# Thomason fibrant replacement and nerve of a localization

The Thomason model structure on the category $$\mathrm{Cat}$$ of small categories is transferred along the right adjoint of the adjunction $$\tau_1 \circ \mathrm{Sd}^2 \colon s\mathrm{Set} \rightleftarrows \mathrm{Cat} \colon \mathrm{Ex}^2 \circ N,$$ where $$\tau_1 \colon s\mathrm{Set} \to \mathrm{Cat}$$ denotes the fundamental category functor, left adjoint to the nerve $$N$$. The functor $$\mathrm{Sd} \colon s\mathrm{Set} \to s\mathrm{Set}$$ denotes the barycentric subdivision, and $$\mathrm{Ex}$$ is its right adjoint.

A functor $$F \colon \mathcal{C} \to \mathcal{D}$$ is a Thomason weak equivalence if and only if it induces a weak equivalence on nerves $$NF \colon N\mathcal{C} \to N\mathcal{D}$$. The adjunction displayed above is a Quillen equivalence.

Question 1. Is the fibrant replacement $$\mathcal{C} \to \mathcal{C}'$$ in the Thomason model structure a localization? Here I mean localization in the $$1$$-categorical sense, i.e., a functor $$\mathcal{C} \to \mathcal{C}[S^{-1}]$$ that inverts a set of maps $$S$$ in $$\mathcal{C}$$.

My hunch is that the answer is no in general, but I'd be interested in situations where the answer is yes.

I've looked at Thomason's original paper [1], this paper by Meier and Ozornova on Thomason-fibrant categories [2], and this paper by Bruckner and Pegel on Thomason-cofibrant categories.

A related topic is what a localization does to the nerve, in particular, when does it preserve the homotopy type.

In Proposition 3.7 of [3], Dwyer and Kan show that if a category is a free product $$\mathcal{C} = \mathcal{D} \ast \mathcal{W}$$, where $$\mathcal{W}$$ is a free category, then the localization $$\mathcal{C} \to \mathcal{C}[\mathcal{W}^{-1}]$$ induces a weak equivalence $$N\mathcal{C} \to N(\mathcal{C}[\mathcal{W}^{-1}])$$ on nerves. Technically, their statement is happening in $$O$$-categories, with a fixed set of objects $$O$$.

Question 2. Are there other conditions on the category $$\mathcal{C}$$ and the set of maps $$S$$ under which the localization $$\mathcal{C} \to \mathcal{C}[S^{-1}]$$ induces a weak equivalence $$N\mathcal{C} \to N(\mathcal{C}[S^{-1}])$$ upon applying the nerve?

[1] Thomason, R. W., Cat as a closed model category, Cah. Topol. Géom. Différ. 21, 305-324 (1980). ZBL0473.18012.

[2] Meier, Lennart; Ozornova, Viktoriya, Fibrancy of partial model categories, Homology Homotopy Appl. 17, No. 2, 53-80 (2015). ZBL1332.18019.

[3] Dwyer, W. G.; Kan, D. M., Simplicial localizations of categories, J. Pure Appl. Algebra 17, 267-284 (1980). ZBL0485.18012.

• The fibrant replacement in the Thomason model structure is precisely the homotopical localization, alias ∞-localization. Thus, Question 1 immediately reduces to Question 2. – Dmitri Pavlov Mar 31 at 14:01
• @Dmitri Pavlov: Thank you for your comment. My question is mostly about distinguishing between 1-localization and $\infty$-localization. You mention the $\infty$-localization of a $1$-category $C$. With respect to which maps? Does a category come with an intrinsic notion of maps to invert? One could try "all maps", whose $\infty$-localization is the $\infty$-groupoidification, i.e., Kan fibrant replacement. This is a more drastic procedure than the Thomason fibrant replacement. – Martin Frankland Mar 31 at 18:19
• (cont'd) Furthermore, the nerve of the $1$-groupoidification is a much more drastic procedure. I think the nerve of the $1$-groupoidification $N(C[C^{-1}])$ is the Postnikov $1$-truncation of the $\infty$-groupoidification of the nerve $(NC)[NC_1^{-1}] \simeq \mathrm{Ex}^{\infty} NC$. Sorting out these issues is the goal of my question. – Martin Frankland Mar 31 at 18:23
• About Question 2: if a functor $f:C\to D$ is smooth or proper with weakly contractible fibres, then it exhibits $D$ as the $(\infty,1)$-localization of $C$ by the maps which are sent to identities in$D$. This is true if $C$ and $D$ are $(\infty,1)$--categories, hence also for nerves of $1$-categories. Examples of smooth (proper) maps are Cartesian (coCartesian) fibrations. A typical instance of this is when $C$ is the category of (semi-)simplices of the nerve of $D$. – Denis-Charles Cisinski Apr 1 at 10:47
• See Prop. 7.1.12 in my book "Higher categories...". There is also Prop. 7.3.8 which provides examples, and Theorem 4.4.36 gives a rather concrete characterization of proper functors. – Denis-Charles Cisinski Apr 1 at 20:25

Regarding Question 1, the only time I can think of when a Thomason fibrant replacement can be taken to be a 1-categorical localization is when $$C$$ has the homotopy type of the classifying space of a discrete groupoid $$G$$, i.e. $$|C|$$ is aspherical.[1] That is:

• For any category $$C$$, we have $$\Pi_1(|C|) \simeq C[C^{-1}]$$, where $$\Pi_1$$ denotes the fundamental groupoid. (Proof: the van Kampen theorem allows one to write down a presentation of $$\Pi_1(|C|)$$ from the 2-skeleton of the nerve of $$C$$, and it is equally a presentation of $$\Pi_1(C)$$.).

• So under the assumption that $$|C| \simeq BG$$, we have $$G = C[C^{-1}]$$, and therefore $$|C| \to |C[C^{-1}]|$$ is a weak homotopy equivalence.

• Moreover, the nerve of a groupoid is a Kan complex (and conversely!), so under the assumption that $$|C|$$ is aspherical, we have that $$C \to C[C^{-1}]$$ i a Thomason fibrant replacement.

In particular, regarding Question 2, a sufficient condition for $$|C| \to |C[S^{-1}]|$$ to be a weak homotopy equivalence is for $$|C|$$ to be aspherical.

This may sound quite restrictive, but in practice it turns out that most of the categories one actually likes to work with qua categories are aspherical.

• For one thing, most of the categories one really cares about are contractible -- a category is contractible if it has a terminal object, if it has binary products, if it is filtered, if it is homotopy sifted, if it admits an adjunction to such a category, etc.

But more generally, there are some natural categorical conditions which imply that a category is aspherical without necessarily being contractible:

• For instance, if $$C$$ has pullbacks, then by a theorem of Pare, $$C$$ has all finite simply-connected limits. Then arguing using the $$Ex^\infty$$ functor, one sees that $$|C|$$ is aspherical.

• For another example, Dwyer and Kan showed (Prop 7.3 -- this paper follows up on the one linked in the question) that if $$C$$ has a left calculus of fractions, then $$|C|$$ is aspherical.[2] More generally, they showed that if $$C$$ has a left calculus of fractions as well as a left calculus of fractions with respect to a class of morphisms $$S$$, then then the $$\infty$$-categorical localization $$L_S C$$ (=simplicial localization) agrees with the 1-categorical localization $$C[S^{-1}]$$.

• There are probably other interesting conditions in this vein which I don't know -- I'd love to hear of more!

Of course, the duals of all of the above statements also hold.

[1] There's a possible confusion here: I'm using "aspherical" in the sense of aspherical space, i.e. a 1-truncated space. But since we're talking about modeling spaces with categories, it's worth bearing in mind that in this context, the Grothendieck school (I'm thinking of Grothendieck, Maltsiniotis, Cisinski, Ara,...) uses the term "aspherical category" to mean something different (namely, "category with a weakly contractible classifying space" -- possibly after performing an appropriate localization if working with respect to a different fundamental localizer).

[2] I'm not sure if this is related to the fact, discussed by Meier and Ozornova in the article linked in the question, that $$C$$ has a left calculus of fractions if and only if $$Ex NC$$ is a Kan complex.