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](https://arxiv.org/abs/1011.1691), 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?



Is there an intrinsic definition of weak equivalence in Cat or RelCat?
There is the following characterization.


[Homotopy Limit Functors on Model Categories and Homotopical Categories](http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf) (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.


