natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category 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 literature, the “Rezk-completeness” condition on such categories is omitted from the definition, and categories that satisfy it are called saturated or univalent.
Particularly useful in the context of such categories are displayed categories.
The relation between Segal completeness (now often “Rezk completeness”) for internal categories in HoTT and the univalence axiom had been pointed out in
A comprehensive discussion for 1-categories is in
Exposition of this includes
Discussion of implementation of this in Coq includes
See also
Generalization to (n,1)-categories is discussed in
and, by different means, in
Emily Riehl, Michael Shulman, A type theory for synthetic $\infty$-categories (arXiv:1705.07442)
Emily Riehl, The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories, talk at Vladimir Voevodsky Memorial Conference 2018 (pdf)
Formalization of bicategories:
A formalization in HoTT-Agda of general (n,r)-categories for $n,r \in \mathbb{N} \sqcup \{\infty\}$, defined as coinductive types of infinity-graphs, with operations defined by induction-coinduction, is implemented in
Last revised on August 27, 2019 at 03:15:42. See the history of this page for a list of all contributions to it.