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Let $\text{Cat}$ be the category of (small) categories and functors. There is a "categorical" (also called "canonical" or "folk") model structure on $\text{Cat}$ in which the weak equivalences are the usual equivalences of categories (and as explained here, it turns out that there is a unique choice of cofibrations and fibrations that come with them). In this model structure, all categories are fibrant and cofibrant.

There is also a natural question of constructing a "geometric" model structure on $\text{Cat}$ for which the weak equivalences are functros that induce weak equivalence of nerves. i.e. there is the nerve functor $N:\text{Cat}\to \text{sSet}$, where $\text{sSet}$ is the category of simplicial sets with the standard model structure, and we would like a model structure on $\text{Cat}$, for which a functor $F:C\to D$ is a weak equivalence if and only if $N(F):N(C)\to N(D)$ is a weak equivalence. There is such a construction by Thomason, in which the model structure on $\text{Cat}$ is "transported" from $\text{sSet}$ along the adjunction $$ \tau_1 \circ Sd^2 :\text{sSet}\leftrightarrows\text{Cat}: Ex^2\circ N $$ Where $\tau_1$ is the left adjoint of $N$ (sometimes called "fundamental category") and $Sd\dashv Ex$ are the barycentric subdivision / Extension adjunction of Kan. In Thomason's model structure, the cofibrations are quite complicated (they are all "Dwyer maps"). In particular a cofibrant category must be a poset.

My question is whether there exist a model structure on $\text{Cat}$, with "geometric" weak equivalences as in Thomason's, but with the same cofibrations as the "categorical" one. In other words, can we obtain a "geometric" model structure on $\text{Cat}$ by localizing the categorical one?

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2 Answers 2

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No, there are far too many "canonical cofibrations" for that. For example, let $A$ be a category with two objects and two parallel arrows between them. Then there is a unique functor $A \to [1]$ that is injective on objects and hence a "canonical cofibration". The homotopy type of $A$ is $S^1$ so the homotopy type of the pushout of $A \to [1]$ with itself should be $S^2$, but it is $[1]$ again.

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Can you comment about what precisely fails when you try to do the obvious thing? I mean, the Thomason model structure is combinatorial so the class of weak equivalences is accessible. The canonical model structure is left proper and combinatorial, so you might try to apply Bousfield localization at the Dwyer maps. The only issue I can see is that perhaps the weak equivalences are $\kappa$-accessible for some $\kappa > \omega$, and Smith's Theorem (HTT A.2.6.13) requires the weak equivalences to be perfect rather than merely accessible. I'd love to see what fails in this argument. –  David White Jul 20 '14 at 8:44
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The problem is exactly the one I explained above. The assumption (2) of [HTT, A.2.6.13] essentially asks for pushouts along generating cofibrations to be homotopy pushouts. My example shows that this not the case for "canonical cofibrations". –  Karol Szumiło Jul 20 '14 at 9:04
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Alternatively, one notes that every object is "canonically cofibrant", so the model structure (if it exists) must be left proper and so pushouts of cofibrations "must" be homotopy pushouts. –  Zhen Lin Jul 20 '14 at 9:27
    
@KarolSzumilo: Okay, I agree A.2.6.13 can't be used. But that's for building a combinatorial model structure from scratch. There's also Smith's theorem that if you already have a left proper, combinatorial model category $M$ and a set of maps $C$ then the left Bousfield localization $L_C(M)$ exists. One might hope that in the presence of accessibility for the Dwyer maps then a set $C$ could be found so that the $C$-local equivalences are the Dwyer maps. I'd like to better understand the limits of the Smith machinery when it comes to finding such a set given an accessible class. –  David White Jul 20 '14 at 9:40
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Ah, I meant the weak equivalences of the Thomason model structure. I think I misread the OP. A reference for the theorem I have in mind is Barwick `On (Enriched) left Bousfield localization' Theorem 2.11. It seems I was right about only needing the accessibility of the weak equivalences (rather than the fact that they're generated by a set) as Barwick's proof only requires his Lemma 2.10. However, condition (3) of his Proposition 1.7 is not satisfied thanks to your example ($W \cap I$-cof is not closed under pushout). Other versions of Smith's Theorem (e.g. HTT) hit the same problem. –  David White Jul 20 '14 at 10:13

Karol's answer is excellent, however, I want to suggest another argument. I claim that a model category on $\mathbf{Cat}$ whose weak equivalences are the Thomason equivalences can never be a localization of a model structure whose weak equivalences are the categorical equivalences.

Indeed, on the one hand the canonical model structure is a simplicial model category whose mapping space is given by $Map(C,D)=N(i \mathrm{Fun}(C,D))$ where $i$ sends a category to its subcategory of isomorphisms. From this description, we see that the mapping spaces are all nerve of groupoids which implies that they are $1$-truncated (i.e. that their homotopy groups of degree at least $2$ are trivial). The important point is that this fact is intrinsic to the weak equivalences in the canonical model structure because these mapping spaces can also be computed using for instance the hammock localization which only depends on the weak equivalences.

On the other hand, if $M\leftrightarrows L_SM$ is a Bousfield localization, the fibrant objects of $L_SM$ are a subclass of the fibrant objects of $M$. Moreover, between fibrant objects of $L_SM$, the derived mapping spaces (computed using a simplicial enrichment or the hammock localization) are the same in $M$ or $L_SM$. This means that the $\infty$-category $L_SM$ is a full sub $\infty$-category of $M$.

The category of categories with the Thomason weak equivalences is a model for the $\infty$-category of spaces. In particular, some of its derived mapping spaces are not $1$-truncated (for instance $Map(*,S^2)$is not $1$-truncated) which means that it cannot embed as a full sub $\infty$-category of the $\infty$-category of categories with the canonical weak equivalences.

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This is a fantastic argument! It really explains what's going on. –  KotelKanim Feb 9 at 10:33
    
I should add that this argument is not due to me. I think I learnt it from Rune Haugseng. –  Geoffroy Horel Feb 9 at 18:12

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