the following structure.

1. A map f: X → Y ∈ 𝒮 is a weak equivalence if and only if its geometric realization |f|: |X| → |Y | is a homotopy equivalence of spaces.

2. A map f: X → Y ∈ 𝒮 is a fibration if and only if it is a Kan fibration.

3. A map f: X → Y ∈ 𝒮 is a cofibration if and only if it is an inclusion.

4. The sets of maps

I = {Δ̇[n] → Δ[n], n ≥ 0}

and

J = {Λ^{k}[n] → Δ[n], n ≥ 1, 0 ≤ k ≤ n}

are generators for the cofibrations and trivial cofibrations respectively.

3.1.8 Model category structure for G-simplicial sets

Let G be a discrete group. The category of G-simplicial sets, denoted by G𝒮, is the category with objects simplicial sets with an action of G on the right, and with morphisms G-equivariant maps. 3.1.9. Proposition. [5] The category G𝒮 of G-simplicial sets is a simplicial closed model category with the following structure.

1. A map f: X → Y ∈ G𝒮 is a weak equivalence (resp. a fibration) if and

only if for each subgroup H ⊆ G the induced map

f^{H}: X^{H} → Y^{H} ∈ 𝒮

on fixed points is a weak equivalence (resp. a fibration) of simplicial sets.

2. A map f: X → Y ∈ G𝒮 is a cofibration if and only if it is an inclusion.

3. The sets of maps

I = {Δ̇[n] × H\G → Δ[n] × H\G, n ≥ 0, H ⊆ G}

and

J = {Λ^{k}[n] × H\G → Δ[n] × H\G, n ≥ 0, 0 ≤ k ≤ n, H ⊆ G} are generators for the cofibrations and trivial cofibrations respectively.

There is another characterization of weak equivalences in G𝒮.

3.1.10. Proposition. A map f: X → Y ∈ G𝒮 is a weak equivalence if and only if the induced map

|f|: |X| → |Y |

of geometric realizations is a G-equivariant homotopy equivalence; i.e., there exists a G-equivariant map g: |Y | → |X| and G-equivariant homotopies |f|g ∼ 1 and g|f| ∼ 1.

To prove Proposition 3.1.10 we need several results about G-spaces. The category G𝒯 of G-spaces is the category having as objects topological spaces with a G-action, and having as maps G-equivariant maps.

3.1.11. Proposition. [5] The category G𝒯 of G-spaces is a simplicial closed model category with the following structure.

1. A map f: X → Y ∈ G𝒯 is a weak equivalence (resp. a fibration) if and only if for each subgroup H ⊆ G the induced map

f^{H}: X^{H} → Y^{H} ∈ 𝒯

on fixed points is a weak equivalence (resp. a Serre fibration) of spaces.

2. The sets of maps

I = {|Δ̇[n]| × H\G → |Δ[n]| × H\G}

and

J = {|Λ^{k}[n]| × H\G → |Δ[n]| × H\G}

are generators for the cofibrations and trivial cofibrations respectively.

3.1.12. Proposition. For X ∈ G𝒮, the adjunction map ϵ_{X}: X → Sing|X| is a weak equivalence in G𝒮.

Proof. We note the easily proved fact that geometric realization of simplicial sets commutes with taking the fixed point set of a finite group action. (In fact, geometric realization preserves all finite limits; see [6, Ch.III,3].) Thus for H ⊆ G, we have

X^{H} → Sing|X|^{H} ≃ Sing|X^{H}|,

which is a weak equivalence.

Proof of Proposition 3.1.10. Since every object in G𝒯 is fibrant, a map between cofibrant objects is a weak equivalence if and only if it is a simplicial homotopy equivalence. Thus, a map f in G𝒮 is a weak equivalence if and only if its realization |f| in G𝒯 is a simplicial homotopy equivalence, and thus a G-equivariant homotopy equivalence.

3.1.13 Model category structure for simplicial R-modules

We recall that the category ℳ_{R} of simplicial R-modules forms a simplicial closed model category.

3.1.14. Proposition. [13] The category ℳ_{R} of simplicial R-modules is a simplicial closed model category with the following structure.

1. A map f: X → Y ∈ ℳ_{R} is a weak equivalence (resp. a fibration) if and only if it is a weak equivalence (resp. a fibration) of the underlying simplicial sets.

2. A map f: X → Y ∈ ℳ_{R} is a cofibration if and only if in each simplicial degree q ≥ 0 the maps f_{q}: X_{q} → Y_{q} are injective and the cokernel of f_{q} is a projective R-module.

3. The sets of maps

I = {RΔ̇[n] → RΔ[n]}

and

J = {RΛ^{k}[n] → RΔ[n]}

are generators for the cofibrations and trivial cofibrations respectively.

Note that every object in ℳ_{R} is fibrant as a simplicial set by [12], and hence is fibrant in ℳ_{R}.

3.1.15 Model category structure for G-equivariant simplicial R-modules

Let G be a discrete group. The category of G-equivariant simplicial R-modules, denoted by Gℳ_{R}, is the category with objects simplicial R-modules with an action of G on the right, and with morphisms G-equivariant maps. Equivalently, Gℳ_{R} is the category of simplicial R[G]-modules, where R[G] is the group ring of G.

3.1.16. Proposition. The category Gℳ_{R} of G-equivariant simplicial R-modules is a simplicial closed model category with the following structure.

1. A map f: X → Y ∈ Gℳ_{R} is a weak equivalence (resp. a fibration) if and only if for each subgroup H ⊆ G the induced map

f^{H}: X^{H} → Y^{H} ∈ ℳ_{R}

on invariants is a weak equivalence (resp. a fibration) of underlying simplicial sets.

2. A map f: X → Y ∈ Gℳ_{R} is a cofibration if and only in each simplicial degree q ≥ 0 the maps f_{q}: X_{q} → Y_{q} are inclusions and the cokernel of f_{q} is a retract of a R[G]-module of the form RS, where S is a G-set.

3. The sets of maps

I = {R(Δ̇[n] × H\G) → R(Δ[n] × H\G)}

and

J = {R(Λ^{k}[n] × H\G) → R(Δ[n] × H\G)}

are generators for the cofibrations and trivial cofibrations respectively.

Proof. Everything except part 2 follows from Proposition 3.1.5 applied to the adjunction

R : G𝒮 ⇄ Gℳ_{R} : U,

where we take E to be the identity functor on ℳ_{R}.

The characterization of cofibrations in part 2 follows from the small object argument.

Note that every object in Gℳ_{R} is fibrant.

3.1.17. Corollary. An object X ∈ Gℳ_{R} is cofibrant if and only if in each simplicial degree q ≥ 0 the module X_{q} is a retract of a R[G]-module of the form RS, where S is a G-set.

3.2 Model categories over simplicial sets

In this section we describe model category structures for Σ-objects, operads, algebras over an operad, and biobjects over an operad, when defined over the category of simplicial sets.

3.2.1 A model category structure for Σ-objects over simplicial sets

Given a subgroup H ⊆ Σ_{m}, we define Z_{H} ∈ Σ𝒮 by

Z_{H}[k] = \begin{cases}H\Σ_{m} & if k = m, \\ ∅ & otherwise. \end{cases}

3.2.2. Proposition. The category Σ𝒮 of Σ-objects over simplicial sets is a simplicial closed model category with the following structure.

1. A map f: X → Y ∈ Σ𝒮 is a weak equivalence (resp. a fibration) if and only if for each m ≥ 0 and each subgroup H ⊆ Σ_{m} the induced map

f[m]^{H}: X[m]^{H} → Y [m]^{H} ∈ 𝒮

on fixed points is a weak equivalence (resp. a fibration) of simplicial sets.

2. A map f: X → Y ∈ Σ𝒮 is a cofibration if and only if each f[m]: X → Y is an inclusion of simplicial sets.

3. The sets of maps

I = {Δ̇[n] ⊗ Z_{H} → Δ[n] ⊗ Z_{H}}

and

J = {Λ^{k}[n] ⊗ Z_{H} → Δ[n] ⊗ Z_{H}}

are generators for the cofibrations and trivial cofibrations respectively.

Proof. This follows from Proposition 3.1.9, since Σ𝒮 is equivalent to a product of categories Σ_{m}𝒮.

3.2.3. Proposition. 1. A map f: X → Y ∈ Σ𝒮 is a weak equivalence if and only if for each m ≥ 0 the induced maps

|f[m]|: |X[m]| → |Y [m]|

of geometric realizations are Σ_{m}-equivariant homotopy equivalences.

2. For X ∈ Σ𝒮, the adjunction map ϵ_{X}: X → Sing|X| is a weak equivalence in Σ𝒮.

Proof. Immediate from Proposition 3.1.10.

Note that the Σ-object I is cofibrant in Σ𝒮.

3.2.4 A model category structure for A-algebras over simplicial sets

3.2.5. Proposition. Let A ∈ oper𝒮 be a simplicial operad. The category 𝒮^{A} of algebras over A is a simplicial closed model category with the following structure.

1. A map f: X → Y of A-algebras is a weak equivalence (resp. a fibration) if it is a weak equivalence (resp. a fibration) of underlying simplicial sets.

2. The sets of maps

I = {A(Δ̇[n]) → A(Δ[n])}

and

J = {A(Λ^{k}[n]) → A(Δ[n])}

generate the cofibrations and trivial cofibrations respectively.

Proof. This follows from Proposition 3.1.5 applied to the adjunction

𝒮 ⇄ 𝒮^{A},

where we take E = Sing|−| and ϵ to be the adjunction map; by Corollary 2.4.5 the functor E is well defined, and by Proposition 3.1.7 the adjunction ϵ is always a weak equivalence in 𝒮.

3.2.6. Remark. Following Quillen [13] we note that a map X → Y ∈ 𝒮^{A} is a cofibration if and only if it is a retract of a free map; a map i: X → Y ∈ 𝒮^{A} is free if Y is the diagonal of a simplicial object in 𝒮^{A} with qth degree X ⨿ A(S_{q}), where the S_{q} are a collection of discrete simplicial sets which are closed under the degeneracy operators, and i is induced by inclusion of the constant simplicial object X.

3.2.7 A model category structure for A, B-biobjects over simplicial sets

3.2.8. Proposition. Let A, B ∈ oper𝒮 be operads. The category (A, B)-biobj𝒮 of A, B-biobjects is a simplicial closed model category with the following structure.

1. A map f: X → Y of A, B-biobjects is a weak equivalence (resp. a fibration) if it is a weak equivalence (resp. a fibration) in Σ𝒮.

2. The sets of maps

I = {A ∘ (Δ̇[n] ⊗ Z_{H}) ∘ B → A ∘ (Δ[n] ⊗ Z_{H}) ∘ B}

and

J = {A ∘ (Λ^{k}[n] ⊗ Z_{H}) ∘ B → A ∘ (Δ[n] ⊗ Z_{H}) ∘ B}

generate the cofibrations and the trivial cofibrations respectively.

Proof. This follows from Proposition 3.1.5 applied to the adjunction

Σ𝒮 ⇄ (A, B)-biobj𝒮,

where we take E = Sing|−| and ϵ to be the adjunction map; by Corollary 2.4.6 the functor E is well defined, and by Proposition 3.2.3 the adjunction ϵ is always a weak equivalence in Σ𝒮.

3.2.9. Remark. Following Quillen [13] we note that a map X → Y ∈ (A, B)-biobj𝒮 is a cofibration if and only if it is a retract of a free map; a map i: X → Y ∈ (A, B)-biobj𝒮 is free if Y is the diagonal of a simplicial object in (A, B)-biobj𝒮 with qth degree X ⨿A∘S_{q} ∘B, where the S_{q} are a collection of discrete Σ-objects which are closed under the degeneracy operators, and i is induced by inclusion of the constant simplicial object X.

3.2.10 A model category structure for operads over simplicial sets

3.2.11. Proposition. The category oper𝒮 of operads is a simplicial closed model category with the following structure.

1. A map f: X → Y of operads is a weak equivalence (resp. a fibration) if it is a weak equivalence (resp. a fibration) in Σ𝒮.

I = {F (Δ̇[n] ⊗ Z_{H}) → F (Δ[n] ⊗ Z_{H})}

and

J = {F (Λ^{k}[n] ⊗ Z_{H}) → F (Δ[n] ⊗ Z_{H})}

generate the cofibrations and trivial cofibrations respectively.

Proof. This follows from Proposition 3.1.5 applied to the adjunction

Σ𝒮 ⇄ oper𝒮,

where we take E = Sing|−| and ϵ to be the adjunction map; by Corollary 2.4.4 the functor E is well defined, and by Proposition 3.2.3 the adjunction ϵ is always a weak equivalence in 𝒮.

3.2.12. Remark. Following Quillen [13] we note that a map X → Y ∈ oper𝒮 is a cofibration if and only if it is a retract of a free map; a map i: X → Y ∈ oper𝒮 is free if Y is the diagonal of a simplicial object in oper𝒮 with qth degree X ⨿ F (S_{q}), where the S_{q} are a collection of discrete Σ-objects which are closed under the degeneracy operators, and i is induced by inclusion of the constant simplicial object X.

3.3 Model category structures over simplicial modules

In this section we describe model category structures for Σ-objects, operads, algebras over an operad, and biobjects over an operad when defined the category of simplicial R-modules.

3.3.1 A model category structure for Σ-objects over simplicial modules

Given a subgroup H ⊆ Σ_{m}, let RZ_{H} ∈ Σℳ_{R} be defined by

RZ_{H}[k] = \begin{cases}R(Σ_{m}/H) & if k = m, \\ 0 & otherwise. \end{cases}

3.3.2. Proposition. The category Σℳ_{R} is a simplicial closed model category, with the following structure.

1. A map f: X → Y is a weak equivalence (resp. a fibration) if for every H ⊆ Σ_{m} the map f[m]^{H}: X[m]^{H} → Y [m]^{H} is a weak equivalence (resp. a fibration) of simplicial sets.

2. The sets of maps

I = {Δ̇[n] ⊗ RZ_{H} → Δ[n] ⊗ RZ_{H}}

and

J = {Λ^{k}[n] ⊗ RZ_{H} → Δ[n] ⊗ RZ_{H}}

generate the cofibrations and trivial cofibrations respectively. Proof. This follows from Proposition 3.1.16, since Σℳ_{R} is equivalent to a product of categories Σ_{m}ℳ_{R}. 3.3.3. Proposition. Let f: X → Y ∈ Σℳ_{R} be a map. The following are equivalent.

1. The map f is a cofibration.

2. The map f is injective and Cok(f) is cofibrant.

3. In each simplicial degree q ≥ 0 the maps f_{q}[m]: X_{q}[m] → Y_{q}[m] are injective and the cokernel of f_{q}[m] is a retract of a Σ_{m}-module of the form RS, where S is a Σ_{m}-set.

Proof. This follows immediately from Proposition 3.1.16 3.3.4. Corollary. An object X ∈ Σℳ_{R} is cofibrant if and only if in each simplicial degree q ≥ 0 the module X_{q}[m] is a retract of a Σ_{m}-module of the form RS, where S is a Σ_{m}-set.

3.3.5 A model category structure for A-algebras over simpli-

cial modules

3.3.6. Proposition. Let A ∈ operℳ_{R} be an operad. The category ℳ_{R}^{A} of algebras over A is a simplicial closed model category with the following structure.

1. A map f: X → Y of A-algebras is a weak equivalence (resp. a fibration) if it is a weak equivalence (resp. a fibration)of underlying simplicial sets.

2. The sets of maps

I = {A(RΔ̇[n]) → A(RΔ[n])}

and

J = {A(RΛ^{k}[n]) → A(RΔ[n])}

generate the cofibrations and trivial cofibrations respectively. Proof. This follows from Proposition 3.1.5 applied to the adjunction

ℳ_{R} ⇄ ℳ_{R}^{A},

where we take E to be the identity functor, since every object in ℳ_{R}^{A} is fibrant as a simplicial set.

3.3.7. Remark. Following Quillen [13] we note that a map X → Y ∈ ℳ_{R}^{A} is a cofibration if and only if it is a retract of a free map; a map i: X → Y ∈ ℳ_{R}^{A} is free if Y is the diagonal of a simplicial object in ℳ_{R}^{A} with qth degree X ⨿ A(RS_{q}), where the S_{q} are a collection of discrete simplicial sets which are closed under the degeneracy operators, and i is induced by inclusion of the constant simplicial object X.

3.3.8 A model category structure for A, B-biobjects on simplicial modules

3.3.9. Proposition. Let A, B ∈ operℳ_{R} be operads. The category (A, B)-biobjℳ_{R} of simplicial A, B-biobjects is a simplicial closed model category with the following structure.

1. A map f: X → Y of A, B-biobjects is a weak equivalence (resp. a fibration) if it is a weak equivalence (resp. a fibration) in Σℳ_{R}.

2. The sets of maps

I = {A ∘ (Δ̇[n] ⊗ RZ_{H}) ∘ B → A ∘ (Δ[n] ⊗ RZ_{H}) ∘ B}

and

J = {A ∘ (Λ^{k}[n] ⊗ RZ_{H}) ∘ B → A ∘ (Δ[n] ⊗ RZ_{H}) ∘ B}

generate the cofibrations and trivial cofibrations respectively.

Proof. This follows from Proposition 3.1.5 applied to the adjunction

Σℳ_{R} ⇄ (A, B)-biobjℳ_{R},

where we take E to be the identity functor, since every object in (A, B)-biobjℳ_{R} is fibrant in Σℳ_{R}.

3.3.10. Remark. Following Quillen [13] we note that a map X → Y ∈ (A, B)-biobjℳ_{R} is a cofibration if and only if it is a retract of a free map; a map i: X → Y ∈ (A, B)-biobjℳ_{R} is free if Y is the diagonal of a simplicial object in (A, B)-biobjℳ_{R} with qth degree X ⨿ A ∘ RS_{q} ∘ B, where the S_{q} are a collection of discrete objects in Σ𝒮 which are closed under the degeneracy operators, and i is induced by inclusion of the constant simplicial object X.

3.3.11 A model category structure for operads on simplicial modules

3.3.12. Proposition. The category operℳ_{R} of operads is a simplicial closed model category with the following structure.

1. A map f: X → Y of operads is a weak equivalence (resp. a fibration) if it is a weak equivalence (resp. a fibration) in Σℳ_{R}.

I = {F (Δ̇[n] ⊗ RZ_{H}) → F (Δ[n] ⊗ RZ_{H})}

and

J = {F (Λ^{k}[n] ⊗ RZ_{H}) → F (Δ[n] ⊗ RZ_{H})}

generate the cofibrations and trivial cofibrations respectively.

Proof. This follows from Proposition 3.1.5 applied to the adjunction

Σℳ_{R} ⇄ operℳ_{R},

where we take E to be the identity functor, since every object in Σℳ_{R} is fibrant.

3.3.13. Remark. Following Quillen [13] we note that a map X → Y ∈ operℳ_{R} is a cofibration if and only if it is a retract of a free map; a map i: X → Y ∈ operℳ_{R} is free if Y is the diagonal of a simplicial object in operℳ_{R} with qth degree X ⨿F (RS_{q}), where the S_{q} are a collection of discrete objects in Σ𝒮 which are closed under the degeneracy operators, and i is induced by inclusion of the constant simplicial object X.

3.4 Compatibility of actions with model category structure

In this section we investigate how well the actions of Σ𝒮 on 𝒮 and Σ𝒮, and the actions of Σℳ_{R} on ℳ_{R} and Σℳ_{R} interact with the closed model category structures. In particular, we show that these actions satisfy axioms which are analogous to, but weaker than, those which characterize a simplicial closed model category structure (3.1.1).

3.4.1 Compatibility axioms

Let M be a monoidal category with a right-closed action on C, and suppose that both M and C are closed model categories. We say that the action of M on C is compatible with the closed model category structure if either of the following equivalent statements are true.

(C1) Let i: X → Y ∈ C be a cofibration between cofibrant objects, and let j: F → G ∈ M be a cofibration. Then the induced maps

f: F (∅) → G(∅)

and

g: F (Y) ⨿_{F (X)} G(X) → G(Y)

are cofibrations in C. If i is also a weak equivalence, then g is a trivial cofibration. If j is also a weak equivalence, then f and g are trivial cofibrations.

(C2) Let i: X → Y ∈ C be a cofibration between cofibrant objects and p: Z → W ∈ C be a fibration. Then the induced maps

f: ℌ𝔬𝔪(∅, Z) → ℌ𝔬𝔪(∅, W)

and

g: ℌ𝔬𝔪(Y, Z) → ℌ𝔬𝔪(X, Z) ×_{ℌ𝔬𝔪(X,W)} ℌ𝔬𝔪(Y, W)

are fibrations in M. If i is also a weak equivalence, then g is a trivial fibration. If p is also a weak equivalence, then f and g are trivial fibrations.

If M is a right-closed monoidal category which is also a closed model category, we say that the monoidal structure on M is compatible with the closed model category structure if the induced right-closed action of M on itself is compatible with the closed model category structure.

Note that if C is a category enriched over simplicial sets having all enriched limits and colimits, then there is a right-closed action of 𝒮 on C. If C is a simplicial closed model category then we see that, in particular, the action of 𝒮 on C is compatible with the model category structure.

3.4.2. Proposition. Let M and C be closed model categories equipped with a rightclosed action of M on C which is compatible with the model category structure.

1. If X ∈ C and A ∈ M are cofibrant objects, then A ∘ X ∈ C is cofibrant.

2. If X → Y ∈ C is a cofibration and A ∈ M is a cofibrant object, then A ∘ X → A ∘ Y ∈ C is a cofibration.

Proof. Statement 2 follows immediately from (C1), and statement 1 follows from statement 2 and the fact that A ∘ ∅ is cofibrant by (C1).

3.4.3 Compatibility of actions of Σ-objects over 𝒮 and ℳ_{R}

3.4.4. Proposition. The action of Σ𝒮 on 𝒮 and the action of Σℳ_{R} on ℳ_{R} are compatible with the closed model category structure.

3.4.5. Proposition. The monoidal structure of Σ𝒮 and the monoidal structure of Σℳ_{R} are compatible with the closed model category structure.

Before we prove Proposition 3.4.4, we will need lemmas on equivariant homotopy. Let C denote either 𝒮 or ℳ_{R}. Note that if X and Y are in C, then for H ⊆ Σ_{n} a subgroup we have that.

ℌ𝔬𝔪(X, Y)[n]^{H} = map_{C}(X^{⊗n}, Y)^{H} = map_{C}((X^{⊗n})_{H}, Y).

3.4.6. Lemma. Let i: X → Y be a cofibration (resp. a trivial cofibration) of simplicial sets, and let H ⊆ Σ_{n} be a subgroup of the symmetric group. Then the induced map j: (X^{×n})_{H} → (Y^{×n})_{H} is a cofibration (resp. a trivial cofibration) of simplicial sets.

Proof. To see that j is a cofibration, it suffices to note that since i is an inclusion, then j is also. Furthermore, if i is a trivial cofibration, then |i|: |X| → |Y | has a homotopy inverse in 𝒯. Hence |i|^{×n} has a Σ_{n}-equivariant homotopy inverse, and thus |j| has a homotopy inverse in 𝒯, and thus in particular j is a weak equivalence.

3.4.7. Lemma. Let i: X → Y be a cofibration (resp. a trivial cofibration) of cofibrant simplicial R-modules, and let H ⊆ Σ_{n} be a subgroup of the symmetric group. Then the induced map j: (X^{⊗n})_{H} → (Y^{⊗n})_{H} is a cofibration (resp. a trivial cofibration) of simplicial R-modules.

Proof. Because X is cofibrant and i is a cofibration, we may assume without loss of generality that X and Cok(i) are degree-wise free R-modules. In other words, X_{q} ≃ R(K_{q}) and Y_{q} ≃ R(K_{q}) ⊕ R(L_{q}) for some sets K_{q} and L_{q}. To show that j: (X^{⊗n})_{H} → (Y^{⊗n})_{H} is a cofibration it suffices to show that the induced maps

(R(K_{q})^{⊗n})_{H} → (R(K_{q}) ⊕ R(L_{q})^{⊗n})_{H}

are inclusions with projective cokernel. This follows easily from the fact that

(K_{q}^{×n})_{H} → ((K_{q} ⨿ L_{q})^{×n})_{H}

is an inclusion of sets.

If i is a trivial cofibration, then since all objects in ℳ_{R} are fibrant there exists a simplicial homotopy inverse to i, and hence j has a simplicial homotopy inverse.

Proof of Proposition 3.4.4. We prove (C2). Let C denote either 𝒮 or ℳ_{R}. To prove the first statement in (C2), it suffices to note that ℌ𝔬𝔪(∅, Z) ≃ Z[0]. Suppose H ⊆ Σ_{n} is a subgroup. Then we see that the induced map g[n]^{H} may be

written as

g[n]^{H}: map_{C}((Y^{⊗n})_{H}, Z) → map_{C}((X^{⊗n})_{H}, Z)×_{map_{C}((X^{⊗n})_{H},W)}map_{C}((Y^{⊗n})_{H}, W ).

By Lemma 3.4.6 or 3.4.7 we have that i_{H}: (X^{⊗n})_{H} → (Y^{⊗n})_{H} is a cofibration (resp. a trivial cofibration) in C whenever i is one, and thus the result follows from the simplicial closed model category structure on C.

We need two preliminary lemmas before we can prove Proposition 3.4.5. Recall that if A and B are in ΣC, then for H ⊆ Σ_{n} a subgroup we have that

F(A, B)[n]^{H} = (∏_{s} map_{C}(A[s, n], B[s])^{Σ_{s}})^{H} ≃ ∏_{s} map_{C}(A[s, n]_{H}, B[s])^{Σ_{s}}.

3.4.8. Lemma. Let i: A → B be a cofibration (resp. a trivial cofibration) in Σ𝒮, and let H ⊆ Σ_{n} be a subgroup of the symmetric group. Then the induced map j: A[s, n]_{H} → B[s, n]_{H} is a cofibration (resp. a trivial cofibration) of Σ_{s}-equivariant simplicial sets.

Proof. To see that j is a cofibration, it suffices to note that since i is an inclusion, then so is j. Furthermore, if i is a trivial cofibration, then the realizations |i[n]|: |A[n]| → |B[n]| have a Σ_{n}-equivariant homotopy inverse in 𝒯. Thus the induced map |A[s, n]| → |B[s, n]| has a Σ_{s} × Σ_{n}-equivariant homotopy inverse in 𝒯, whence |j| has a Σ_{s}- equivariant homotopy inverse in 𝒯. Thus, in particular, j is a weak equivalence.

3.4.9. Lemma. Let i: A → B be a cofibration (resp. a trivial cofibration) between cofibrant objects in Σℳ_{R}, and let H ⊆ Σ_{n} be a subgroup of the symmetric group. Then the induced map j: A[s, n]_{H} → B[s, n]_{H} is a cofibration (resp. a trivial cofibration) of Σ_{s}-equivariant simplicial R-modules.

Proof. Because A is cofibrant and i is a cofibration, we may assume without loss of generality using Proposition 3.3.3 that A and Cok(i) are degree-wise free over R. In other words, A_{q} ≃ R(K_{q}) and B_{q} ≃ R(K_{q}) ⊕ R(L_{q}) for some Σ-objects K_{q} and L_{q} on the category of sets. To show that j: A[s, n]_{H} → B[s, n]_{H} is a cofibration it suffices to show that the induced maps

R(K_{q})[s, n]_{H} → R(K_{q} ⨿ L_{q})[s, n]_{H}

are inclusions such that the cokernel is a free R-module on a Σ_{s}-set. This follows easily from the fact that

K_{q}[s, n]_{H} → (K_{q} ⨿ L_{q})[s, n]_{H}

is an inclusion of Σ_{s}-sets.

If i is a trivial cofibration, then since all objects in Σℳ_{R} are fibrant there exists a simplicial homotopy inverse to i in Σℳ_{R}, and hence j has a simplicial homotopy inverse in Σ_{s}ℳ_{R}.

Proof of Proposition 3.4.5. We prove (C2). Let C denote either 𝒮 or ℳ_{R}. To prove the first statement it suffices to note that

F(∅, Z) ≃ ∏_{s≥0} Z[s]^{Σ_{s}}.

Suppose H ⊆ Σ_{n}. Then we see that the induced map g[n]^{H} may be written as

∏_{s} map_{C}(G[s, n]_{H}, Z[s]) ⟶^{g[n]^{H}}

∏_{s} map_{C}(F [s, n]_{H}, Z[s]) ×_{map_{C}(F [s,n]_{H},W [s])} map_{C}(G[s, n]_{H}, W [s]).

Since by Lemma 3.4.8 or 3.4.9 we have that F [s, n]_{H} → G[s, n]_{H} is a cofibration (resp. a trivial cofibration) of Σ_{s}-equivariant objects whenever i is one in ΣC, the result follows from the simplicial closed model category structure of C.

3.5 Cofibrant objects and forgetful functors

In this section we study how well forgetful functors preserve cofibrancy. For any A ∈ operC we say that an A-algebra X ∈ C^{A} is C-cofibrant if it is cofibrant in the underlying category C. We say that an A-biobject is Σ-cofibrant if it is cofibrant in the underlying category of Σ-objects. We say that an operad A is Σ-cofibrant if the unit map η: I → A is a cofibration in the underlying category of Σ-objects.

Over the category of simplicial sets, cofibrancy is preserved by these forgetful functors for trivial reasons.

3.5.1. Proposition. Every operad in oper𝒮 is Σ-cofibrant, and for any A ∈ oper𝒮 every object in 𝒮^{A} (resp. A-biobj𝒮) is 𝒮-cofibrant (resp. Σ-cofibrant).

Proof. Every object in 𝒮 and Σ𝒮 is cofibrant, and for any operad A the unit map I → A is an inclusion, and thus is a cofibration from a cofibrant object.

Over the category of simplicial R-modules we need to be more careful. First note that over a field of characteristic zero things become simple.

3.5.2. Proposition.

1.If R is a field, then for any A ∈ operℳ_{R} every object in ℳ_{R}^{A} is Σ-cofibrant.

2.If R is a field of characteristic zero, then for an operad A ∈ operℳ_{R} every object in A-biobjℳ_{R} is Σℳ_{R}-cofibrant, and if A ≠ 0 then the operad A is Σℳ_{R}-cofibrant.

Proof. If R is a field, then every R-module is projective, and thus every object in ℳ_{R} is cofibrant. If R is a field of characteristic zero then the group ring R[Σ_{n}] is semisimple, and thus every R[Σ_{n}]-module is a retract of a free one, and thus every object in Σℳ_{R} is cofibrant. If A is a non-zero operad, then the map I → A is necessarily an inclusion, and thus A is Σ-cofibrant.

More generally, we have the following.

3.5.3. Proposition. Every cofibrant operad in Σℳ_{R} is Σ-cofibrant.

We begin by proving the following lemma.

3.5.4. Lemma. If A ∈ Σℳ_{R} is cofibrant, then the free operad F A is Σ-cofibrant.

Proof. We use the construction of the free operad given in Section A. Thus, F A is the colimit of a sequence of maps i_{n}: F_{n}A → F_{n+1}A, where F_{n+1}A = I ⨿ A ∘ F_{n}A and where i_{0} : I → I ⨿ A and i_{n} = I ⨿ A ∘ i_{n−1} for n > 0. Thus it suffices to show that each i_{n} is a cofibration of Σ-objects.

Since A and I are cofibrant, i_{0} is a cofibration between cofibrant objects. Now suppose we have shown that i_{n} is a cofibration between cofibrant objects. Then by Propositions 3.4.2 and 3.4.5 we see that A ∘ i_{n} is a cofibration between cofibrant objects, and it follows that i_{n+1} is a cofibration between cofibrant objects, as desired.

Proof of 3.5.3. By the construction of the model category structure on operads, a cofibrant operad is a retract of one which is obtained as the diagonal of a simplicial object in operℳ_{R} which in degree n has the form F A_{n}, where A_{n} ∈ Σℳ_{R} is a cofibrant Σ-object. The result now follows by Lemma 3.5.4 and Proposition 3.3.3.

3.5.5. Proposition. If A ∈ operℳ_{R} is a Σ-cofibrant operad, then every cofibrant A-algebra is ℳ_{R}-cofibrant.

Proof. Since A is Σ-cofibrant it follows from Proposition 3.4.4 that A(RS) is also Σ-cofibrant, where S ∈ 𝒮 is a simplicial set. Thus by Proposition 3.1.14 and Remark 3.3.7, we see that a cofibrant A-algebra is ℳ_{R}-cofibrant.

3.5.6. Proposition. If A, B ∈ operℳ_{R} are Σ-cofibrant operads, then every cofibrant A, B-biobject is Σ-cofibrant.

Proof. Since A and B are Σ-cofibrant it follows from Proposition 3.4.5 that A ∘ (RS) ∘ B is also Σ-cofibrant, where S ∈ Σ𝒮 is a Σ-object over simplicial sets. Thus by Corollary 3.3.4 and Remark 3.3.10, we see that a cofibrant A, B-biobject is Σcofibrant.

3.6 Relations between homotopy categories

In this section we show that under good conditions weakly equivalent operads induce equivalent homotopy categories of algebras and biobjects. Let C denote either 𝒮 or ℳ_{R}.

3.6.1 Homotopy categories of algebras

Let f: A → B be a map of operads on C. Then as noted in 2.3.17, f induces a pair of adjoint functors

f_{∗} : C^{A} ⇄ C^{B} : f^{∗},

where f^{∗} denotes the forgetful functor and f_{∗}X = B ∘_{A} (X).

3.6.2. Proposition.

1.A map j in C^{B} is a weak equivalence (resp. a fibration) in C^{B} if and only if f^{∗}j is a weak equivalence (resp. a fibration) in C^{A}.

2.The functor f_{∗} takes cofibrations (resp. trivial cofibrations) in C^{A} to cofibrations (resp. trivial cofibrations) in C^{B}.

3.The functor f_{∗} takes weak equivalences between cofibrant objects in C^{A} to weak equivalences between cofibrant objects in C^{B}.

Proof. Part (1) is immediate. Parts (2) and (3) follow from (1) and the characterization of weak equivalences and fibrations in 𝒮 and ℳ_{R}.

3.6.3. Corollary. [13] If f: A → B ∈ operC, then the adjoint functors f_{∗}, f^{∗} induce a pair of adjoint functors on homotopy categories

Ho C^{A} ⇄ Ho C^{B}.

3.6.4. Proposition. If X is a cofibrant A-algebra, and f: A → B is a weak equivalence of Σ-cofibrant operads, then the adjunction map i: X → f^{∗}f_{∗}X is a weak equivalence of A-algebras.

Proof. Because X is cofibrant in 𝒮^{A} (resp. ℳ_{R}^{A}, by Remarks 3.2.6 (resp. 3.3.7) we may without loss of generality assume that X is a retract of an A-algebra of the form diag A(S_{•}), where A(S_{•}) is a simplicial object in 𝒮^{A} (resp. ℳ_{R}^{A}) such that each S_{q} is a discrete set (resp. a free R-module on a discrete set) and the degeneracy operators are induced by maps between the S_{q}’s. Thus it suffices to show that the map

diag A(S_{•}) → f^{∗}f_{∗} diag A(S_{•}) ≃ B(S_{•})

is a weak equivalence, but this is clear because each map A(S_{q}) → B(S_{q}) is a weak equivalence by Proposition 3.4.4.

3.6.5. Corollary. Suppose f: A → B is a weak equivalence of Σ-cofibrant operads. Then the adjoint pair of functors f_{∗} and f^{∗} induces an equivalence of homotopy theories in the sense of Quillen between C^{A} and C^{B}. In particular, there is an induced equivalence of homotopy categories

Ho C^{A} ≃ Ho C^{B}.

Proof. It suffices to prove Quillen’s criterion [13]: that for cofibrant X ∈ C^{A} and fibrant Y ∈ C^{B}, a map j: f_{∗}X → Y is a weak equivalence if and only if its adjoint k: X → f^{∗}Y is a weak equivalence. In fact, k factors into

k: X ⟶^{i}f^{∗}f_{∗}X ⟶^{f^{∗}j} f^{∗}Y,

where the unit i of the adjunction is a weak equivalence by Proposition 3.6.4, since X is cofibrant. By Proposition 3.6.2 (1) the map f^{∗}j is a weak equivalence if and only if j is, and the Corollary follows.

3.6.6 Homotopy categories of biobjects

Let f: A → B and g: A′ → B′ be maps of operads on C. Then as noted in 2.3.17, f and g induce a pair of adjoint functors

(f, g)_{∗} : (A, A′)-biobjC ⇄ (B, B′)-biobjC : (f, g)^{∗},

where (f, g)^{∗} denotes the forgetful functor and (f, g)_{∗}M = B ∘_{A} M ∘_{A′} B′.

3.6.7. Proposition.

1.A map j in B-biobjC is a weak equivalence (resp. a fibration) in B-biobjC if and only if f^{∗}j is a weak equivalence (resp. a fibration) in A-biobjC.

2.The functor f_{∗} takes cofibrations (resp. trivial cofibrations) in A-biobjC to cofibrations (resp. trivial cofibrations) in B-biobjC.

3.The functor f_{∗} takes weak equivalences between cofibrant objects in A-biobjC to weak equivalences between cofibrant objects in B-biobjC.

Proof. Part (1) is immediate. Parts (2) and (3) follow from (1) and the characterization of weak equivalences and fibrations in Σ𝒮 and Σℳ_{R}.

3.6.8. Proposition. If M is a cofibrant (A, A′)-biobject, and f: A → B and g: A′ → B′ are weak equivalences of Σ-cofibrant operads, then the adjunction map i: M → (f, g)^{∗}(f, g)_{∗}M is a weak equivalence of (A, A′)-biobjects.

Proof. Because M is cofibrant in (A, A′)-biobj𝒮 (resp. (A, A′)-biobjℳ_{R}, by Remarks 3.2.9 (resp. 3.3.10) we may without loss of generality assume that M is a retract of an A, A′-algebra of the form diag A ∘ S_{•} ∘ A′, where A ∘ S_{•} ∘ A′ is a simplicial object in (A, A′)-biobj𝒮 (resp. (A, A′)-biobjℳ_{R}) such that each S_{q} is a Σ-object such that each S_{q}[n] is a discrete Σ_{n}-set (resp. a free R-module on a discrete Σ_{n}-set) and the degeneracy operators are induced by maps between the S_{q}’s. Thus it suffices to show that the map

diag A ∘ S_{•} ∘ A′ → f^{∗}f_{∗} diag A ∘ S_{•} ∘ A′ ≃ B ∘ S_{•} ∘ B′

is a weak equivalence, but this is clear because each map A ∘ S_{q} ∘ A′ → B ∘ S_{q} ∘ B′ is a weak equivalence by Proposition 3.4.5.

3.6.9. Corollary. Suppose f: A → B and g: A′ → B′ are weak equivalences of Σcofibrant operads. Then the adjoint pair of functors (f, g)_{∗} and (f, g)^{∗} induces an equivalence of homotopy theories in the sense of Quillen between (A, A′)-biobjC and (B, B′)-biobjC. In particular, there is an induced equivalence of homotopy categories

Ho(A, A′)-biobjC ≃ Ho(B, B′)-biobjC.

Proof. This is similar to the proof of Corollary 3.6.5.

3.7 Standard resolutions

In this section we define notions of bar complexes involving operads, and show that under mild hypotheses they can be used to produce cofibrant resolutions of algebras or biobjects.

3.7.1 Bar complexes

Let A and B be operads on some simplicial symmetric monoidal category C. Given X ∈ C^{A} and M ∈ (B, A)-biobjC, define the bar complex B(M, A, X) to be the simplicial object in C^{B} with nth degree M ∘ A^{n}(X), along with the obvious face and degeneracy maps. Then the diagonal diag B(M, A, X) is an object in C^{B}, and there is an augmentation map diag B(M, A, X) → M ∘_{A} (X) in S^{B}.

Likewise, if A, B, and C are operads on some simplicial symmetric monoidal category C, and given M ∈ (B, A)-biobjC and N ∈ (A, C)-biobjC, define the bar complex B(M, A, N) to be the simplicial object in (B, C)-biobjC with nth degree M ∘A^{∘n}∘N, along with the obvious face and degeneracy maps. Then diag B(M, A, N) is an object in (B, C)-biobjC, and there is an augmentation map diag B(M, A, N) → M ∘_{A} N in (B, C)-biobjC.

3.7.2 Realizations of bar complexes

For the remainder of this section, we shall let C denote either the category of simplicial sets or the category of simplicial R-modules.

Given a category N enriched over simplicial sets, let sN denote the category of simplicial objects in N. Then there is a realization functor diag: sN → N, which is left adjoint to the functor N → sN which takes X to the simplicial object [n] ↦ X^{Δ[n]}.

3.7.3. Proposition. If A ∈ operC is is Σ-cofibrant, then given a C-cofibrant Aalgebra X ∈ C^{A}, and a Σ-cofibrant B, A-biobject M ∈ (B, A)-biobjC, the B-algebra diag B(M, A, X) is a cofibrant B-algebra.

3.7.4. Corollary. If A ∈ operC is Σ-cofibrant, and X ∈ C^{A} is C-cofibrant, then diag B(A, A, X) → X is a cofibrant resolution of X as an A-algebra.

Proof. This follows from Proposition 3.7.3 and the existence of a contracting homotopy in the underlying category C given by η: A^{n}(X) → A(A^{n}(X)) = A^{n+1}(X), which shows that the augmentation map is a weak equivalence of A-algebras.

3.7.5. Proposition. If A ∈ operC is Σ-cofibrant, then given a Σ-cofibrant B, Abiobject M ∈ (B, A)-biobjC and a Σ-cofibrant A, C-biobject N ∈ (A, C)-biobjC, the B, C-biobject diag B(M, A, N) is a cofibrant B, C-biobject.

3.7.6. Corollary. If A ∈ operC is Σ-cofibrant, then diag B(A, A, A) is a cofibrant resolution of A as an A-biobject.

Proof. This follows from Proposition 3.7.5 and the existence of a contracting homotopy in the underlying category ΣC given by η: ∘1: A^{∘(n+1)} → A∘A^{∘(n+1)} = A^{∘(n+2)}, which shows that the augmentation map is a weak equivalence of A, A-biobjects.

To prove these, we need to describe the realization functor diag: sN → N more explicitly. Let N be a simplicial category, and consider a simplicial object Y_{•} in sN. The latching object L_{n}Y_{•} is defined to be the coequalizer in

∐_{0≤i<j≤n} Y_{n−1} ⇉ ∐_{0≤k≤n} Y_{n} → L_{n}Y_{•},

where the top map takes the (i, j)th summand to the jth summand by s_{i}, and the bottom map takes the (i, j)th summand to the ith summand by s_{j−1}. Then it follows from the simplicial identities that there is a canonical map L_{n}Y_{•} → Y_{n+1}, induced by the map which takes the ith summand in ∐ Y_{n} to Y_{n+1} by s_{i}. We let L_{−1} = ∅.

It is well known that the realization diag Y_{•} can be recovered from information about the latching spaces. In particular, if we let diag_{0} Y_{•} = Y_{0}, and if we construct diag_{n} Y_{•} inductively by a pushout square

\xymatrix{Δ[n] ⊗ L_{n−1}Y_{•} ⨿_{Δ̇[n]⊗L_{n−1}Y_{•}} Δ̇[n] ⊗ Y_{n} \ar[r] \ar[d] & Δ[n] ⊗ Y_{n} \ar[d] \\ diag_{n−1} Y_{•} \ar[r] & diag_{n} Y_{•},}

then diag Y_{•} = colim_{n} diag_{n} Y_{•}. This leads to the following proposition.

3.7.7. Proposition. If each map L_{n−1}Y_{•} → Y_{n} for n ≥ 0 is a cofibration, then diag Y_{•} is cofibrant.

Let A be an operad on C. Write s_{i}: A^{∘n} → A^{∘(n+1)} for the maps s_{i} = 1∘⋯∘η∘⋯∘1 induced by the unit map η: 1 → A. Thus, s_{0} = η ∘ 1 ∘ ⋯ ∘ 1 and s_{n} = 1 ∘ ⋯ ∘ 1 ∘ η. Define an object K_{n} ∈ ΣC to be the coequalizer in

∐_{0≤i<j≤n} A^{∘(n−1)} ⇉ ∐_{0≤k≤n} A^{∘n} → K_{n},

where the top map takes the (i, j)th summand to the jth summand by s_{i}, and the bottom map takes the (i, j)th summand to the ith summand by s_{j−1}. It follows from the simplicial identities that there is a canonical map k: K_{n} → A^{∘(n+1)}. We let K_{−1} = ∅.

3.7.8. Lemma. For each n ≥ 0, there is a pushout square in ΣC of the form

[figure: K_{n} ∘ I; 1∘η; k∘1; K_{n} ∘ A; A^{∘(n+1)} ∘ I; K_{n+1}]

such that the maps 1 ∘ η: A^{∘(n+1)} ∘ I → A^{∘(n+2)} and k ∘ 1: K_{n} ∘ A → A^{∘(n+2)} induce k: K_{n+1} → A^{∘(n+2)}.

Proof. The proof is by induction on n ≥ −1. The case n = −1 is clear, since K_{0} = I.

For n ≥ 0 we consider the following diagram:

[figure: ∐_{0≤i<j≤n} A^{∘(n−1)} ∘ A; ∐_{0≤k≤n} A^{∘n} ∘ A; K_{n} ∘ A; ∐_{0≤i<j≤n} A^{∘(n−1)}; 1∘η; ∐_{0≤k≤n} A^{∘n}; 1∘η; K_{n} ∘ I; 1∘η; k∘1; A^{∘(n+1)}; A^{∘(n+1)} ∘ I.]

The middle row is the coequalizer which defines K_{n}. The top row is − ∘ A applied to the middle row; since − ∘ A preserves colimits in ΣC the top row is a coequalizer. The colimit of the diagram consisting of all objects in the two left-most columns and all maps shown between them is clearly K_{n+1}. This colimit is also clearly the same as the colimit of the right-hand column, which is the desired push-out.

3.7.9. Lemma. If A is a Σ-cofibrant operad, then the map k: K_{n} → A^{∘(n+1)} is a cofibration between cofibrant objects in ΣC for each n ≥ 0.

Proof. The proof proceeds by induction on n. For n = 0, it’s clear since K_{0} = I → A is a cofibration between cofibrant objects in ΣC since A is Σ-cofibrant. Suppose now we have that K_{n−1} → A^{∘n} is a cofibration between cofibrant objects in ΣC. Then since I → A is also cofibration between cofibrant objects, the result follows from Lemma 3.7.8 and Proposition 3.4.5.

3.7.10. Lemma. If A is a Σ-cofibrant operad, X a C-cofibrant algebra, and M a Σ-cofibrant B, A-biobject, then the natural map L_{n−1}B(M, A, X) → M ∘ A^{n}(X) in C^{B} is a cofibration of B-algebras.

Proof. First, note that L_{n}B(M, A, X) ≃ M ∘ K_{n}(X). The map K_{n}(X) → A^{n}(X) is a cofibration in C by Lemma 3.7.9 and Proposition 3.4.4, and the lemma follows by applying M ∘ − and using Proposition 3.4.4.

Proof of Proposition 3.7.3. From Lemma 3.7.10 and Proposition 3.7.7, we see that diag B(M, A, X) is cofibrant in C^{A}.

3.7.11. Lemma. The natural map L_{n−1}B(M, A, N) → M ∘A^{∘n}∘N in (B, C)-biobjC is a cofibration of B, C-biobjects.

Proof. First, note that L_{n}B(M, A, N) ≃ M ∘ K_{n} ∘ N. The lemma then follows from Lemma 3.7.9.

Proof of Proposition 3.7.5. From Lemma 3.7.11 and Proposition 3.7.7, we see that diag B(M, A, N) is cofibrant in (B, C)-biobjC.

Moduli spaces of algebra structures

4.1 Moduli spaces of algebra structures

In this section we define the endomorphism operad of a diagram, and hence the moduli space of algebra structures of a diagram. We then show that in certain cases maps between such moduli spaces are fibrations or equivalences.

As usual, C denotes either 𝒮 or ℳ_{R}.

4.1.1 Endomorphism operads for diagrams

In Section 2.2 we defined the endomorphism operad E_{X} of an object X ∈ C. The endomorphism operad classifies algebra structures on X.

Let D be a small category, and let X: D → C denote a functor. An A-algebra structure on X consists of for each object D ∈ D an A-algebra structure on XD such that for each map d: D → D′ ∈ D the induced map Xd: XD → XD′ is a map of A-algebras. Given a functor F: B → D we see that an A-algebra structure on X naturally restricts to an A-algebra structure on XF, where XF denotes the composite functor XF: B → D.

Let D be a small category, and let D_{0} denote the set of objects in D. Given a diagram X: D → C, define the object E_{X} in ΣC to be the equalizer of the diagram

E_{X} → ∏_{α∈D_{0}} ℌ𝔬𝔪(X_{α}, X_{α}) ⇉ ∏_{α→β∈D[α,β]} ℌ𝔬𝔪(X_{α}, X_{β}).

4.1.2. Proposition. Let X: D → C be a diagram. The object E_{X} in ΣC has the structure of an operad, induced as a subobject of the operad ∏_{α∈D_{0}} ℌ𝔬𝔪(X_{α}, X_{α}). Maps A → E_{X} of operads are in one-to-one correspondence with A-algebra structures on the diagram X.

The operad E_{X} is called the endomorphism operad of the diagram X. In particular, if D is the category with a single object α and a single morphism, then E_{X} is just the endomorphism operad of the space X_{α}, as defined in Section 2.2.

4.1.3. Proposition. Given a diagram X: D → C and a functor F: B → D there is an induced map E_{X} → E_{XF}.

Given an operad A, let A{X} denote the simplicial set defined by

A{X} = map_{operC}(A, E_{X}).

We call A{X} the moduli space of A-algebra structures on X.

Given a map f: X → Y ∈ C, we let E_{f} or E_{(X⟶^{f}Y)} denote the endomorphism

operad of the diagram (• → •) → C which takes the unique non-identity arrow of (• → •) to f, and let A{f} or A{X ⟶^{f} Y } denote the corresponding space of Aalgebra structures. Likewise, given a string of maps X_{n} → ⋯ → X_{0} ∈ C, we write E_{(X_{n}→···→X_{0})} for the endomorphism operad and A{X_{n} → ⋯ → X_{0}} for the space of A-algebra structures on this diagram.

4.1.4. Proposition. Given a map f: X → Y in C, we have that

E_{f} ≃ ℌ𝔬𝔪(X, X) ×_{ℌ𝔬𝔪(X,Y)} ℌ𝔬𝔪(Y, Y).

More generally, we have the following.

4.1.5. Proposition. Given a sequence of maps X_{n} → ⋯ → X_{0} in C, we have that

E_{(X_{n}→···→X_{0})} ≃ ℌ𝔬𝔪(X_{n}, X_{n}) ×_{ℌ𝔬𝔪(X_{n},X_{n−1})} ⋯ ×_{ℌ𝔬𝔪(X_{1},X_{0})} ℌ𝔬𝔪(X_{0}, X_{0}).

4.1.6 Homotopical properties of endomorphism operads for diagrams

4.1.7. Proposition. If p: X → Y is a fibration between cofibrant-fibrant objects in C, then in

E_{X} ⟵^{f} E_{p} ⟶^{g}E_{Y} ,

the map g is a fibration. If furthermore p is a trivial fibration between cofibrant-fibrant objects, then f and g are also weak equivalences.

Proof. Given Proposition 4.1.4, we see that to show that g is a fibration (resp. a trivial fibration), it suffices to show that the map ℌ𝔬𝔪(X, p): ℌ𝔬𝔪(X, X) → ℌ𝔬𝔪(X, Y) is. That ℌ𝔬𝔪(X, p) is a fibration (resp. a trivial fibration) follows from Corollary 3.4.4.

Given Proposition 4.1.4 and the fact that ℌ𝔬𝔪(X, p) is a fibration, we see that to show that f is a weak equivalence it suffices to show that the map

ℌ𝔬𝔪(p, Y): ℌ𝔬𝔪(Y, Y) → ℌ𝔬𝔪(X, Y)

is. That ℌ𝔬𝔪(p, Y) is a weak equivalence follows from Corollary 3.4.4.

4.1.8. Proposition. If i: X → Y is a cofibration between cofibrant-fibrant objects in C, then in

E_{X} ⟵^{f} E_{i} ⟶^{g}E_{Y} ,

the map f is a fibration. If furthermore i is a trivial cofibration between cofibrantfibrant objects, then f and g are also weak equivalences.

Proof. Given Proposition 4.1.4, we see that to show that f is a fibration (resp. a trivial fibration), it suffices to show that the map ℌ𝔬𝔪(i, Y): ℌ𝔬𝔪(Y, Y) → ℌ𝔬𝔪(X, Y) is. That ℌ𝔬𝔪(i, Y) is a fibration (resp. a trivial fibration) follows from Corollary 3.4.4.

Given Proposition 4.1.4 and the fact that ℌ𝔬𝔪(i, Y) is a fibration, we see that to show that g is a weak equivalence it suffices to show that the map

ℌ𝔬𝔪(X, i): ℌ𝔬𝔪(X, X) → ℌ𝔬𝔪(X, Y)

is. That ℌ𝔬𝔪(X, i) is a weak equivalence follows from Corollary 3.4.4.

We will actually need the following generalization of Proposition 4.1.7

4.1.9. Proposition. If X_{n} → ⋯ → X_{0} are fibrations between cofibrant-fibrant objects in C, then for any k such that 0 ≤ k ≤ n we have that in

E_{(X_{n}→···→X_{k+1})} ⟵^{f} E_{(X_{n}→···→X_{0})} ⟶^{g}E_{(X_{k}→···→X_{0})},

the map g is a fibration. If furthermore all the maps in the sequence are trivial fibrations, then f and g are also weak equivalences.

Proof. Similar to the proof of Proposition 4.1.7.

4.1.10 Homotopical properties of moduli spaces of algebra structures on diagrams

Propositions 4.1.7, 4.1.8, and 4.1.9 imply the following results about spaces of algebra structures.

4.1.11. Proposition. If i: X → Y is a cofibration between cofibrant-fibrant objects in C, and A is a cofibrant operad, then in

A{X} ⟵^{f} A{i} ⟶^{g}A{Y },

the map f is a fibration. If furthermore i is a trivial cofibration between cofibrantfibrant objects, then f and g are also weak equivalences.

4.1.12. Proposition. If p: X → Y is a fibration between cofibrant-fibrant objects in C, and A is a cofibrant operad, then in

A{X} ⟵^{f} A{p} ⟶^{g}A{Y },

the map g is a fibration. If furthermore p is a trivial fibration between cofibrant-fibrant objects, then f and g are also weak equivalences.

4.1.13. Proposition. If the maps X_{n} → ⋯ → X_{0} are trivial fibrations between cofibrant-fibrant objects in C, and A is a cofibrant operad, then for any k such that 0 ≤ k ≤ n we have that in

A{X_{n} → ⋯ → X_{k+1}} ⟵^{f} A{X_{n} → ⋯ → X_{0}} ⟶^{g}A{X_{k} → ⋯ → X_{0}},

the map f is a weak equivalence and g is a trivial fibration.

4.1.14 An application to C-cofibrancy

We make use of the space of A-algebra structures to prove the following handy result, which will be used in Section 4.2.

4.1.15. Proposition. Let A be a cofibrant operad. If f: X → Y is a cofibration of A-algebras from a C-cofibrant object X, then Y is also C-cofibrant.

4.1.16. Lemma. Consider X ⟶^{i} Z ⟶^{p} Y ∈ C such that i is a cofibration, p is a fibration, and X is cofibrant in C. Then the induced map of endomorphism operads

g: E_{(X→Z→Y)} ⟶ E_{(X→Y)}

is a fibration. Furthermore g is a weak equivalence if either i or p is.

Proof. By Corollary 3.4.4 we see that the induced map

h: ℌ𝔬𝔪(Z, Z) ⟶ ℌ𝔬𝔪(X, Z) ×_{ℌ𝔬𝔪(X,Y)} ℌ𝔬𝔪(Z, Y)

is a fibration, and is furthermore a weak equivalence if either i or p is. The result follows because g is a pull-back of h, namely

g = ℌ𝔬𝔪(X, X) ×_{ℌ𝔬𝔪(X,Z)} h ×_{ℌ𝔬𝔪(Z,Y)} ℌ𝔬𝔪(Y, Y).

Proof of Proposition 4.1.15. Choose a factorization X ⟶^{i} Z ⟶^{p} Y in C of the map f: X → Y ∈ C^{A} into a cofibration i followed by a trivial fibration p. By Lemma 4.1.16 and the fact that A is a cofibrant operad we see that there is a trivial fibration

A{X ⟶^{i}Z ⟶^{p}Y } ⟶ A{X ⟶^{f} Y }.

In particular, this map is a surjective map of simplicial sets! Thus there exists an A-algebra structure on Z making X ⟶^{i} Z ⟶^{p} Y into a diagram in C^{A}. Since p is therefore a trivial fibration of A-algebras there is a lift in the square

[figure: X; i; f; Z; p; Y; Y ,]

showing that Y is a retract of Z in C^{A}, hence a retract in C. Thus Y is cofibrant in C since Z is.

4.2 Relation between moduli space and classification spaces

In this section we prove Theorems 1.2.10 and 1.2.15. We let C denote either the category of simplicial sets or the category of simplicial R-modules.

Given a closed model category N, we will let wN denote the subcategory of weak equivalences of N. We also let wN denote the nerve of this category. Note that any functor π: N → N′ which takes weak equivalences to weak equivalences induces a corresponding map π: wN → wN′.

The proof proceeds in several steps. The first step is to replace the categories wC and wC^{A} by more convenient subcategories. Let w′C denote the subcategory of wC having as objects the fibrant and cofibrant objects of C, and having as morphisms all trivial fibrations between such objects. Let w′C^{A} ⊂ wC^{A} denote the pre-image of w′C under π. Thus w′C^{A} is the subcategory of wC^{A} having as objects all fibrant A-algebras X such that πX ∈ C is cofibrant, and having as morphisms all trivial fibrations between such objects.

4.2.1. Lemma. If A is a cofibrant operad, the horizontal maps in the diagram

\xymatrix{w′C^{A} \ar[r] \ar[d] & wC^{A} \ar[d] \\ w′C \ar[r] & wC}

are weak equivalences.

Lemma 4.2.1 is proved below.

Let wC^{Δ[−]⊗A} denote the bisimplicial set of 1.2.8. Since A is a cofibrant operad, then so are Δ[t] ⊗ A for t ≥ 0, and thus by Lemma 4.2.1 the inclusion w′C^{Δ[t]⊗A} → wC^{Δ[t]⊗A} is a weak equivalence.

Furthermore, each map Δ[t] ⊗ A → Δ[t′] ⊗ A induced by a map Δ[t] → Δ[t′] of simplicial sets is a weak equivalence of cofibrant operads, and thus by Corollary 3.6.5 induces an equivalence of homotopy theories in the sense of Quillen, and thus by a result of Dwyer-Kan [3] induces an equivalence of simplicial sets wC^{Δ[t′]⊗A} → wC^{Δ[t]⊗A}. In particular, there are natural equivalences

diag w′C^{Δ[−]⊗A} ⟶^{∼} diag wC^{Δ[−]⊗A} ≃ wC^{A}.

The third step is to get a hold of the homotopy fiber of the map

diag w′C^{Δ[−]⊗A} → w′C,

using the following variation on Quillen’s Theorem B [15].

4.2.2. Lemma. Let M^{•} be a simplicial object in the category of categories, let N be a category, and let π: M^{•} → N be a functor to N considered as a constant simplicial object. Then the diagram of simplicial sets

\xymatrix{diag (π ↓ X) \ar[r] \ar[d] & diag M^{•} \ar[d] \\ (N ↓ X) \ar[r] & N}

will be a homotopy pull-back if each map diag (π ↓ X) → diag (π ↓ X′) induced by f: X → X′ ∈ N is a weak equivalence.

Lemma 4.2.2 is proved below.

Thus it will suffice to show that for each map X → X′ ∈ w′C, the induced map diag (π ↓ X) → diag (π ↓ X′) is an equivalence, and that diag (π ↓ X) is weakly equivalent to A{X}.

Note that the bisimplicial set (π ↓ X) has (s, t)-bisimplices given by

(π ↓ X)_{st} = ∐_{Y_{s}→···→Y_{0}→X∈w′C} A{Y_{s} → ⋯ → Y_{0}}_{t}.

We see from Proposition 4.1.13 that the vertical maps in

\xymatrix{∐_{Y_{s}→···→Y_{0}→X∈w′C} A\{Y^{s} → ⋯ → Y^{0}\} \\ ∐_{Y_{s}→···→Y_{0}→X∈w′C} A\{Y_{s} → ⋯ → Y_{0} → X\}\} \ar[u] \ar[d] \\ _{∐_{Y_{s}→···→Y_{0}→X∈w′C} A\{X\}}}

are weak equivalences, and we see that the bottom space is just (w′C ↓ X)×A{X} ≃ A{X}; thus, each fiber (π ↓ X) is weakly equivalent to A{X}.

To see what a map f: X → X′ ∈ w′C induces, we look at another diagram:

[figure: ∐_{Y_{s}→···→X} A{Y^{s} → ⋯ → Y^{0}}; b; ∐_{Y_{s}→···→X′} A{Y^{s} → ⋯ → Y^{0}}; _{∐_{Y_{s}→···→X} A{Y_{s} → ⋯ → Y_{0} → X ⟶^{f} X′}}; ∼; ∼; _{∐_{Y_{s}→···→X′} A{Y_{s} → ⋯ → Y_{0} → X′}}; ∼; ∼; _{∐_{Y_{s}→···→X} A{X ⟶^{f} X′}}; a; _{∐_{Y_{s}→···→X′} A{X′}.}]

The maps marked by ∼ are weak equivalences by Proposition 4.1.13, and a is just the map

a: (w′C ↓ X) × A{X ⟶^{f} X′} → (w′C ↓ X′) × A{X′},

which is a weak equivalence by Proposition 4.1.11, and because (w′C ↓ X) and (w′C ↓ X′) are contractible. Thus b is a weak equivalence as desired. The proof is complete.

4.2.3 Proofs of technical lemmas

We still need to prove several lemmas. In the following, let N be a closed model category. Let N_{f} denote the full subcategory of N which contains all fibrant objects. Let N_{u} denote a full subcategory of N such that

1. N_{u} contains all cofibrant objects of N,

2. N_{u} is stable under cofibrations. That is, if X ∈ N_{u} and i: X → Y ∈ N is a cofibration, then Y ∈ N_{u}.

Let N_{uf} = N_{u} ∩ N_{f}.

If N′ is a full subcategory of a model category N, let wN′ ⊆ N′ denote the subcategory consisting of all the objects and all weak equivalences between them. Let fwN′ ⊆ N′ denote the subcategory consisting of all the objects and all trivial fibrations between them.

4.2.4. Lemma. The map wN_{uf} → wN is a weak equivalence.

4.2.5. Lemma. The map π: fwN_{uf} → wN_{uf} is a weak equivalence.

Proof of Lemma 4.2.1. To show that w′C → wC is a weak equivalence, let N = C and let N_{u} denote the subcategory of cofibrant objects. Then w′C = fwN_{uf} and Lemmas 4.2.4 and 4.2.5 apply.

To show that w′C^{A} → wC^{A} is a weak equivalence, let N = C^{A} and let N_{u} denote the subcategory of C-cofibrant objects. Then since A is assumed to be a cofibrant operad, Proposition 4.1.15 shows that N_{u} satisfies the desired properties. Thus w′C^{A} = fwN_{uf} and Lemmas 4.2.4 and 4.2.5 apply.

Proof of Lemma 4.2.4. We first prove that the inclusion i: wN_{f} → wN is a weak equivalence. Using the functorial factorization of Proposition 3.1.5 we can define a functor j: wN → wN_{f} and natural transformations 1 → ij and 1 → ji, which show that i is a simplicial homotopy equivalence.

To show that the inclusion i′: wN_{uf} → wN_{f} is a weak equivalence, we again use the functorial factorization to define a functor j′: wN_{f} → wN_{uf} and natural transformations i′j′ → 1 and j′i′ → 1, which show that i′ is a simplicial homotopy equivalence. This part of the proof makes use of the fact that N_{u} contains all cofibrant objects.

Proof of Lemma 4.2.5. By Quillen’s Theorem A it suffices to show that for each X ∈ N_{uf} the nerve of (X ↓ π) is contractible.

Each category (X ↓ N) is a closed model category, with fibrant initial object X. It is easy to see that (X ↓ π) is equivalent to the full subcategory of fw(X ↓ N)_{uf} consisting of objects which are weakly equivalent to X. Here (X ↓ N)_{u} denotes the full subcategory of objects X → Y such that Y ∈ N_{u}. Thus (X ↓ N)_{u} is stable under cofibrations. The result now follows from the following lemma, taking (X ↓ N) for N and (X ↓ N)_{u} for N_{u}.

4.2.6. Lemma. Let N be a closed model category with fibrant initial object X, and with a subcategory N_{u} stable under cofibrations which contains X. Then the full subcategory N_{0} of fwN_{uf} containing the initial object has contractible nerve.

Proof. It suffices to show that any functor F: D → N_{0} from the subdivision category of a simplicial set induces a null homotopic map on nerves. The subdivision D of a simplicial set K is the category whose objects are the non-degenerate simplices k ∈ K_{n} for some n, and whose morphisms k → l correspond to pairs of non-degenerate simplices such that l is a face of k. Note that D is actually a partial order on the set of non-degenerate simplices of K, with the 0-simplices of K as the maximal elements

The category N^{D} of D-diagrams in N admits a model category structure with the following properties.

1. A map of diagrams F → G is a weak equivalence if and only if each object d ∈ D is taken to a weak equivalence f(d) → g(d) in N.

2. A map of diagrams F → G is a cofibration if and only if each object d ∈ D is taken to a cofibration f(d) → g(d) in N.

Furthermore, I claim that if i: D′ → D is the inclusion of a subcategory of D with the property that every non-empty subset of the set of objects of D′ has a unique maximal element, then the restriction of a fibration F → G of D-diagrams to D′ is a fibration of D′-diagrams.

Let X̄: D → N denote the initial diagram, defined by X̄(d) = X. Given a diagram F: D → N_{0} ⊆ N, let F × X̄: D → N be the functor defined on objects by

(F × X̄)(d) = F (d) × X.

This functor actually lands in fN_{f}, since X is fibrant. Factor the unique map X̄ → F × X̄ into maps of diagrams

X̄ ⟶^{i}G ⟶^{p=p_{1}×p_{2}} F × X̄

where i is a trivial cofibration and p is a fibration of D-diagrams. We claim that the diagram G and the maps of diagrams X̄ ⟵^{p_{2}} G ⟶^{p_{1}} F land in N_{0}, and thus define a functor

D × (• ← • → •) → N_{0}

which homotops F to the constant functor.

First, we show that for each object d in D, the maps p_{1}(d): G(d) → F (d) and p_{2}(d): G(d) → X are trivial fibrations. Since F (d) and X are fibrant, it is enough to show that p(d): G(d) → F (d) × X is a fibration. Let D′ be the subcategory of D consisting of the single object d and the identity map. Then the restriction of p to D′ is a fibration of diagrams, hence p(d) is a fibration as desired.

Next, we show that for each map f: d_{0} → d_{1} in D, the map G(f): G(d_{0}) → G(d_{1}) is a trivial fibration. Since G(f) is by construction a weak equivalence, it is enough to show that it is a fibration. Let D′ = (d_{0} ⟶^{f} d_{1}) be a subcategory of D. Then the restriction of p: G → F′ to D′ is a fibration of D′ diagrams, hence the map

G(d_{0}) → G(d_{1}) ×_{F (d_{1})×X} (F (d_{0}) × X)

is a fibration. Since by hypothesis the map F (f) is a fibration and X is fibrant, this implies that G(f) is a fibration as desired.

Finally, we note that G is fibrant since there is a fibration p to a fibrant object, and that G lies in the subcategory N_{u}, since there is a cofibration i from an object in N_{u}.

Proof of Lemma 4.2.2. The key element of Quillen’s proof of Theorem B [15]is the fact that for any functor X: D → 𝒮 such that for each d: D → D′ ∈ D the induced map Xd is a weak equivalence, the square

\xymatrix{XD \ar[r] \ar[d] & Y \ar[d] \\ \{D\} \ar[r] & D}

is a homotopy pull-back, where Y denotes the diagonal of the simplicial object in 𝒮 defined by

t ↦ ∐_{D_{0}→...→D_{t}∈D} XD_{0}.

Using this one may easily show that, given a functor X_{•}: D → 𝒮^{Δ^{op}} to bisimplicial sets, the corresponding square

\xymatrix{diag X_{•}D \ar[r] \ar[d] & diag Y_{•} \ar[d] \\ \{D\} \ar[r] & D}

is also a weak equivalence. The lemma now follows by an argument similar to Quillen’s [15, §1].

Cohomology of operads

5.1 Abelian group objects in simplicial R-module operads

In this section we describe abelian group objects in the category of biobjects over an operad, and show the existence of an abelianization functor.

5.1.1 Abelian group objects and pointed objects

Let D be a complete category, and let X be an object in D. An abelian group object in D over X is a map π: K → X ∈ D together with addition, unit, and inverse maps +: K ×_{X} K → K, (−1): K → K, 0: X → K in (D ↓ X), which satisfy the usual abelian group axioms. We let ab(D ↓ X) denote the category of abelian group objects in D over X.

Again, suppose X is an object in some category D. A pointed object in D over X is a map π: K → X together with a map s: X → K such that πs = 1_{X}. We let pt(D ↓ X) denote the category of pointed objects over X. There is an obvious forgetful functor ab(D ↓ X) → pt(D ↓ X).

Let X ∈ Σℳ_{R} be a Σ-object on simplicial R-modules. An abelian group object in Σℳ_{R} over X is a Σ-object K equipped with a direct sum splitting K = X ⊕ K̄; the X-summand maps in via the 0-section, and the K̄ summand is the kernel of the projection π: K → X. Conversely, every Σ-object K over X equipped with such a direct sum splitting in an abelian group object in a unique way. Thus the functor

ab(Σℳ_{R} ↓ X) → Σℳ_{R}

sending K ↦ K̄ is an equivalence of categories.

5.1.2 Abelian group objects in categories of biobjects

Let A and B be operads over simplicial R-modules. Let M be an A, B-biobject. A map π: K → M ∈ (A, B)-biobjℳ_{R} equipped with a section 0_{K}: M → K ∈ (A, B)-biobjℳ_{R} can be an abelian group object over M in at most one way. Thus the forgetful functor

ab((A, B)-biobjℳ_{R} ↓ M) → pt((A, B)-biobjℳ_{R} ↓ M)

is inclusion of a full subcategory. We give a characterization of the objects in this full subcategory.

An A, B-biobject is equivalent to a Σ-object K equipped with maps

λ: A[n] ⊗ K[i_{1}] ⊗ ⋯ ⊗ K[i_{n}] → K[m]

and

ρ: K[n] ⊗ B[i_{1}] ⊗ ⋯ ⊗ B[i_{n}] → K[m]

for n ≥ 0, i_{j} ≥ 0, m = ∑ i_{j}, which satisfy certain unit, equivariance, and associativity relations. An ideal I ⊆ K is a collection of submodules {I[n] ⊆ K[n]} such that

1. I[n] is preserved by the action of Σ_{n} on K[n],

2. if a ∈ A[n] and k_{1} ∈ K[i_{1}], …, k_{n} ∈ K[i_{n}] such that k_{j} ∈ I[i_{j}] for at least one value of j where 1 ≤ j ≤ n, then λ(a ⊗ k_{1} ⊗ ⋯ ⊗ k_{n}) ∈ I[m],

3. if k ∈ I[n] and b_{1} ∈ B[i_{1}], …, b_{n} ∈ B[i_{n}], then ρ(k ⊗ b_{1} ⊗ ⋯ ⊗ b_{n}) ∈ I[m].

It is not hard to see that if I ⊆ K is an ideal, then the Σ-object K/I defined by

(K/I)[n] = K[n]/I[n]

has a canonical A, B-biobject structure making the quotient map K → K/I into a map of A, B-biobjects. Conversely, the kernel of a map of A, B-biobjects is an ideal.

Suppose π: K → M is a map of A, B-biobjects; let I be the kernel of this map. Let J_{K} ⊆ K denote the set of elements of K which are of the form λ(a⊗k_{1} ⊗⋯⊗k_{n}), where k_{j} ∈ I[i_{j}] for at least two distinct indices j in the set {1, …, n}. Let (J_{K}) ⊆ K denote the smallest ideal containing J_{K}.

5.1.3. Proposition. An object K ∈ pt((A, B)-biobjℳ_{R} ↓ M) is an abelian group object if and only if J_{K} = 0.

5.1.4. Corollary. There is an abelianization functor

A: ((A, B)-biobjℳ_{R} ↓ M) → ab((A, B)-biobjℳ_{R} ↓ M)

which is left adjoint to the forgetful functor.

Proof. We define A as follows. Given K → M ∈ (A, B)-biobjℳ_{R}, let M ∗ K ∈ (A, B)-biobjℳ_{R} denote the coproduct in the category of A, B-biobjects of M and K. Clearly M ∗ K is equipped with a projection map M ∗ K → M and a section M → M ∗ K. Define

A(K) = M ∗ K/(J_{M∗K}).

5.1.5. Proposition. Limits and colimits exist in ab((A, B)-biobjℳ_{R} ↓ M), and are computed in pt(Σℳ_{R} ↓ M).

Proof. The statement about limits is clear. For formal reasons, in any category of abelian group objects finite coproducts are isomorphic to finite products; thus finite coproducts exist and are computed in the underlying category (Σℳ_{R} ↓ M).

Filtered colimits and reflexive coequalizers may also be computed in the underlying category: we already know that filtered colimits and reflexive coequalizers in (A, B)-biobjℳ_{R} are computed in Σ-objects, and since these colimits commute with finite limits we see that the filtered colimit (resp. reflexive coequalizer) in (A, B)-biobjℳ_{R} of a diagram of abelian group objects is itself an abelian group object.

The proposition follows, since any colimit can be built out of finite coproducts, filtered colimits, and reflexive coequalizers.

5.1.6. Corollary. The forgetful functor

ab((A, B)-biobjℳ_{R} ↓ M) → ab(Σℳ_{R} ↓ M)

is exact.

5.1.7 Abelian group objects in the category of operads

In Section 5.2 we will show that for an operad A over simplicial R-modules, the category ab(operℳ_{R} ↓ A) is naturally equivalent to the category ab(A-biobjℳ_{R} ↓ A).

5.1.8. Proposition. There is an abelianization functor

D: (operℳ_{R} ↓ A) → ab(operℳ_{R} ↓ A)

which is left adjoint to the forgetful functor.

5.2 The fundamental exact sequence

Let M denote a complete right-closed monoidal category.

5.2.1 An equivalence of categories of abelian group objects

Let K be an abelian group object in (monM ↓ A). Altogether, K has the following structure maps:

η_{A} : I ⟶ A,
μ_{A} : A ∘ A ⟶ A,

which make A into a monoid,

η_{K} : I ⟶ K,
μ_{K} : K ∘ K ⟶ K,

which make K into a monoid,

π : K ⟶ A,

which is a map of monoids, and

0 : A ⟶ K,
(−1) : K ⟶ K,
+ : K ×_{A} K ⟶ K,

which are maps of monoids which make K into an abelian group object as a monoid over A.

Recall that if K is an abelian group object over A, the functor (M ↓ A)[−, K] lands in abelian groups. In particular, given an object X ∈ (M ↓ A) and maps f, g ∈ (M ↓ A)[X, K], we can add them, obtaining f + g : X → K; the map 0π_{X} is the zero element of this abelian group, where π_{X} : X → A. Furthermore, given maps f_{1}, f_{2} : X → K and g_{1}, g_{2} : Y → K in (M ↓ A) we have an equality

μ_{K}(f_{1} ∘ g_{1}) + μ_{K}(f_{2} ∘ g_{2}) = μ_{K}((f_{1} + f_{2}) ∘ (g_{1} + g_{2})),

where both sides are elements in (M ↓ A)[X ∘ Y, K]; this equality follows from the fact that + is a map of monoids over A.

Let ℓ : A ∘ K → K and r : K ∘ A → K be defined by ℓ = μ_{K}(0 ∘ 1) and r = μ_{K}(1 ∘ 0) respectively. The maps ℓ and r make K into a A-biobject over A, and since + commutes with ℓ and r we see that K is an abelian group object in (_{A}M_{A} ↓ A).

5.2.2. Lemma. Given K in ab(monM ↓ A), the identities

μ_{K} = ℓ(π ∘ 1) + r(1 ∘ π) (5.2.3)

and

η_{K} = 0η_{A} (5.2.4)

hold.

Proof. To show 5.2.3, note that we have that

ℓ(π ∘ 1) = μ_{K}(0 ∘ 1)(π ∘ 1) = μ_{K}(0π ∘ 1)

and

r(1 ∘ π) = μ_{K}(1 ∘ 0)(1 ∘ π) = μ_{K}(1 ∘ 0π),

whence

ℓ(π ∘ 1) + r(1 ∘ π) = μ_{K}(0π ∘ 1) + μ_{K}(1 ∘ 0π)
= μ_{K}((0π + 1) ∘ (1 + 0π))
= μ_{K}(1 ∘ 1)
= μ_{K}.

Equation 5.2.4 from the fact that 0 is a map of monoids over A.

5.2.5. Theorem. The functor

ab(monM ↓ A) ⟶ ab(_{A}M_{A} ↓ A)

which forgets structure is an equivalence of categories.

Proof. It is easy to see that given K in ab(_{A}M_{A} ↓ A) we can give K the structure of a monoid using μ_{K} = ℓ(π ∘ 1) + r(1 ∘ π) and η_{K} = 0η_{A}; this produces a functor ab(_{A}M_{A} ↓ A) → ab(monM ↓ A). Now by Lemma 5.2.2 we see that we get an equivalence of categories.

5.2.6 Derivations

Given a monoid A in M and an abelian group object K ∈ ab(monM ↓ A), a derivation from A to K is a map f ∈ (M ↓ A)[A, K] such that

fμ_{A} = ℓ(1 ∘ f) + r(f ∘ 1).

We write Der(A, K) for the set of derivations from A to K.

5.2.7. Lemma. Let K be an object in ab(monM ↓ A). Then

Der(A, K) ≃ (monM ↓ A)[A, K].

Proof. By Lemma 5.2.2 we see that

μ_{K}(f ∘ f) = (ℓ(π ∘ 1) + r(1 ∘ π))(f ∘ f)
= ℓ(πf ∘ f) + r(f ∘ πf)
= ℓ(1 ∘ f) + r(f ∘ 1).

It follows that f is a derivation if and only if fμ_{A} = μ_{K}(f ∘ f). In particular, a map of monoids is a derivation.

Conversely, if f is a derivation, we have

fη_{A} = fμ_{A}(η_{A} ∘ η_{A})
= (ℓ(1 ∘ f) + r(f ∘ 1))(η_{A} ∘ η_{A})
= ℓ(η_{A} ∘ fη_{A}) + r(fη_{A} ∘ η_{A})
= fη_{A} + fη_{A},

whence fη_{A} = 0η_{A} = η_{K}, and thus f is a map of monoids.

5.2.8 Fundamental exact sequence

We suppose there are adjoint pairs

A : (M ↓ A) ⇄ ab(M ↓ A) : U,
A_{ℓ} : (_{A}M ↓ A) ⇄ ab(_{A}M ↓ A) : U_{ℓ},
A_{r} : (M_{A} ↓ A) ⇄ ab(M_{A} ↓ A) : U_{r},

and

A_{b} : (_{A}M_{A} ↓ A) ⇄ ab(_{A}M_{A} ↓ A) : U_{b},

as well as

D : (monM ↓ A) ⇄ ab(monM ↓ A) : U_{mon},

where U_{.} denote the obvious forgetful functors.

Let us also assume that the forgetful functors

ab(_{A}M_{A} ↓ A) → ab(_{A}M ↓ A) → ab(M ↓ A)

and

ab(_{A}M_{A} ↓ A) → ab(M_{A} ↓ A) → ab(M ↓ A)

are exact.

The above conditions are satisfied in the case when M is Σℳ_{R}, as was shown in Section 5.1.

5.2.9. Proposition. In ab(_{A}M_{A} ↓ A) there is an exact sequence

A_{b}(A ∘ A ∘ A ∘ A) ⟶^{d}A_{b}(A ∘ A ∘ A) ⟶ D(A) ⟶ 0.

Proof. Define d by d = A_{b}(d_{0}) − A_{b}(d_{1}) + A_{b}(d_{2}), where d_{0} = μ ∘ 1 ∘ 1, d_{1} = 1 ∘ μ ∘ 1 and d_{2} = 1 ∘ 1 ∘ μ. Let K be an abelian group object in (_{A}M_{A} ↓ A). If we let d^{∗} = ab(_{A}M_{A} ↓ A)[d, K], then by standard adjunctions we get an exact sequence

(M ↓ A)[A ∘ A, K] ⟵^{d^{∗}} (M ↓ A)[A, K] ⟵ ab(_{A}M_{A} ↓ A)[Cok d, K] ⟵ 0,

where d^{∗}(f) = ℓ(1 ∘ f) − fμ_{A} + r(f ∘ 1). Now using Lemma 5.2.7 we see that Cok d has the same universal property as D(A).

Recall from Section 2.1 that since M is right-closed, so is (M ↓ A), where the function object functor is denoted F_{A}(−, −).

5.2.10. Proposition. The functor F_{A}(−, −) : (M ↓ A)^{op} × (M ↓ A) → (M ↓ A) underlies a functor

F_{A}(−, −) : (_{A}M ↓ A)^{op} × (_{A}M ↓ A) ⟶ (_{A}M_{A} ↓ A)

which has the property that for X in (_{A}M_{A} ↓ A) and Y in (_{A}M ↓ A), then

(_{A}M ↓ A)[X, Y] ≃ (_{A}M_{A} ↓ A)[X, F_{A}(A, Y)].

Proof. This is immediate from Proposition 2.1.23.

5.2.11. Proposition. Taking F_{A}(−, −) as in 5.2.10, if X and Y are abelian group objects in (_{A}M_{A} ↓ A) and (_{A}M ↓ A) respectively, then F_{A}(A, Y) is naturally an abelian group object, and we have a bijection

ab(_{A}M ↓ A)[X, Y] ≃ ab(_{A}M_{A} ↓ A)[X, F_{A}(A, Y)].

Proof. This follows from Proposition 5.2.10 and the fact that

F_{A}(A, Y) ×_{A} F_{A}(A, Y) ≃ F_{A}(A, Y ×_{A} Y).

5.2.12. Proposition. Given X in ab(_{A}M_{A} ↓ A), the natural map f : A_{ℓ}(X) → A_{b}(X) is an isomorphism.

Proof. Using Propositions 5.2.10 and 5.2.11 we see that, given an abelian group object Y in (_{A}M ↓ A), there are natural bijections

ab(_{A}M ↓ A)[A_{b}(X), Y] ≃ ab(_{A}M_{A} ↓ A)[A_{b}(X), F_{A}(A, Y)]
≃ (_{A}M_{A} ↓ A)[X, F_{A}(A, Y)]
≃ (_{A}M ↓ A)[X, Y].

Thus A_{b} has the universal property which characterizes A_{ℓ}.

Let X_{•} be the simplicial object in ab(_{A}M_{A} ↓ A) defined by

X = A_{b}B(A, A, A).

5.2.13. Proposition. The augmentation map A_{b}B(A, A, A) → A_{b}(A) has a contracting homotopy in the underlying category ab(_{A}M ↓ A), and hence the chain complex

… ⟶ A_{b}(A ∘ A ∘ A ∘ A) → A_{b}(A ∘ A ∘ A) → A_{b}(A ∘ A) → A_{b}(A) → 0

in ab(_{A}M_{A} ↓ A) is acyclic.

Proof. Recalling Proposition 5.2.12, define h : X_{n} → X_{n+1} by h = A_{ℓ}(1 ∘ ⋯ ∘ 1 ∘ η), where η : I → A is the unit of the monoid A. Then it is easy to check that h is the desired contracting homotopy.

5.2.14. Theorem. There is a short exact sequence

0 → D(A) ⟶ A_{b}(A ∘ A) ⟶ A_{b}(A) → 0.

Proof. This is an immediate consequence of Propositions 5.2.13 and 5.2.9.

5.2.15. Example. Let us illustrate Theorem 5.2.14 in the “classical” case of associative algebras. Thus, let M denote the category of R-modules, which is monoidal under tensor product over R. A monoid A in M is thus an associative R-algebra. The category _{A}M_{A} is then just the category of A-bimodules, the category of abelian group objects ab(_{A}M_{A} ↓ A) is equivalent to _{A}M_{A}, and the abelianization functor A_{b}: (_{A}M_{A} ↓ A) → _{A}M_{A} takes a bimodule M → A over A to M. Then Theorem 5.2.14 reduces to a short exact sequence

0 → D(A) → A ⊗ A → A → 0.

5.3 Model category structures for abelian group objects

Let A be an operad on simplicial R-modules.

5.3.1. Proposition. There is a simplicial model category structure on the category ab(operℳ_{R} ↓ A) (which by Proposition 5.2.5 is equivalent to ab(operℳ_{R} ↓ A)) with the following structure.

1. A map f: K → L is a weak equivalence (resp. a fibration) if for every subgroup H ⊆ Σ_{n} the induced map on invariants f[m]^{H}: K[m]^{H} → L[m]^{H} is a weak equivalence (resp. a fibration) of the underlying simplicial sets.

2. The cofibrations (resp. trivial cofibrations) are generated by the abelianizations of the generators for (operℳ_{R} ↓ A), which are the same as the abelianizations of the generators for (A-biobjℳ_{R} ↓ A).

Proof. Every object in ab(operℳ_{R} ↓ A) is a map K → A of Σ-objects; because of the existence of a zero-section A → K, each map K[m]^{H} → A[m]^{H} of fixed point sets is surjective, and hence K → A is a fibration of Σ-objects. Thus we may apply Proposition 3.1.5 to the adjunction

D : (operℳ_{R} ↓ A) ⇄ ab(operℳ_{R} ↓ A) : U

where we take E to be the identity functor on ab(operℳ_{R} ↓ A).

The fact that the abelianizations of the generators of (operℳ_{R} ↓ A) are the same as the abelianizations of the generators of (A-biobjℳ_{R} ↓ A) follows from a simple adjunction argument. If K ∈ ab(A-biobjℳ_{R} ↓ A) = ab(operℳ_{R} ↓ A), then

ab(A-biobjℳ_{R} ↓ A)[A_{b}(A ∘ X ∘ A), K] ≃ (A-biobjℳ_{R} ↓ A)[A ∘ X ∘ A, K]
≃ (Σℳ_{R} ↓ A)[X, K]
≃ (operℳ_{R} ↓ A)[F X, K]
≃ ab(operℳ_{R} ↓ A)[D(F X), K].

5.3.2. Proposition. A map i: K → L in ab(operℳ_{R} ↓ A) is a cofibration if and only if i is injective and the cokernel of i is cofibrant.

5.3.3. Proposition. In the adjoint pairs of functors

A_{b} : (A-biobjℳ_{R} ↓ M) ⇄ ab(A-biobjℳ_{R} ↓ M) : U_{b}

and

D : (operℳ_{R} ↓ A) ⇄ ab(operℳ_{R} ↓ A) : U

the right adjoints preserve fibrations and weak equivalences and the left adjoints preserve cofibrations and trivial cofibrations.

Proof. This is clear, because the right adjoints preserve fibrations and weak equivalences.

5.4 Equivalence of Quillen and Hochschild cohomologies

In this section we prove Theorem 1.3.8 which states that Quillen cohomology and Hochschild cohomology of an operad are essentially the same up to a dimension shift.

5.4.1 Computing Quillen cohomology of operads

Let A ∈ operℳ_{R} be an operad, and let U be a coefficient system over A. We compute the Quillen cohomology group H_{oper}^{t}(A, U) by choosing a weak equivalence p: B → A ∈ operℳ_{R} from a cofibrant operad B. Since all abelian group objects are fibrant, we have that

H_{oper}^{t}(A, U) = Ho (operℳ_{R} ↓ A)(B, K(U, t))
≃ π_{0} map_{(operℳ_{R} ↓A)}(B, K(U, t)).

5.4.2 Quillen cohomology of biobjects

Let A ∈ operℳ_{R} be an operad, let M be an A-biobject, and let U be a coefficient system over M. We compute the Quillen cohomology group H_{A-biobj}^{t}(M, U) by choosing a weak equivalence p: N → M ∈ operℳ_{R} from a cofibrant A-biobject N. Since all abelian group objects are fibrant, we have that

H_{A-biobj}^{t}(M, U) = Ho (A-biobjℳ_{R} ↓ M)(N, K(U, t))
≃ π_{0} map_{(A-biobjℳ_{R} ↓M)}(N, K(U, t)).

5.4.3. Proposition. If A is a Σ-cofibrant operad, then diag B(A, A, A) → A is a weak equivalence of A-biobjects in which the domain is a cofibrant biobject. More generally, if f: A → B is a weak equivalence of Σ-cofibrant operads, then

g: diag B(B, A, B) → B

is a weak equivalence of B-biobjects in which the domain is a cofibrant biobject.

Proof. The first part is just Corollary 3.7.6. The second part follows from Proposition 3.7.5 and from factoring g into diag B(B, A, B) → diag B(B, B, B) → B; the first map is an equivalence by Proposition 3.4.5 and the fact that A and B are Σ-cofibrant, and the second map is an equivalence by Corollary 3.7.6.

5.4.4. Corollary. If f: A → B is a weak equivalence of Σ-cofibrant operads, then there is a natural isomorphism Hoch^{∗}(A, f^{∗}U) ≃ Hoch^{∗}(B, U).

5.4.5. Proposition. The Hochschild cohomology of the trivial operad I is given by

Hoch^{n}(I, U) = \begin{cases}U[1] & if n = 0, \\ 0 & if n > 0. \end{cases}

5.4.6 Proof of Theorem 1.3.8

Choose a cofibrant operad B and weak equivalence p: B → A. Since cofibrant operads are Σ-cofibrant by Proposition 3.5.3, it follows from Corollary 5.4.4 that Hoch^{∗}(B, p^{∗}U) ≃ Hoch^{∗}(A, U). Thus without loss of generality we may assume that A is a cofibrant operad.

The result now follows from Theorem 5.2.14 and the following Lemma.

5.4.7. Lemma. If A is a cofibrant operad, then there is a diagram in the category ab((A-biobjℳ_{R} ↓ A)) of the form

[figure: 0; D(A); i; P; f; A_{b}(diag B(A, A, A)); g; 0; 0; D(A); A_{b}(A ∘ A); A_{b}(A); 0,]

where P = A_{b}(diag B(A, A, A)) ×_{A_{b}(A)} A_{b}(A ∘ A) is the pullback of the right-hand square, the rows are exact, the objects in top row are cofibrant, i is a cofibration, and f and g are weak equivalences.

Proof. The object diag B(A, A, A) is a cofibrant A-biobject by Proposition 3.7.6, since A is Σ-cofibrant by Proposition 3.5.3. Thus A_{b}(diag B(A, A, A)) is cofibrant by Proposition 5.3.3. Hence it follows by Proposition 5.3.2 that i is a cofibration. Likewise D(A) is cofibrant by Proposition 5.3.3, and hence P is cofibrant.

By Proposition 5.2.13 the map g has a contracting homotopy in the underlying category ab(Σℳ_{R} ↓ A), and hence the pullback f of this map has one also. Thus f and g are weak equivalences as desired.

Appendix A

Free monoids

In this appendix we describe a construction of the free operad. We present this construction as a special case of the construction of a free monoid in a monoidal category M which satisfies the left distributivity law

(A ⨿ B) ∘ C ≃ (A ∘ C) ⨿ (B ∘ C),

and such that the monoidal product ∘ commutes with countable directed colimits in each variable.

Our construction is a generalization of the Barr-Wells construction of a free triple [1], which corresponds to the case of a free monoid in the monoidal category M = C^{C} of endofunctors on C. Our proof is in fact an improvement on theirs; they construct the free triple on a functor A as at best a retract of the object we call F A.

There is another construction of the free operad as a “sum over trees”, which is described in [8], [7].

For the sake of readability, in what follows we will write “+” for the coproduct in the category M, and “∘” will take precedence over “+”; e.g., A ∘ B + C ∘ D should be read as (A ∘ B) + (C ∘ D).

Construction of the Free Monoid. We first construct functors F_{n}: M → M inductively, by

F_{0}A = I,
F_{1}A = I + A,

and in general

F_{n+1}A = I + A ∘ F_{n}A.

We let η_{n}: I → F_{n}A and e_{n}: A ∘ F_{n−1}A → F_{n}A denote the inclusions of these two summands.

The functors F_{n} come with maps i_{n}: F_{n}A → F_{n+1}A defined inductively by

i_{0} : F_{0}A = I ⟶ I + A = F_{1}A,

and

i_{n} = I + A ∘ i_{n−1} : I + A ∘ F_{n−1}A ⟶ I + A ∘ F_{n}A.

Then we can define a functor F: M → M as the colimit

F A = colim_{n}(F_{0}A ⟶^{i_{0}} F_{1}A ⟶^{i_{1}} F_{2}A ⟶^{i_{2}} ⋯).

A.0.1. Remark. Note that if the monoidal product in M distributes over coproducts on both sides, then this construction simplifies to

F A = I + A + A^{∘2} + A^{∘3} + · · · .

A.0.2. Proposition. Let M be a cocomplete monoidal category satisfying the left distributivity law, with the property that the functor − ∘ −: M × M → M commutes with countable directed colimits in each variable. Then the above functor F: M → M is the free monoid triple. Furthermore, the free monoid F A on A satisfies a “recursive formula”

F A ≃ I + A ∘ F A.

Proof. Let A ∈ M. We have already constructed F A. We shall show below that F A has the structure of a monoid, and is in fact the free monoid on A.

It is easy to show that the diagrams

[figure: I; η_{n}; I; η_{n+1}; A ∘ F_{n−1}A; A∘i_{n−1}; e_{n−1}; A ∘ F_{n}A; e_{n}; F_{n}A; i_{n}; F_{n+1}A; F_{n}A; i_{n}; F_{n+1}A]

commute. Thus the η’s and e’s induce maps η: I → F A and e: A ∘ F A → F A, and there is an isomorphism

η + e: I + A ∘ F A ⟶^{∼} F A;

this is the “recursion formula”.

To define a product map, we inductively define maps

μ_{mn}: F_{m}A ∘ F_{n}A → F_{m+n}A

by setting

μ_{0n} = id: F_{n}A → F_{n}A

and

μ_{m+1,n}: F_{m+1}A ∘ F_{n}A = F_{n}A + A ∘ F_{m}A ∘ F_{n}A ⟶^{F_{n}A+A∘μ_{mn}}

F_{n}A + A ∘ F_{m+n}A ⟶^{(i,e)} F_{m+n+1}A.

It is straightforward to show that the diagrams

[figure: F_{m}A ∘ F_{n}A; i_{m}∘F_{n}A; μ_{mn}; F_{m+1}A ∘ F_{n}A; μ_{m+1,n}; F_{m}A ∘ F_{n}A; F_{m}A∘i_{n}; μ_{mn}; F_{m} ∘ F_{n+1}A; μ_{m,n+1}; F_{m+n}A; i_{m+n}; F_{m+n+1}A; F_{m+n}A; i_{m+n}; F_{m+n+1}A]

commute. Since F A ∘ F A ≃ colim_{m,n} F_{m}A ∘ F_{n}A the μ’s fit together to give a map μ: F A ∘ F A → F A.

We claim that (F A, η, μ) is a monoid. The unit axioms follow from the commutative diagrams above. To prove the associativity axiom, it suffices to show that for all m, n, p ≥ 0 the two ways of going around the square

[figure: F_{m}A ∘ F_{n}A ∘ F_{p}A; μ_{mn}∘F_{p}A; F_{m}A∘μ_{np}; F_{m+n}A ∘ F_{p}A; μ_{m+n,p}; F_{m}A ∘ F_{n+p}A; μ_{m,n+p}; F_{m+n+p}A]

are the same. The proof is by induction on m. The case when m = 0 is straightforward. For general m we note that the above square is the coproduct of the two squares:

[figure: I ∘ F_{n}A ∘ F_{p}A; ∼; 1∘μ_{n,p}; F_{n}A ∘ F_{p}A; μ_{n,p}; I ∘ F_{n+p}A; ∼; F_{n+p},]

and

[figure: A ∘ F_{m−1}A ∘ F_{n}A ∘ F_{p}A; A∘μ_{m−1,n}∘F_{p}A; A∘F_{m−1}A∘μ_{n,p}; A ∘ F_{m+n−1}A ∘ F_{p}A; A∘μ_{m+n−1,p}; A ∘ F_{m−1}A ∘ F_{n+p}A; A∘μ_{m−1,n+p}; A ∘ F_{m+n+p−1}A.]

The commutativity of each of these squares follows from the inductive hypothesis.

Finally, we show that F A is the free monoid on A. Let α: A → F A be the map defined by

α: A ⟶^{e_{1}} I + A = F_{1}A ⟶^{i}F A.

To show that F A is a free monoid on A it suffices to show that for any monoid B the map

ϕ: monM[F A, B] → M[A, B]

which sends g ↦ gα is a bijection.

We show that ϕ is an injection. Suppose g: F A → B is a map of monoids, and let f = gα. Let g_{n} denote the composite F_{n}A → F A ⟶^{g}B; we will show by induction on n that the maps g_{n} are determined by the value of f, and hence so is g. Clearly the map g_{1} = (η, f): F_{1}A = I + A → B is determined by f. Now suppose g_{n} is given.

Then from the commutativity of the diagram

[figure: F_{1}A ∘ F_{n}A; g_{1}∘g_{n}; μ_{1n}; B ∘ B; μ; I + A ∘ F_{n}A; (η_{1}∘η_{n},e_{1}∘F_{n}A); ∼; F_{n+1}A; g_{n+1}; B]

it follows that g_{n+1} is determined.

We show that ϕ is a surjection. Thus, given f: A → B, we want to construct a map of monoids g: F A → B such that gα = f. We build maps g_{n}: F_{n}A → B inductively by setting

g_{1} : F_{1}A = I + A ⟶^{(η,f)} B

and

g_{n+1} : F_{n+1}A = I + A ∘ F_{n}A ⟶^{I+f∘g_{n}} I + B ∘ B ⟶^{(η,μ)} B.

By induction on n one shows that g_{n+1}i_{n} = g_{n}. Let g: F A → B be the map induced by the g_{n}’s. To show that g is indeed a map of monoids, we must check that the square

[figure: F_{m}A ∘ F_{n}A; g_{m}∘g_{n}; μ_{mn}; B ∘ B; μ; F_{m+n}A; g_{m+n}; B]

commutes. The proof is by induction on m.

Bibliography

[1] M. Barr and C. Wells. Toposes, Triples and Theories. Springer-Verlag, 1984.

[2] W. G. Dwyer and D. M. Kan. General abstract homotopy theory. in preperation.

[3] W. G. Dwyer and D. M. Kan. Function complexes in homotopical algebra. Topology, 19:427–440, 1980.

[4] W. G. Dwyer and D. M. Kan. A classification theorem for diagrams of simplicial sets. Topology, 23:139–155, 1984.

[5] W. G. Dwyer and D. M. Kan. Singular functors and realization functors. Indag. Math., 46, 1984.

[6] P. Gabriel and M. Zisman. Calculus of Fractions and Homotopy Theory. Springer-Verlag, 1967.

[7] E. Getzler and J. D. S. Jones. Operads, homotopy algebra, and iterated integrals for double loop spaces. Preprint (hep-th/9403054).

[8] V. Ginzburg and M. Kapranov. Koszul duality for operads. Duke Mathematical Journal, 76:203–272, 1994.

[9] G. M. Kelly. Basic Concepts of Enriched Category Theory. London Mathematical Society Lecture Note Series 64. Cambridge University Press, 1982.

[10] S. Mac Lane. Categories for the Working Mathematician. Springer-Verlag, 1971.

[11] L. G. Lewis, J. P. May, and M. Steinberger. Equivariant Stable Homotopy Theory. Lecture Notes in Mathematics 1213. Springer-Verlag, 1986.

[12] J. P. May. Simplicial Objects in Algebraic Topology. University of Chicago Press, 1967.

[13] D. G. Quillen. Homotopical Algebra. Lecture Notes in Mathematics 43. Springer-Verlag, 1967.

[14] D. G. Quillen. Rational homotopy theory. Annals of Math, 90:65–87, 1969.

[15] D. G. Quillen. Higher algebraic K-theory I. In Higher K-theories, Vol. I, Lecture Notes in Mathematics 341, pages 85–147. Springer-Verlag, 1973.

[16] A. Robinson. Obstruction theory and the strict associativity of Morava Ktheories. In Advances in Homotopy Theory, pages 143–152. Cambridge University Press, 1989.