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It seems like we could somehow by means of black magic construct the localization functor $\gamma: C\to W^{-1}C$ where $W$ satisfies the two-out of six axiom and contains all identities by messing around with the nerve in the following way:
Let $U:Cat\to Quiv$ be the forgetful functor sending categories to quivers, let $F:Quiv\to Cat$ denote the free category functor, and let $N:Cat\to Set_\Delta$ denote the nerve functor. Then it seems like the first few steps of the localization procedure can be described as taking the nerve of the free category of the underlying quiver of C. This gives us the simplicial set where the n-simplices are zig-zags of morphisms. We can look at the simplicial subset of this guy given by the restricted zig-zags (I haven’t worked out an exact characterization of this, but it seems plausible that we could, but this is the weakest part of the idea). Then we construct the pre-hom-sets between vertices A and B to be the simplices whose initial vertex is A and terminal vertex is B. This seems like it should give a category, with composition given by concatenation. Now, the other weak part of the idea is how exactly to impose the equivalence relations in a functorial way (perhaps there’s a way to do this before we turn our simplicial set back into a category?).
Is there anything to this, or will it not work (or even if it will work, will it be stupid and contrived)?
Then it seems like the first few steps of the localization procedure can be described as taking the nerve of the free category of the underlying quiver of C. This gives us the simplicial set where the n-simplices are zig-zags of morphisms.
Hm, the n-simplices in the nerve are not zig-zags, but just sequences of morphisms. And for the simplicial localization it is important that every zag in a zig-zag is in $W$. Where is that condition in your prescription?
It’s more like this:
you form something like the Kan fibrant replacement of the nerve, but subject to the condition that certain 1-cells are in $W$.
Ah, thanks. I meant to say take the underlying undirected graph, but you’re right, it doesn’t work anyway.
A useful exercise is this:
consider the special case where $W$ contains all morphisms. So we have a category $C$ in which every morphisms is supposed to be a weak equivalence. This is suposed to present an $(\infty,1)$-category that is in fact an $\infty$-groupoid.
So one can form the nerve $N(C)$ and then its Kan fibrant replacement $Ex^\infty N(C)$ to get a Kan complex.
Alternatively, one can form the Dwyer-Kan simplicial localization $L_W C$ to obtain a Kan-complex enriched category.
Under homotopy coherent nerve, these two constructions ought to give equivalent results
$N_{hc} L_W C \simeq Ex^\infty N(C) \,.$This should be a good exercise to do in order to warm up for understanding the question that you are getting at.
(I need to do that exercise myself.)
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