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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?

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The comments in… give an answer in terms of homology for group completions of monoids. – Benjamin Steinberg Jul 5 '13 at 20:32
Thanks, @BenjaminSteinberg! I assume you are referring to the following paper which gives a surprising (to me) answer for group completions of monoids: Fiedorowicz, Z. Classifying spaces of topological monoids and categories. Amer. J. Math. 106 (1984) – Vidit Nanda Jul 5 '13 at 22:01
up vote 7 down vote accepted

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.)

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Thank you so much for the DK reference! – Vidit Nanda Jul 6 '13 at 14:24

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