nLab context extension

Contents

Context

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
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
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
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 setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
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

semantics

Contents

Idea

In dependent type theory, context extension introduces new free variables into the context.

Definition

If TT is a type in a context Γ\Gamma, then the extension of Γ\Gamma by (a free variable of) the type TT is the context denoted

Γ,x:T \Gamma, x\colon T

(where xx is a new variable).

(We have said ‘the’ extension of Γ\Gamma by TT using the generalised the; but it may literally be unique using certain conventions for handling alpha equivalence.)

Categorical semantics

The categorical semantics of context extension is the inverse image of the base change geometric morphism (or its analog for hyperdoctrines) along the projection morphism TΓT \to \Gamma in the slice H /Γ\mathbf{H}_{/\Gamma}

(H /Γ) /T x:T()×T x:TH /Γ (\mathbf{H}_{/\Gamma})_{/T} \stackrel{\overset{\prod_{x : T}}{\longrightarrow}}{\stackrel{\overset{(-)\times T}{\longleftarrow}}{\underset{\sum_{x : T}}{\longrightarrow}}} \mathbf{H}_{/\Gamma}

Generally speaking, a morphism ΔΓ\Delta \to \Gamma in the category of contexts (an interpretation of Γ\Gamma in Δ\Delta) is a display morphism iff there is an isomorphism ΔΘ\Delta \leftrightarrow \Theta where Θ\Theta is an extension of Γ\Gamma. (This might not actually be true in all type theories, or maybe it should be taken as the definition of ‘display morphism’; I'm not sure.)

Notions of pullback:

References

The observation that context extension forms an adjoint pair/adjoint triple with quantifiers is due to

  • Bill Lawvere, Adjointness in Foundations, (TAC), Dialectica 23 (1969), 281-296

and further developed in

  • Bill Lawvere, Equality in hyperdoctrines and

    comprehension schema as an adjoint functor_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.

Last revised on December 2, 2020 at 17:30:15. See the history of this page for a list of all contributions to it.