When does localization preserve homotopy type of classifying spaces?

Question

Let $\mathcal{C}$ be a small category and $\Sigma$ a collection of morphisms in $\mathcal{C}$. Denote by $F_\Sigma:\mathcal{C} \to \mathcal{C}[\Sigma^{-1}]$ the usual quotient functor from $\mathcal{C}$ to its localization about $\Sigma$.

Are there conditions on $\Sigma$ which guarantee that $F_\Sigma$ induces a homotopy equivalence $B\mathcal{C} \sim B\mathcal{C}[\Sigma^{-1}]$ of classifying spaces?

For example: if $\mathcal{C}$ consists of two objects with a single arrow from one to the other, then localization about that single arrow preserves homotopy type of classifying spaces: everything is contractible before and after localization. On the other hand, see this paper for a counter-example to the conjecture that the group completion of a monoid has the same classifying space as that monoid. Clearly, we can't just shamelessly start inverting arrows all over the place without destroying homotopy type.

One always has Quillen's Theorem A: if the under categories $F_\Sigma \downarrow c$ are all contractible, then $BF_\Sigma$ is a homotopy-equivalence of classifying spaces. So, one possible answer would highlight those conditions on $\Sigma$ which magically give contractible over/under categories. Is there a known result that does the trick?

Answer 1

You can find some sufficient conditions in terms of simplicial localization of Dwyer and Kan. In Prop. 3.7 of Simplicial Localizations of Categories they prove that it holds when $\Sigma$ is free and $\mathcal{C} = \mathcal{D} * \Sigma$ where $*$ denotes the coproduct of categories with a fixed set of objects (aka the free product). More generally it follows from 4.3 of the same paper that a sufficient condition is that $L^H(\mathcal{C}, \Sigma) \to \mathcal{C}[\Sigma^{-1}]$ is a DK-equivalence i.e. that $L^H(\mathcal{C}, \Sigma)$ has homotopy discrete mapping spaces. (As pointed out in Prop 7.2 of Calculating Simplicial Localizations this is implied by a calculus of fractions which generalizes the classical criterion of a monoid satisfying the Ore conditions.)