Is there an intrinsic definition of weak equivalence in Cat or RelCat?

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 $W$.

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.