nLab cubical-type model category

Cubical-type model categories


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/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) 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


Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras



for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Cubical-type model categories


These exist a class of model category structures on certain categories of cubical sets, due to Christian Sattler, motivated by cubical type theory.


Let \mathbb{C} be a small category with finite products. Assume there is a (cartesian) interval object 𝕀\mathbb{I}, i.e., an object 𝕀\mathbb{I} with two parallel morphisms 0,1:e𝕀0,1 \colon e \to \mathbb{I}, where ee is the terminal object in \mathbb{C}.

Assume further that we have connection maps ,:𝕀×𝕀𝕀\vee,\wedge : \mathbb{I}\times\mathbb{I} \to \mathbb{I} such that x0=0x=xx \vee 0 = 0 \vee x = x and x1=1x=xx \wedge 1 = 1 \wedge x = x.

We also need a face lattice 𝔽\mathbb{F}, that is, a sub-lattice of the subobject classifier Ω\Omega in the presheaf category cSet= opSetcSet = \mathbb{C}^{op} \to Set containing the endpoints of each cylinder object J +=J×𝕀J^+ = J \times \mathbb{I} for each object JJ of \mathbb{C}, and an operation :𝔽 𝕀𝔽\forall \colon \mathbb{F}^\mathbb{I} \to \mathbb{F} right adjoint to the projection in the sense that ψδ\psi \le \forall \delta iff ψpδ\psi p \le \delta, where p:𝔽×𝕀𝔽p : \mathbb{F} \times \mathbb{I} \to \mathbb{F}.

If we want the model to be effective, we also require that each proposition ψ=1\psi = 1 with ψ𝔽(I)\psi \in \mathbb{F}(I) is decidable. (This rules out taking 𝔽=Ω\mathbb{F} = \Omega.)

Instances of the assumptions

A simple and central example is the full subcategory of the category of posets PosPos on powers of 𝕀=(0<1)\mathbb{I} = (0 \lt 1). This is also the Lawvere theory of distributive lattices. The idempotent completion is the full subcategory of PosPos on finite lattices.

Other prominent examples are the Lawvere theories of de Morgan, Kleene, and boolean algebras.

For computational purposes we can take 𝔽\mathbb{F} to be the smallest lattice containing the cylinder endpoints (the faces of cubes). To get Cisinski model structures we can take 𝔽=Ω\mathbb{F} = \Omega.

Equivalence with simplicial sets

One may wonder whether these models structures are equivalent to the model in simplicial sets. This is not the case for Cartesian cubes; see mailing-list. However, this model does form a Grothendieck (infinity,1)-topos, because it satisfies the Giraud axioms. For deMorgan cubical sets the geometric realization is not a Quillen equivalence (because of the reversal map); the counterexample is the unit interval quotient by the symmetry); see Sattler. Whether they are equivalent by another map is not yet excluded. The question whether the geometric realization for the cubical sets based on distributive lattices is an equivalence is still open.


  • Thierry Coquand, A model structure on some presheaf categories, pdf

  • Thierry Coquand, Some examples of complete Cisinski model structures, pdf

  • Christian Sattler, Do cubical models of type theory also model homotopy types, lecture at Hausdorff Trimester Program: Types, Sets and Constructions, youtube

Last revised on June 4, 2021 at 05:47:01. See the history of this page for a list of all contributions to it.