4. Enriched categories and their modules
4.3. Homotopy theory of right modules over enriched categories . 167
For a CI-category O, a (right) O-module is a contravariant C-functor from O to C.
Explicitly anO-moduleM consists of objectsM(i) inC fori∈I and (sinceCis a closed monoidal category and since it is enriched over itself) ofC-morphisms
M(j)⊗O(i, j)−!M(i)
which are associative and unital. Amorphism ofO-modules, Φ:M!M0, is an enriched natural transformation, i.e., a collection of C-morphisms Φ(i):M(i)!M0(i) satisfying the usual naturality requirements. Such a morphism ofO-module is aweak equivalence if each Φ(i) is a weak equivalence in C. We denote by Mod-O the category of right O-modules and natural transformations.
Let Ψ:O!Rbe a morphism of CI-categories. Clearly, Ψ induces a restriction of scalars functor on module categories
Ψ∗:Mod-R −!Mod-O, M7−!MΨ.
As explained in [19, p. 323], the functor Ψ∗ has a left adjoint functor Ψ∗, also denoted
−⊗OR
(one can think of Ψ∗as the left Kan extension). Schwede and Shipley [19, Theorem 6.1]
prove that under some technical hypotheses on C, the categoryMod-O has a Quillen model structure, and moreover, if Ψ is a weak equivalence ofCI-categories, then the pair (Ψ∗,Ψ∗) induces a Quillen equivalence of module categories.
We will need this result in the caseC=Ch. In keeping with our notation, we use ChI-categories to denote categories enriched over chain complexes, with object set I.
Note that the category of modules over a ChI-category admits coproducts (i.e. direct sums).
Theorem 4.1. (Schwede–Shipley, [19])
(1) Let O be a ChI-category. Then Mod-O has a cofibrantly generated Quillen model structure, with fibrations and weak equivalences defined objectwise.
(2) Let Ψ:O!Rbe a weak equivalence of ChI-categories. Then (Ψ∗,Ψ∗)induce a Quillen equivalence of the associated module categories.
Proof. General conditions on C that guarantee the result are given in [19, Theo-rem 6.1]. It is straightforward to check that the conditions are satisfied by the category of chain complexes (the authors of [19] verify them for various categories of spectra, and the verification for chain complexes is strictly easier).
LetOandRbeCI-categories and letM andN be right modules overOandR, re-spectively. A morphism of pairs (O, M)!(R, N) consists of a morphism ofCI-categories Ψ:O!Rand a morphism ofO-modules Φ:M!Ψ∗(N). The corresponding category of pairs (O, M) is called theCI-module category.
A morphism (Ψ,Φ) in aCI-module is called aweak equivalenceif both Ψ and Φ are weak equivalences. Two objects of a CI-module are calledweakly equivalent if they are linked by a chain of weak equivalences, pointing in either direction.
In our study of the formality of the little balls operad, we will consider certain splittings of O-modules into direct sums. The following homotopy invariance property of such a splitting will be important.
Proposition 4.2. Let (O, M) and (O0, M0) be weakly equivalent ChI-modules.
If M is weakly equivalent as an O-module to a direct sum L
nMn, then M0 is weakly equivalent as an O0-module to a direct sum L
nMn0 such that (O, Mn)is weakly equiva-lent to (O0, Mn0)for each n.
Proof. It is enough to prove that for a direct weak equivalence (Ψ,Φ): (O, M)−'!(R, N),
M splits as a direct sum if and only ifN splits in a compatible way.
In one direction, suppose that N'L
nNn as R-modules. It is clear that the re-striction of the scalars functor Ψ∗ preserves direct sums and weak equivalences (quasi-isomorphisms). Therefore Ψ∗(N)'L
nΨ∗(Nn). Since by hypothesisM is weakly equiv-alent to Ψ∗(N), we have the required splitting ofM.
In the other direction suppose that theO-moduleM is weakly equivalent toL
nMn. We can assume that each Mn is cofibrant, hence so is L
nMn. Moreover Ψ∗(N) is fibrant because every O-module is. Therefore, since M is weakly equivalent to Ψ∗(N), there exists a direct weak equivalenceγ:L
nMn
−'!Ψ∗(N). Since (Ψ∗,Ψ∗) is a Quillen equivalence, the weak equivalenceγ induces an adjoint weak equivalence
γ[: Ψ∗
As a left adjoint, Ψ∗ commutes with coproducts, therefore we get the splitting M
n
Ψ∗(Mn)−−'!N.
Moreover, we have a weak equivalenceMn−'!Ψ∗Ψ∗(Mn), because it is the adjoint of the identity map on Ψ∗(Mn),Mn is cofibrant, and (Ψ∗,Ψ∗) is a Quillen equivalence. Thus this splitting ofN is compatible with the given splitting ofM.
4.4. Lax monoidal functors, enriched categories, and their modules
LetC andDbe two symmetric monoidal categories. Alax symmetric monoidal functor F:C!Dis a (non-enriched) functor, together with morphisms
1D−!F(1C) and F(X)⊗F(Y)−!F(X⊗Y),
natural inX, Y∈C, that satisfy the obvious unit, associativity, and symmetry relations.
In this paper, we will sometimes use “monoidal” to mean “lax symmetric monoidal”, as this is the only notion of monoidality that we will consider.
Such a lax symmetric monoidal functor F induces a functor (which we will still denote byF) fromCI-categories toDI-categories. Explicitly ifOis aCI-category then F(O) is theD-category whose set of objects isI and morphisms are
(F(O))(i, j) :=F(O(i, j)).
Moreover,F induces a functor fromMod-OtoMod-F(O). We will denote this functor byF as well.
The main examples that we will consider are those from§2.0.1 and§2.0.2, and their composites:
(1) Homology: H∗: (Ch,⊗,K)!(Ch,⊗,K);
(2) Normalized singular chains: C∗: (Top,×,∗)!(Ch,⊗,K),X7!C∗(X).
The fact that the normalized chains functor is lax monoidal, and equivalent to the un-normalized chains functor, is explained in [19,§2]. As is customary, we often abbreviate the composite H∗C∗ as H∗.
Recall that we also use the functor Hn: (Ch,⊗,K)!(Ch,⊗,K), where Hn(C, d) is seen as a chain complex concentrated in degreen. The functor Hn is not monoidal for n>0. However, H0 is monoidal.
Thus, ifBis a small TopI-category then C∗(B) and H∗(B) are ChI-categories. Also if B:B!Top is a B-module then C∗(B) is a C∗(B)-module and H∗(B) is an H∗ (B)-module. We also have the ChI-category H0(B).
4.5. Discretization of enriched categories
When we want to emphasize that a category isnot enriched (or, equivalently, enriched over Set), we will use the termdiscrete category. When we speak of anA-diagram we always assume thatAis a discrete category.
LetC be a closed symmetric monoidal category. Consider the forgetful functor φ:C −!Set,
C7−!homC(1, C).
It is immediate from the definitions thatφis a monoidal functor. Therefore, it induces a functor from categories enriched overC to discrete categories. We will call this induced functor thediscretization functor. LetO be a category enriched overC. The discretiza-tion of O will be denotedOδ. It has the same objects as O, and its sets of morphisms are given by the discretization of morphisms inO. For example, Top can be either the Top-enriched category or the associated discrete category. For Ch, the set of morphisms between two chain complexes X∗ and Y∗ in the discretization of Ch is the set of cycles of degree 0 in the chain complex Ch(X∗, Y∗), i.e. the set of chain maps. It is easy to see that ifCis a closed symmetric monoidal category, then the discretization ofCis the same as C, considered as a discrete category. We will not use special notation to distinguish betweenC and its underlying discrete category.
Let M:O!R be a C-functor between two C-categories. The underlying discrete functor is the functorMδ:Oδ!Rδ induced in the obvious way fromM. More precisely, ifi is an object of O thenMδ(i)=M(i). If j is another object andf∈Oδ(i, j), that is f:1!O(i, j), thenMδ(f)∈Rδ(Mδ(i), Mδ(j)) is defined as the composite
1−−f!O(i, j)−−−−−M(i,j)!R(M(i), M(j)).
Similarly if Φ:M!M0 is an enriched natural transformation between enriched functors, we have an induced discrete natural transformation Φδ:Mδ!(M0)δ. In particular, an O-moduleM induces an Oδ-diagram Mδ in C and a morphism ofO-modules induces a morphism ofOδ-diagrams.
LetF:C!Dbe a lax symmetric monoidal functor, letObe aCI-category, and let M:O!Cbe anO-module. As explained before, we have an inducedDI-categoryF(O), and anF(O)-moduleF(M). We may compareOδ andF(O)δ by means of a functor
FOδ:Oδ−!F(O)δ
which is the identity on objects and iff:1C!O(i, j) is a morphism inOδ, thenFOδ(f) is the composite1D!F(1C)−−−F(f)!F(O(i, j)).
It is straightforward to verify the following two properties of discretization.
Lemma 4.3. Let F:C!D be a lax symmetric monoidal functor, let O be a CI -category and let M be an O-module. The following diagram of discrete functors com-mutes
Oδ
FOδ
Mδ //C
F
F(O)δ F(M)
δ //D.
Lemma 4.4. Let C be a monoidal model category and let O be a CI-category. If Φ:M−'!M0 is a weak equivalence of O-modules then Φδ:Mδ−'!(M0)δ is a weak equiv-alence of Oδ-diagrams.