nLab set truncation

Redirected from "set truncations".

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
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

semantics

Contents

Idea

Set truncation is an operation in type theory which turns a type into an h-set. The idea is that given a type TT, the type of connected components of TT is a set.

Definition

With propositional truncations and quotient sets

Suppose that the dependent type theory has propositional truncations and quotient sets. Then given a type TT, the type family R(x,y)R(x, y) indexed by x:Tx:T and y:Ty:T defined by the propositional truncation of the identity type of TT, R(x,y)[Id T(x,y)] 1R(x, y) \equiv [\mathrm{Id}_T(x, y)]_{-1} is an equivalence relation. The set truncation of TT is the quotient set of TT by RR: [T] 0T/R[T]_0 \equiv T / R.

With localizations and the circle type

Suppose that dependent type theory has localizations of types and the circle type S 1S^1. Then the set truncation of a type TT is the localization of TT at S 1S^1: [T] 0L S 1(T)[T]_0 \equiv L_{S^1}(T)

As a higher inductive type

As a higher inductive type, the set truncation of a type AA is a type [A] 0[A]_0 inductively generated by

  • A function [] 0:A[A] 0[-]_0:A \to [A]_0

  • For each point x:[A] 0x:[A]_0 and path p:x= [A] 0xp:x =_{[A]_0} x, a 2-path K [A] 0(x,p):p= x= [A] 0xrefl A(x)K_{[A]_0}(x, p):p =_{x =_{[A]_0} x} \mathrm{refl}_A(x)

Equivalently, it is the higher inductive type generated by

  • A function [] 0:A[A] 0[-]_0:A \to [A]_0

  • For each function f:S 1[A] 0f:S^1 \to [A]_0 from the circle type, a 2-path K f:ap f(loop)= f(base)= [A] 0f(base)ap f(refl S 1(base))K_f:\mathrm{ap}_f(\mathrm{loop}) =_{f(\mathrm{base}) =_{[A]_0} f(\mathrm{base})}\mathrm{ap}_f(\mathrm{refl}_{S^1}(\mathrm{base}))

Definitional set truncations

There are also definitional or judgmental versions of set truncations, similar to how there are definitional and propositional versions of bracket types and squash types:

The definitional set truncation or judgmental set truncation of a type AA is a type [A] 0[A]_0 inductively generated by

  • A function [] 0:A[A] 0[-]_0:A \to [A]_0

  • For each point x:[A] 0x:[A]_0 and path p:x= [A] 0xp:x =_{[A]_0} x, a definitional equality prefl [A] 0(x):x= [A] 0xp \equiv \mathrm{refl}_{[A]_0}(x):x =_{[A]_0} x

The recursion principle of definitional set truncations says that given a type BB with

  • A function f:ABf:A \to B

  • For each point y:By:B and path q:y= Byq:y =_{B} y, a definitional equality qrefl B(y):y= Byq \equiv \mathrm{refl}_B(y):y =_{B} y

there exists a unique function u B:[A] 0Bu_B:[A]_0 \to B such that u B([a] 0)=f(a)u_B([a]_0) = f(a) for all a:Aa:A.

Equivalently, the definitional set truncation of a type AA is generated by

  • A function [] 0:A[A] 0[-]_0:A \to [A]_0

  • For each function l:S 1[A] 0l:S^1 \to [A]_0 from the circle type, a definitional equality

ap l(loop)ap l(refl S 1(base)):l(base)= [A] 0l(base)\mathrm{ap}_l(\mathrm{loop}) \equiv \mathrm{ap}_l(\mathrm{refl}_{S^1}(\mathrm{base})):l(\mathrm{base}) =_{[A]_0} l(\mathrm{base})

Similarly, the recursion principle of this kind of definitional set truncations says that given a type BB with

  • A function f:ABf:A \to B

  • For each function l f:S 1Bl_f:S^1 \to B from the circle type, a definitional equality

ap l f(loop)ap l f(refl S 1(base)):l f(base)= Bl f(base)\mathrm{ap}_{l_f}(\mathrm{loop}) \equiv \mathrm{ap}_{l_f}(\mathrm{refl}_{S^1}(\mathrm{base})):l_f(\mathrm{base}) =_{B} l_f(\mathrm{base})

there exists a unique function u B:[A] 0Bu_B:[A]_0 \to B such that u B([a] 0)=f(a)u_B([a]_0) = f(a) for all a:Aa:A.

Inference rules for set truncations

Formation rules for set truncations:

ΓAtypeΓ[A] 0type\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash [A]_0 \; \mathrm{type}}

Introduction rules for set truncations:

ΓAtypeΓsettrunc A:A[A] 0type\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{settrunc}_A:A \to [A]_0 \; \mathrm{type}}

Dependent universal property of set truncations:

ΓAtypeΓ,x:[A] 0B(x)typeΓdup [A] 0 B:( x:[A] 0isSet(B(x)))isEquiv(λf: x:[A] 0B(x).fsettrunc A)\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:[A]_0 \vdash B(x) \; \mathrm{type}}{\Gamma \vdash \mathrm{dup}_{[A]_0}^{B}:\left(\prod_{x:[A]_0} \mathrm{isSet}(B(x))\right) \to \mathrm{isEquiv}\left(\lambda f:\prod_{x:[A]_0} B(x).f \circ \mathrm{settrunc}_A\right)}

With a choice of unique representatives

A choice of unique representatives for propositional truncation consists of a type family C(x)C(x) indexed by x:Ax:A, and an element of type

x:A!y:A.C(x)×[x= Ay]\prod_{x:A} \exists!y:A.C(x) \times [x =_A y]

Then the type x:AC(x)\sum_{x:A} C(x) is the set truncation of AA.

Equivalently, one could use functions instead of type families, and say that a choice of unique representatives is a type CC with a function f:CAf:C \to A and an element of type

x:A!y:A.( z:Cf(z)= Ax)×[x= Ay]\prod_{x:A} \exists!y:A.\left(\sum_{z:C} f(z) =_A x\right) \times [x =_A y]

Then

C x:A z:Cf(z)= AxC \simeq \sum_{x:A} \sum_{z:C} f(z) =_A x

is the set truncation of AA.

Properties

 Universal property of set truncations

The universal property of set truncations says that given a type AA, a set BB, and a function f:ABf:A \to B, there exists a unique function u:[A]Bu:[A] \to B such that for all x:Ax:A, f(x)=u([x] 0)f(x) = u([x]_0).

Power sets

Given a type AA and a subtype P:APropP:A \to \mathrm{Prop}, by propositional extensionality and the universal property of set truncations, one can construct a subtype [P] 0:[A] 0Prop[P]_0:[A]_0 \to \mathrm{Prop} of the set truncation of AA such that for all x:Ax:A, P(x)=[P] 0([x] 0)P(x) = [P]_0([x]_0), .

References

For set truncations in dependent type theory, see section 18.5 of:

and section 6.9 and 7.3 of:

see also:

Last revised on September 13, 2024 at 22:52:38. See the history of this page for a list of all contributions to it.