nLab localization of a type at a family of functions

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

Induction

Contents

Idea

On localization in homotopy type theory.

Definition

Consider a family F: (i:I)S(i)T(i)F:\prod_{(i:I)} S(i) \to T(i) of functions. We say that a type XX is FF-local if the function

λg.g(F(i)):(S(i)X)(T(i)X)\lambda g . g (F(i)) : (S(i) \to X) \to (T(i) \to X)

is an equivalence for all (i : I).

The following higher inductive type can be shown to be a reflection of all types into the local types, constructing the localization of the category of types at the given family of functions.

Inductive localize X :=
| to_local : X -> localize X
| local_extend : forall (i:I) (h : S i -> localize X),
    T i -> localize X
| local_extension : forall (i:I) (h : S i -> localize X) (s : S i),
    local_extend i h (f i s) == h s
| local_unextension : forall (i:I) (g : T i -> localize X) (t : T i),
    local_extend i (g o f i) t == g t
| local_triangle : forall (i:I) (g : T i -> localize X) (s : S i),
    local_unextension i g (f i s) == local_extension i (g o f i) s.

The first constructor gives a function from X to localize X, while the other four specify exactly that localize X is local (by giving adjoint equivalence data to the function that we want to become an equivalence). See this blog post for details. This construction is also already interesting in extensional type theory.

 See also

References

Last revised on June 9, 2022 at 02:40:23. See the history of this page for a list of all contributions to it.