Homotopy Type Theory



One may consider internal categories in homotopy type theory. Under the interpretation of HoTT in an (infinity,1)-topos, this corresponds to the concept of a category object in an (infinity,1)-category. The general idea is presented there at Homotopy Type Theory Formulation.

For internal 1-categories in HoTT (as opposed to more general internal (infinity,1)-categories) a comprehensive discussion was given in (Ahrens-Kapulkin-Shulman-13).

In some of the literature, the “Rezk-completeness” condition on such categories is omitted from the definition, and categories that satisfy it are called saturated or univalent.

Similarly to the univalence axiom we make two notions of sameness the same. This leads to some nice concequences.


A category is a precategory such that for all a,b:Aa,b:A, the function idtoiso a,bidtoiso_{a,b} from Lemma 9.1.4 (see precategory) is an equivalence.

The inverse of idtoisoidtoiso is denoted isotoidisotoid.


Note: All precategories given can become categories via the Rezk completion.


Lemma 9.1.8

In a category, the type of objects is a 1-type.

Proof. It suffices to show that for any a,b:Aa,b:A, the type a=ba=b is a set. But a=ba=b is equivalent to aba \cong b which is a set. \square

There is a canonical way to turn a precategory into a category via the Rezk completion.

Lemma 9.1.9

See also

Category theory


Coq code formalizing the concept of 1-categories includes the following:

category: category theory