nLab localization of a type


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





On localization in homotopy type theory.


Localization at a function

Consider a function f:STf:S \to T. We say that a type XX is ff-local if the function

λg.g(f):(TX)(SX)\lambda g . g (f) : (T \to X) \to (S \to X)

is an equivalence of types.

The localization of a type XX at ff, L f(X)L_f(X), is the higher inductive type generated by

  • the function tolocal:XL f(X)\mathrm{tolocal}:X \to L_f(X)
  • the function localextend:(SL f(X))(TL f(X))\mathrm{localextend}: (S \to L_f(X)) \to (T \to L_f(X))
  • the dependent function
    localextension: h:SL f(X) s:Slocalextend(h)(f(s))=h(s)\mathrm{localextension}:\prod_{h:S \to L_f(X)} \prod_{s:S} \mathrm{localextend}(h)(f(s)) = h(s)
  • the dependent function
    localunextension: g:TL f(X) t:Tlocalextend(gf)(t)=g(t)\mathrm{localunextension}:\prod_{g:T \to L_f(X)} \prod_{t:T} \mathrm{localextend}(g \circ f)(t) = g(t)
  • the dependent function
    localtriangle: g:TL f(X) s:Slocalunextension(g)(f(s))=localextension(gf)(s)\mathrm{localtriangle}:\prod_{g:T \to L_f(X)} \prod_{s:S} \mathrm{localunextension}(g)(f(s)) = \mathrm{localextension}(g \circ f)(s)

Localization at a type

The localization L S(X)L_S(X) of a type XX at a type SS is the localization L u S 𝟙L_{u_S^\mathbb{1}} of XX at the unique function u S 𝟙:S𝟙u_S^\mathbb{1}:S \to \mathbb{1} from SS to the unit type 𝟙\mathbb{1}, or equivalently, is the type L S(X)L_S(X) for which the canonical function const:L S(X)(SL S(X))\mathrm{const}:L_S(X) \to (S \to L_S(X)) is an equivalence of types.

Localization at a family of functions

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 of types 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.

Localization at a family of types

The localization L F(X)L_F(X) of a type XX at a family of types F: i:IS(i)F:\prod_{i:I} S(i) is the family of localizations L F() i:IL u S(i) 𝟙()L_F(-) \coloneqq \prod_{i:I} L_{u_{S(i)}^\mathbb{1}}(-) of XX at the family of unique functions u S(i) 𝟙:S(i)𝟙u_{S(i)}^\mathbb{1}:S(i) \to \mathbb{1} from S(i)S(i) to the unit type 𝟙\mathbb{1}, or equivalently, is the family of types L S(i)(X)L_{S(i)}(X) for which each canonical function const:L S(i)(X)(S(i)L S(i)(X))\mathrm{const}:L_{S(i)}(X) \to (S(i) \to L_{S(i)}(X)) is an equivalence of types.


 See also


Last revised on December 26, 2023 at 18:00:33. See the history of this page for a list of all contributions to it.