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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeDec 26th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea and definition If $C$ is a [[quasicategory]] and $W$ is a class of morphisms in $C$, then the __quasicategory of fractions__ is a functor $L\colon C\to C[W^{-1}]$ that has the following universal property: precomposition with $L$ identifies the quasicategory of functors $C[W^{-1}]\to D$ with the full subquasicategory of functors $C\to D$ that sends elements of $W$ to invertible morphisms in $D$. ## Terminology In the existing literature on [[quasicategories]], some sources (like Lurie's [[Higher Topos Theory]]) do not assign a specific name to this concept and simply talk about it by saying that “$S^{-1}C$ is obtained from $C$ by inverting the morphisms of $S$”. (See Warning 5.2.7.3 in [[Higher Topos Theory]].) Other sources, like [[Cisinski]]'s [[Higher Categories and Homotopical Algebra]], use the term “localization” to refer to this concept, and refer to localizations in the sense of Lurie as “left Bousfield localizations”. <a href="https://ncatlab.org/nlab/revision/quasicategory+of+fractions/1">v1</a>, <a href="https://ncatlab.org/nlab/show/quasicategory+of+fractions">current</a>

   Created:

   Idea and definition

   If $C$ is a quasicategory and $W$ is a class of morphisms in $C$, then the quasicategory of fractions is a functor $L\colon C\to C[W^{-1}]$ that has the following universal property: precomposition with $L$ identifies the quasicategory of functors $C[W^{-1}]\to D$ with the full subquasicategory of functors $C\to D$ that sends elements of $W$ to invertible morphisms in $D$.

   Terminology

   In the existing literature on quasicategories, some sources (like Lurie’s Higher Topos Theory) do not assign a specific name to this concept and simply talk about it by saying that “$S^{-1}C$ is obtained from $C$ by inverting the morphisms of $S$”. (See Warning 5.2.7.3 in Higher Topos Theory.)

   Other sources, like Cisinski’s Higher Categories and Homotopical Algebra, use the term “localization” to refer to this concept, and refer to localizations in the sense of Lurie as “left Bousfield localizations”.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthornLab edit announcer
   • CommentTimeDec 26th 2023
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexadding a link to [[localization of an (infinity,1)-category]] Edward Wickham <a href="https://ncatlab.org/nlab/revision/diff/quasicategory+of+fractions/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/quasicategory+of+fractions/2">v2</a>, <a href="https://ncatlab.org/nlab/show/quasicategory+of+fractions">current</a>

   adding a link to localization of an (infinity,1)-category

   Edward Wickham

   diff, v2, current