# Localisation in a quasi-category

Let $W$ be a family of arrows in a category $\mathcal{C}$, there is a nature notion of localisation w.r.t. $W$. And if $W$ satisfies some nice properties, we have calculus of fraction.

Now consider a quasi-category(i.e., weak Kan complex) $\mathcal{C}$, in which every $k$-arrows are invertible if $k>1$. Given a family of $1$-arrows $W$, is there a notion of localisation w.r.t. $W$, that is a universal way to invert all arrows in $W$? Furthermore, if $W$ is nice enough, does the notion of the calculus of fraction have a sensible generalization?

When I search localisation + quasicategory online, it yields mostly simplicial Localisation, Hammock localisation...which state how to localise a category rather than a quasi-catgory.

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I can't remember the details off the top of my head, but I'm somewhat certain this is covered in Lurie's Higher Topos Theory (arxiv.org/pdf/math/0608040v4.pdf) I believe on page 295. Might just be a good place to start. – Jon Beardsley Jun 19 '13 at 14:05
Thanks a lot. I got it. – Ma Ming Jun 19 '13 at 14:13

## 2 Answers

Some aspects and plenty of further pointers (mostly to Jacob Lurie's book, of course) are collected here:

http://ncatlab.org/nlab/show/localization+of+an+(infinity,1)-category

http://ncatlab.org/nlab/show/reflective+sub-(infinity,1)-category

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@Urs Thanks so much. – Ma Ming Jun 22 '13 at 8:23

You asked about a category of fractions, so it sounds to me like you really want to formally invert the arrows in $W$. For that, you want something different than what Urs is explaining, however, there is still a way to do this hidden in HTT. Given a quasicategory $C$ with a class $W$ of morphisms, consider the pair $\left(C,W\right)$ as a marked simplicial set. The model category of marked simplicial sets is Quillen equivalent to simplicial sets with the Joyal model structure, and the fibrant objects are of the form $\left(C,inv.\right)$ where $C$ is a quasicategory $inv.$ stands for the class of all equivalences in $C$. The fibrant replacement of $\left(C,W\right)$ is $\left(C[W^{-1}],inv.\right)$- the "homotopy category" of $C$ you are searching for.

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