cubical type 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




Cubical type theory is a version of homotopy type theory in which univalence is not just an axiom but a theorem, hence, since this is constructive, has “computational content”. Cubical type theory models the infinity-groupoid-structure implied by Martin-Löf identity types on constructive cubical sets, whence the name.

The first constructive account of the univalence axiom was given in (Coquand 13, Bezem-Coquand-Huber 17), called the “BCH-model”.

The BCH model, unfortunately, has some problems that make it unsuitable for general HoTT (in particular, it is not known how to model higher inductive types). The problem is that the BCH model is based on presheaves on the ‘symmetric monoidal cube category’, which is basically the free PROP generated by an interval. In particular, the base category’s maps are generated by face maps and permutative renamings of dimension variables (this is where the ‘symmetric monoidal’ part comes in). For somewhat technical reasons, this doesn’t work out when you want to define the elimination rules for higher inductive types (like for the circle).

To account for HITs, you seem to need diagonals in the base category; if you add these, then you have the ‘cartesian cube category’. This is done in (Cohen-Coquand-Huber-Moertberg 17), called the “CCHM model”. This model has a much richer cube category, the free De Morgan algebra generated by an interval. In addition to diagonals, this includes what are called ‘reversals’ and ‘connections’.

The CCHM model validates both univalence and can be used to model a variety of HITs.

One thing to be cautious about is that while it is possible to model the Martin-Löf identity type in both the BCH model and the CCHM model, it does not coincide with the paths in the model. But it is equivalent to the path types.


Last revised on April 24, 2018 at 08:09:26. See the history of this page for a list of all contributions to it.