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It's known that Cat with the Thomason model structure serves as a model for $\infty\mathrm{Grpd}$, and that RelCat has a corresponding model structure that serves as a model for $\infty\mathrm{Cat}$. (with Cat embedding in RelCat as those relative categories where everything is a weak equivalence)

In Barwick and Kan's paper, they define a homotopy relation generated by the natural transformations whose components are weak equivalences.

While appealing, this is inadequate; if I understand anything at all, the infinite zigzag category $Z$ depicted as

$$ \ldots \leftarrow \bullet \to \bullet \leftarrow \bullet \to \bullet \leftarrow \bullet \to \ldots $$

where all arrows are weak equivalences is supposed to have geometric realization homeomorphic to $\mathbb{R}$, and thus have the homotopy type of a point... but $Z$ is not homotopy equivalent to the terminal category, since any homotopy from $1_Z$ can only take a fixed, finite number of steps, but arbitrarily large steps are needed to connect every object to a specified one.

Every exposition on the topic I have seen simply punts the question over to simplicial sets or bisimplicial sets or similar: that whether or not a map is a weak equivalence is determined by the map it induces on nerves.

Is there a description of weak equivalences that can be phrased entirely within Cat or RelCat without taking a detour through simplicial sets or topological spaces?

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    $\begingroup$ Perhaps it is of some use that a minimal fundamental localizer (in the sense of Grothendieck) for Cat is $\mathcal{W}_\infty$, the one associated to Thomason's model structure. In other words, if you want the minimal non-trivial model structure on Cat you have the one whose weak equivalences are those induced by weak equivalences of the nerves. This fact makes this choice less as-hoc. For details, see Cisinski, "Le localisateur fondamental minimal", numdam.org/item/CTGDC_2004__45_2_109_0 . $\endgroup$ – Leo Alonso Apr 27 '17 at 9:27
  • $\begingroup$ @LeoAlonso: Ah great! And now that I have a phrase to look for, I can find a summary in english at ncatlab, at least for Cat. $\endgroup$ – Hurkyl Apr 28 '17 at 5:13
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Following up @LeoAlonso's comment, in the case of Cat, the page on basic localizer at ncatlab, Cisinski's theorem states that the weak equivalences in Cat are the smallest class $W$ satisfying

  • $W$ contains the identities, has the 2-out-of-3 property, and is closed under retracts
  • If $A$ has a terminal object, $A \to 1$ is in $W$
  • Given $A \to B \to C$, if the map of comma categories $(A \downarrow c) \to (B\downarrow c)$ is in $W$ for every object $c$, then $A \to B$ is in $W$.
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There is the following characterization.

Homotopy Limit Functors on Model Categories and Homotopical Categories (DKHS) gives, for any saturated relative category C with objects $x$ and $y$, a category $\mathbf{Gr}(\mathbf{C})^\mathbf{T}(x,y)$ of zigzags from $x$ to $y$. These categories assemble into a strict $2$-category $\mathbf{Gr}(\mathbf{C})^\mathbf{T}$ called its Grothendieck construction.

DKHS show that the Grothendieck construction is the correct $\mathbf{Cat}_{\mathrm{Thomason}}$-enriched model for C in the sense that taking nerves of the hom-categories of $\mathbf{Gr}(\mathbf{C})^\mathbf{T}$ gives a simplicially enriched category weakly equivalent to the Hammock localization $L^H\mathbf{C}$.

Thus, we have

Let $F : \mathbf{C} \to \mathbf{D}$ be a functor between saturated relative categories. It is a weak equivalence if and only if:

  • $\mathbf{Ho}(F) : \mathbf{Ho}(\mathbf{C}) \to \mathbf{Ho}(\mathbf{D})$ is an equivalence of ordinary categories
  • For every pair of objects $x,y$, it induces a weak equivalence $\mathbf{Gr}(\mathbf{C})^\mathbf{T}(x,y) \to \mathbf{Gr}(\mathbf{D})^\mathbf{T}(F(x), F(y))$

Furthermore, on $\mathbf{Cat}_{\mathrm{Thomason}}$ (viewed as the subcategory of RelCat of categores where every arrow is a weak equivalence), we can carry out the definition of weak equivalence as isomorphisms on homotopy groups.

Let $F : \mathbf{C} \to \mathbf{D}$ be a functor between categories. In the Thomason model structure, it is

  • A $0$-equivalence if it induces a bijection on connected components
  • An $n$-equivalence if, for every pair of objects $x,y$, it induces $(n-1)$-equivalences $\mathbf{Gr}(\mathbf{C})^\mathbf{T}(x,y) \to \mathbf{Gr}(\mathbf{D})^\mathbf{T}(F(x), F(y))$
  • A weak equivalence if it is an $n$-equivalence for every $n$

If $x,y$ are in the same connected component, then $\mathbf{Gr}(\mathbf{C})^\mathbf{T}(x,x) \simeq \mathbf{Gr}(\mathbf{C})^\mathbf{T}(x,y)$, so the middle condition only needs checked on one endomorphism category per connected component.

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