nLab quasicategory of fractions

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”.

Related concepts

• category of fractions, homotopy category

• reflective localization

• localization of an (infinity,1)-category