(∞,1)-category

Remarks

Other standard models are:

1. simplicial categories, that is, categories enriched over simplicial sets, whose hom-objects are all Kan complexes;
2. complete Segal spaces, which are certain simplicial objects in simplicial sets;
3. Segal categories.

Each model carries a model category structure, and these model categories are connected by Quillen equivalences, so the models give the same homotopy theory of $(\infty,1)$-categories. Example: the nerve of an ordinary category is a quasi-category, so every category is an $(\infty,1)$-category with only identity higher morphisms. Example: Kan complexes are the $(\infty,1)$-categories in which every morphism is invertible, the $\infty$-groupoids. Non-example: $\mathbf{Cat}$, viewed as a strict $2$-category, is not an $(\infty,1)$-category in its own right. It has non-invertible natural transformations as $2$-morphisms, while the $2$-morphisms of an $(\infty,1)$-category must be invertible.