nLab
context extension

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

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
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

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}}{\to}}{\stackrel{\overset{(-)\times T}{\leftarrow}}{\underset{\sum_{x : T}}{\to}}} \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.)

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.

Revised on November 23, 2012 02:26:33 by Urs Schreiber (82.169.65.155)