nLab contextual fibrancy


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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
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


Category theory



In a categorical model of dependent type theory, a type TT in context Γ\Gamma is called fibrant if the projection morphism p:Γ.TΓp : \Gamma.T \to \Gamma from the extended context, is a fibration.

When we want to study or use fibrancy within type theory, we may be interested in the following properties, which do not necessarily hold:

  1. The Π-type is fibrant if its codomain is,

  2. The fibrant replacement monad commutes with substitution,

  3. A type in context Γ\Gamma is fibrant if and only if its [pullback]] (substitution) along any morphism/cell γ:yWΓ\gamma : y W \to \Gamma from a representable context yWy W, is fibrant.

The first property is just practical. The second property allows the fibrant replacement to be internalized as a monad on types, so that fibrancy can be defined internally as being an algebra for that monad. The third property allows a direct construction of a Hofmann-Streicher universe? (Hofmann and Streicher) of fibrant types. It should be straightforward to prove that this is equivalent with the second property, and with the property that the fibrant replacement can be taken cellwise.

One way to guarantee that properties 1-2 hold, is by defining fibrations as the right class of an algebraic weak factorization system generated by a class of generating left maps whose pullback is again a left map (Nuyts 2020). (So a fibration is any morphism that right-lifts any generating left map, and a left map is any morphism that left-lifts any fibration.)

Important notions of fibrancy violate the properties above:

  • Kan fibrancy as used in models of HoTT. The problem is that a filling of a horn or open box takes place above a cell that is not part of that horn or open box.
  • Segal fibrancy as one would like to use in models of directed type theory, for the same reason.

For this reason, Boulier and Tabareau introduced contextual Kan fibrancy. A type TT in a context Γ.Θ\Gamma.\Theta is contextually Kan fibrant in context Γ\Gamma if it has fillers for open boxes whose “open” dimension is Γ\Gamma-homogeneous, i.e. if it lifts diagrams of the following shape:

This is an instance of a damped algebraic weak factorization system (Nuyts 2020). We can also build a damped AWFS for the Segal condition, where at most one of the morphisms-to-be-composed can be Γ\Gamma-heterogeneous: Both damped AWFSs have the property that they are generated by a class of generating left “damped arrows” (which are just sequences of two composable arrows) whose pullbacks are again left damped arrows, and the result is that we get a form of properties 1-2 (Nuyts 2018, 2020) for these notions of contextual fibrancy.


Last revised on February 15, 2023 at 11:45:18. See the history of this page for a list of all contributions to it.