Simplicial localization

In category theory, a branch of mathematics, the simplicial localization of a category C with respect to a class W of morphisms of C is a simplicial category LC whose ${\displaystyle \pi _{0}}$ is the localization ${\displaystyle C[W^{-1}]}$ of C with respect to W; that is, ${\displaystyle \pi _{0}LC(x,y)=C[W^{-1}](x,y)}$ for any objects x, y in C. The notion is due to Dwyer and Kan.