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?

Catis $\mathcal{W}_\infty$, the one associated to Thomason's model structure. In other words, if you want the minimal non-trivial model structure onCatyou 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 . $\endgroup$Cat. $\endgroup$