(∞,1)-category

Summary

An $(\infty,1)$-category is a higher category with $k$-morphisms for every $k \ge 1$, in which all $k$-morphisms with $k \ge 2$ are invertible. It behaves like a category whose hom-sets are replaced by spaces, taken up to homotopy. Several precise models exist, such as quasi-categories, categories enriched in Kan complexes, and complete Segal spaces, and these models are known to be equivalent. $(\infty,1)$-categories are the natural setting for homotopy theory, derived algebraic geometry and stable homotopy theory.