The Thomason model structure on the category $\mathrm{Cat}$ of small categories is transferred along the right adjoint of the adjunction $$\tau_1 \circ \mathrm{Sd}^2 \colon s\mathrm{Set} \rightleftarrows \mathrm{Cat} \colon \mathrm{Ex}^2 \circ N,$$ where $\tau_1 \colon s\mathrm{Set} \to \mathrm{Cat}$ denotes the fundamental category functor, left adjoint to the nerve $N$. The functor $\mathrm{Sd} \colon s\mathrm{Set} \to s\mathrm{Set}$ denotes the barycentric subdivision, and $\mathrm{Ex}$ is its right adjoint.

A functor $F \colon \mathcal{C} \to \mathcal{D}$ is a Thomason weak equivalence if and only if it induces a weak equivalence on nerves $NF \colon N\mathcal{C} \to N\mathcal{D}$. The adjunction displayed above is a Quillen equivalence.

Question 1.Is the fibrant replacement $\mathcal{C} \to \mathcal{C}'$ in the Thomason model structure a localization? Here I mean localization in the $1$-categorical sense, i.e., a functor $\mathcal{C} \to \mathcal{C}[S^{-1}]$ that inverts a set of maps $S$ in $\mathcal{C}$.

My hunch is that the answer is no in general, but I'd be interested in situations where the answer is yes.

I've looked at Thomason's original paper [1], this paper by Meier and Ozornova on Thomason-fibrant categories [2], and this paper by Bruckner and Pegel on Thomason-cofibrant categories.

A related topic is what a localization does to the nerve, in particular, when does it preserve the homotopy type.

In Proposition 3.7 of [3], Dwyer and Kan show that if a category is a free product $\mathcal{C} = \mathcal{D} \ast \mathcal{W}$, where $\mathcal{W}$ is a *free* category, then the localization $\mathcal{C} \to \mathcal{C}[\mathcal{W}^{-1}]$ induces a weak equivalence $$N\mathcal{C} \to N(\mathcal{C}[\mathcal{W}^{-1}])$$ on nerves. Technically, their statement is happening in $O$-categories, with a fixed set of objects $O$.

Question 2.Are there other conditions on the category $\mathcal{C}$ and the set of maps $S$ under which the localization $\mathcal{C} \to \mathcal{C}[S^{-1}]$ induces a weak equivalence $N\mathcal{C} \to N(\mathcal{C}[S^{-1}])$ upon applying the nerve?

[1] *Thomason, R. W.*, **Cat as a closed model category**, Cah. Topol. Géom. Différ. 21, 305-324 (1980). ZBL0473.18012.

[2] *Meier, Lennart; Ozornova, Viktoriya*, **Fibrancy of partial model categories**, Homology Homotopy Appl. 17, No. 2, 53-80 (2015). ZBL1332.18019.

[3] *Dwyer, W. G.; Kan, D. M.*, **Simplicial localizations of categories**, J. Pure Appl. Algebra 17, 267-284 (1980). ZBL0485.18012.