Simplicial
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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.

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 \(C\). □

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. □
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 zigzags
\[X \to X_1\leftarrow X_2 \rightarrow X_3 \dots Y\]
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:
Where \(W[W^{1}] = GW\) is the groupoid generated by \(W\).

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. □
10.2 DwyerKan 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.
The following lemma is quite easy from definition:

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 \(L(V,V)\).

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 levelwise localization localization functor, computed by the deformation given by diagonalizing the standard free resolution.
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 pushoutproduct construction. □
Let \(\Cat _{\square }\) denote the category of categories enriched in cubical sets. Define \(P,H,E\) via the pushout squares below.
\(P\) 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.

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. □

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. □