Simplicial

10 Simplicial localization

The goal of this section is to show that the invertibility hypothesis is redundant in the definition of an excellent model category. This actually involves proving it in the case of simplicial categories.

10.1 Free Categories and Localization

Let \(O\Cat \) be the category of categories with a fixed set of objects \(O\) and morphisms maps that fix the set objects. A simplicial category with objects \(O\) is a simplicial object in \(O\Cat \), and we denote this category \(sO\Cat \). There is a simplicial nerve functor \(N\) taking a simplicial category to the diagonal of the bisimplicial set obtained by taking the nerve levelwise. By homotopy invariance of the diagonal, it follows that an equivalence of simplcial categories induces an equivalence on nerves.

A category \(C\) in \(O\Cat \) is free if there exists a set of morphisms \(S\) such that every nonidentity map in \(C\) is a unique composite of maps in \(C\). There is a forgetful map from \(U:\Cat \to \DiGrph \), the category of directed graphs, and the free categories are the essential image of the left adjoint. When we fix the set of objects, every free category is a coproduct of free categories on a single map. The free forgetful adjunction induces the free monad \(F\) on \(O\Cat \), taking a category to the free category on its nonidentity morphisms. This monad gives a simplicial resolution \(F_*C\) of an object in \(O\Cat \), which is equipped with a map \(F_*C \to C\). Similarly, given a simplicial category, one can take levelwise free resolutions and take the diagonal simplicial category.

Finally, observe that \(sO\Cat \) and \(O\Cat \) inherit a model structure from \(\Cat _{\Delta }, \Cat \), since they are subcategories of the slice category under the discrete category with objects \(O\) that are closed under all the operations needed to inherit the model structure.

Lemma 10.1. The map \(F_*C \to C\) is an equivalence.

Proof. By homotopy invariance of the diagonal, it suffices to consider \(C\) an ordinary category. For any objects \(X, Y\), the map taking \(x \in F_{n}C\) to \([x] \in F_{n+1}C\) is a contracting homotopy of \(F_*C\) onto \(C\). □

Lemma 10.2. If \(\star \) denotes the free product in \(sO\Cat \), then \(\star \) is homotopical, and computes the homotopy coproduct.

Proof. It suffices to show that \(A \star B \to A' \star B\) is an equivalence if \(A \to A'\) is. The argument in Lemma 10.1 shows that \(A\star F_*B \to A \star B\) is an equivalence, so we can assume \(B\) is free in each degree. In the degreewise free case, by observing that \(A\star F_*B\) is the diagonal of the bisimplicial category given by \(A\star F_nB\), we can reduce to the case \(B\) is discrete and free. In this case, the map \(O\to B\) is a cofibration, so since \(\star \) is the pushout relative to \(O\), it is a homotopy pushout by left properness. □

Lemma 10.3. If \(C\) is a free category, the inclusion \(\sk _kNC\to NC\) is an equivalence for \(k \geq 1\).

Proof. It’s easy to see that the \(k+1\)-skeleton deformation retracts onto the \(k\)-skeleton, because \(C\) is free. □

Let \(C \in O\Cat \) and \(W\) a subcategory of \(C\). \(C[W^{-1}]\) is the localization in the classical sense: its objects are the same, and morphisms are zig-zags

\[X \to X_1\leftarrow X_2 \rightarrow X_3 \dots Y\]

where the left arrows are in \(W\), modulo the obvious equivalence relation. The map \(C \to C[W^{-1}]\) universally sends \(W\) to isomorphisms. Note that this is also the pushout of categories:

(-tikz- diagram)

Where \(W[W^{-1}] = GW\) is the groupoid generated by \(W\).

Lemma 10.4. Let \(C = D\star E\). Then the inclusion \(ND \cup _O NE \to ND\star E\) is an equivalence.

Proof. For a free categories \(D,E\), this follows from Lemma 10.3. In general, we take a free resolution and use Lemma 10.1, Lemma 10.2, and the fact that the diagonal of a bisimplicial set is homotopical. □

Lemma 10.5. If \(C = D\star W \in O\Cat \), where \(W\) is free, then the map \(N(D\star W) = NC \to NC[W^{-1}]\) is an equivalence.

Proof. By transfinite composition and Lemma 10.4, it suffices to show it for \(C=W\) free on one generator. This is easy to see beacause both nerves are contractible. □

10.2 Dwyer-Kan Localization

Let \(C \in O\Cat \) with a subcategory \(W\). The standard simplicial localization is defined by \(L(C,W) = F_*C[F_*W^{-1}]\). The notation will often be shortened to \(LC\). The goal will be to show that this is a well behaved left derived functor of the localization.

The first thing to observe is that \(hLC = C[W^{-1}]\), since the homotopy category is a left adjoint, so preserves colimits.

Lemma 10.6. The simplicial localization preserves the homotopy type of the nerve.

Proof. Using Lemma 10.5 and Lemma 10.1 we see that \(NC \leftarrow NF_*C\rightarrow NLC\) gives a zig-zag of equivalences between the two. □

The following lemma is quite easy from definition:

Lemma 10.7. A map \(u:X \to Y \in C\) is in \(W\) iff it induces an isomorphism on covariant and contravariant mapping spaces.

Proposition 10.8. Suppose \(W = C\), and \(NC\) is connected. Then \(LC\) is a simplicial groupoid, all the mapping spaces are equivalent, and the endomorphism spaces \(LC(X,X)\) are simplicial groups. Moreover, the homotopy type of \(LC(X,X)\) is that of the loop space of \(NC\).

Proof. The fact that the endomorphisms are simplicial groups is trivial. To prove the last statement, first replace \(NC\) with \(NLC\) by Lemma 10.6. Now for every \(k\), \(F_kC[F_kC^{-1}]\) is a connected groupoid, so let \(UF_kC[F_kC^{-1}]\) be its universal cover, which is contractible, and has fibre \(LC(X,X)_k\). Then applying Theorem 7.23, we obtain the result. □

Now we generalize to simplicial localizations of a simplicial category. Given a simplicial category \(B \in sO\Cat \) and \(V\subset B\) a subcategory, the standard simplicial localization is the diagonal of \(F_*B[F_*V^{-1}]\), which we denote \(L(B,V)\) or \(LB\) when it isn’t confusing.

Again, upon taking homotopy categories, it agrees with the localization of the homotopy category. The groupoid completion of a simplicial category \(V\) is \(L(V,V)\).

Lemma 10.9. The map \(V \to L(V,V)\) induces an equivalence on nerves.

Proof. This follows because the diagonal is homotopy invariant and Lemma 10.6. □

Proposition 10.10. A map \(U \to V\) induces a weak equivalence on groupoid completions iff it induces a weak homotopy equivalence on nerves.

Proof. WLOG we can assume \(U,V\) are connected, in which case this follows from Lemma 10.9 and the fact that by the argument in Proposition 10.8 shows that \(LU(X,Y)\) has the homotopy type of the loopspace of \(NU\). □

It follows that the \(L(U,U)\) is the left derived functor of the naive level-wise localization localization functor, computed by the deformation given by diagonalizing the standard free resolution.

Proposition 10.11. Let \(V \in sO\Cat \). Then the natural map \(V \to L(V,V)\) is an equivalence iff \(hV\) is a groupoid.

Proof. The only if part is trivial since \(hL(V,V)\) is a groupoid. For the if, □

10.3 Reduction to Cubical Sets

The strategy will be to reduce the invertibility hypothesis to the case of cubical sets via its universal property below, and then use the work of Dwyer and Kan on simplicial localization. We use the standard model structure on \(S\)-enriched categories.

Proposition 10.12. Suppose \(C\) is a monoidal model category, \(F:\square ^{\leq 1} \to C\) is a functor, and \(\tilde {F}\) is its cocontinuous monoidal extension to \(\Set _{\square }\). Then \(\tilde {F}\) is a left Quillen functor iff \(F\) expresses \(F(\square ^1)\) as a cylinder object for the unit of \(F\), meaning the inclusions \(F(\square ^0)\) are trivial cofibrations. In particular, such a monoidal left Quillen functor exists.

Proof. \(F\) applied to the inclusions \(j_0,j_1:\square ^0 \to \square ^1\) must be a trivial cofibration. The map \(j_0 \cup j_1= i:\square ^0\cup \square ^0 \to \square ^1\) must also be sent to a cofibration. Note that \(\tilde {F}\) applied to the inclusion \(\phi \to \square ^0\) is a cofibration, which amounts to the unit being cofibrant, which is part of the monoidal model category axioms.

Conversely, if \(F\) satisfies those conditions, the generating (trivial) cofibrations are sent to (trivial) cofibrations because \(\otimes \) is a left Quillen bifunctor, and the generating (trivial) cofibrations are built from \(i,j_\ee \) using the pushout-product construction. □

Let \(\Cat _{\square }\) denote the category of categories enriched in cubical sets. Define \(P,H,E\) via the pushout squares below.

(-tikz- diagram)

\(P\) is defined so maps out of it are exactly a pair of morphisms in opposite direction, and \(H\) is in addition a homotopy from \(v\circ u\) to the identity, and \(E\) is a choice of left and right homotopy inverse.

Lemma 10.13. Suppose \(C \in \Cat _{\square }\) is fibrant and \(f:[1]_{\square } \to C\) is a map which becomes an isomorphism in the homotopy category. Then \(f\) extends to a map from \(E\).

Proof. The mapping spaces of \(C\) are all fibrant since \(C\) is fibrant. It follows that we can make the homotopies desired in the mapping spaces, extending the map to \(E\) by its universal property. □

Proposition 10.14. Let \(E[f^{-1}]\) denote the localization of \(E\) at the map \(f\) (the one with the choice of homotopy inverses). Then the map \(E \to E[f^{-1}] \to [1]^{\sim }_{\square }\) are equivalences.

Proof. There is a monoidal left Quillen equivalence (Theorem 9.80) between \(\Set _{\square }\) and \(\SSet \) preserving the localization, giving a left Quillen equivalence between categories of enriched categories. Thus it suffices to show the corresponding statement for simplicial categories.

But then \(f\) is a morphism that is already invertible, so by [DWYERKAN], it follows that \(E \to E[f^{-1}]\) is an equivalence. We still need to show that \(E[f^{-1}] \to [1]^{\sim }_{\square }\) is an equivalence. To do this, since the two objects in \(E[f^{-1}]\) are equivalent, it suffices to show that either of them has contractible endomorphisms. However, this mapping space from the construction of pushouts is the free monoid on the based space \(\Delta ^1\vee \Delta ^1\), which is contractible since \(\Delta ^1\vee \Delta ^1\) is. □

Corollary 10.15. The map \([1]_{\square ^0} \to E\) is a cofibrant replacement for \([1]_{\square ^0}\to [1]^{\sim }_{\square }\).

Theorem 10.16. Suppose that \(S\) is a monoidal model category satisfying \((A1)-(A4)\) of Definition 5.10. Then \(S\) satisfies the invertibility hypothesis.

Proof. Suppose that \(f:[1]_S \to C\) is a cofibration that is an equivalence in the homotopy category. By Remark 5.7, we can assume that \(C\) is fibrant. By Proposition 10.12, there is a left Quillen functor \(L: \Set _\square \to S\), with right adjoint \(R\). This induces a Quillen adjunction \(\adjunction {L}{\Cat _{\square }}{\Cat _S}{R}\)

Because \(L\) sends the unit to the unit, \(R\) preserves homotopy categories. Thus the image of \(f\) in the homotopy category after applying \(R\) extends to a map \(E \to RC\), which is adjoint to a map \(L(E) \to C\). We can assume WLOG that this map is a cofibration by modifying \(C\).

Then we can consider the diagram of pushouts:

The middle vertical map is an equivalence by Proposition 10.14. Thus the right vertical map is an equivalence by left properness. □

Corollary 10.17. A symmetric monoidal model category satisfying \((A1)-(A4)\) is excellent.