internal category in homotopy type theory


Category theory

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




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).

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

Revised on May 12, 2017 03:50:33 by Mike Shulman (