nLab enriched type

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

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

semantics

Enriched category theory

(,1)(\infty,1)-Category theory

Contents

Idea

In homotopy type theory, types represent \infty -groupoids, so there should be something similar to enriched \infty-groupoids in homotopy type theory. The following definition is an experimental definition of such an object:

Definition

Let CC be the subtype of types in a type universe 𝒰\mathcal{U} which satisfy a certain predicate PP: for all types T:CT:C, there is a term p:P(T)p:P(T).

Then a type TT is CC-enriched if for all terms a:Ta:T and b:Tb:T, there is a dependent term p(a,b):P(a=b)p(a,b):P(a = b), where a=ba = b is the identity type of aa and bb in TT.

Examples

  • One could define Ab Ab -enriched types, which are types whose identity types are all abelian groups (usually defined with a predicate isAbelianGroupisAbelianGroup).

  • Similarly, sets are Prop Prop -enriched types, and n n -types are (n1)(n - 1)Type-enriched types.

  • A (extended) lower Dedekind cut could be defined as a set U:𝒰U:\mathcal{U} with an injection i:Ui:U \hookrightarrow \mathbb{Q} which satisfy certain axioms, which are detailed in the article on Dedekind cuts. Then, one could define a (extended) Richman premetric space as a type whose identity types are (extended) lower Dedekind cuts, which in classical mathematics is the same as a (extended) metric space.

Categorical semantics

Under the relation between type theory and category theory, enriched types should correspond to enriched \infty -groupoids in higher category theory.

See also

Last revised on June 16, 2022 at 07:25:59. See the history of this page for a list of all contributions to it.