nLab W-suspension

Redirected from "W-suspensions".

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

A W-suspension is similar to a W-type, except instead of being inductively generated by term constructors and function constructors, W-suspensions are inductively generated by term constructors and identification constructors.

 Definition

There are multiple possible definitions of a (directed, pseudo-) graph in dependent type theory:

  • a graph consists of a type AA and a binary type family R(x,y)R(x, y) indexed by x:Ax:A and y:Ay:A. AA is the type of vertices of a graph and each R(x,y)R(x, y) is the type of edges of a graph.

  • a graph consists of types AA and BB, a type family R(z)R(z) indexed by z:Bz:B, and functions s:BAs:B \to A and t:BAt:B \to A. AA is the type of vertices of a graph, BB is the type of endpoint configurations of edges in the graph, and R(x)R(x) is the type of edges in the graph given an endpoint configuration x:Bx:B. Given elements x:Ax:A and y:Ay:A, the type of edges between x:Ax:A and y:Ay:A in the graph is the type

z:BR(z)×(s(z)= Ax)×(t(z)= Ay)\sum_{z:B} R(z) \times (s(z) =_A x) \times (t(z) =_A y)
  • a graph consists of a type AA and a type RR and two functions s:RAs:R \to A and t:RAt:R \to A. AA is the type of vertices in AA, and the type of edges the type of edges between x:Ax:A and y:Ay:A in the graph is the type
r:R(s(r)= Ax)×(t(r)= Ay)\sum_{r:R} (s(r) =_A x) \times (t(r) =_A y)

These definitions of graph result in different possible definitions of a W-suspension, whose points are generated by the vertices of the graph and whose paths are generated by the edges of the graph.

W-suspensions of binary type families

The first notion of graph leads to the following higher inductive type:

Given a graph (A,R)(A, R), one could construct the W-suspension Wsusx:A,y:AR(x,y)\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y) which is generated by the graph in the following sense: there are point constructors

points:AWsusx:A,y:AR(x,y)\mathrm{points}:A \to \underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)

and for each x:Ax:A and y:Ay:A, path constructors

paths(x,y):R(x,y)(points(x)= Wsusx:A,y:AR(x,y)points(y))\mathrm{paths}(x, y):R(x, y) \to \left(\mathrm{points}(x) =_{\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y)} \mathrm{points}(y)\right)

The recursion principle of Wsusx:A,y:AR(x,y)\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y) says that given a type CC together with

  • a function c points:ACc_\mathrm{points}:A \to C
  • a dependent function c paths: x:A y:AR(x,y)(c points(x)= Cc points(y))c_\mathrm{paths}:\prod_{x:A} \prod_{y:A} R(x, y) \to (c_\mathrm{points}(x) =_C c_\mathrm{points}(y))

there exists a function c:(Wsusx:A,y:AR(x,y))Cc:\left(\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y)\right) \to C such that

  • for all x:Ax:A, c(points(x))c points(x)c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x), and
  • for all x:Ax:A, y:Ay:A, and r:R(x,y)r:R(x, y), ap c(paths(x,y,r))c paths(x,y,r)\mathrm{ap}_c(\mathrm{paths}(x,y,r)) \equiv c_\mathrm{paths}(x, y, r)

Similarly, the induction principle of Wsusx:A,y:AR(x,y)\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y) says that given a type family C(z)C(z) indexed by z:Wsusx:A,y:AR(x,y)z:\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y) together with

  • a dependent function c points: x:AC(points(x))c_\mathrm{points}:\prod_{x:A} C(\mathrm{points}(x))
  • a dependent function c paths: x:A y:A r:R(x,y)c points(x)= C paths(x,y,r)c points(y)c_\mathrm{paths}:\prod_{x:A} \prod_{y:A} \prod_{r:R(x, y)} c_\mathrm{points}(x) =_C^{\mathrm{paths}(x,y,r)} c_\mathrm{points}(y)

there exists a dependent function c: z:Wsusx:A,y:AR(x,y)C(z)c:\prod_{z:\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y)} C(z) such that

  • for all x:Ax:A, c(points(x))c points(x)c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x), and
  • for all x:Ax:A, y:Ay:A, and r:R(x,y)r:R(x, y), apd c(paths(x,y,r))c paths(x,y,r)\mathrm{apd}_c(\mathrm{paths}(x,y,r)) \equiv c_\mathrm{paths}(x, y, r)

The large recursion principle of Wsusx:A,y:AR(x,y)\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y) says that given

  • a type family x:AC(x)x:A \vdash C(x)
  • a dependent function c equiv: x:A y:AR(x,y)(C(x)C(y))c_\mathrm{equiv}:\prod_{x:A} \prod_{y:A} R(x, y) \to (C(x) \simeq C(y))

there exists a type family z:(Wsusx:A,y:AR(x,y))D(z)z:\left(\underset{x:A, y:A}{\mathrm{Wsus}} R(x, y)\right) \vdash D(z) such that

  • for all x:Ax:A, D(c(x))C(x)D(c(x)) \equiv C(x), and
  • for all x:Ax:A, y:Ay:A, and r:R(x,y)r:R(x, y), tr C(paths(x,y,r))c equiv(x,y,r):C(x)C(y)\mathrm{tr}_C(\mathrm{paths}(x,y,r)) \equiv c_\mathrm{equiv}(x, y, r):C(x) \simeq C(y)

Sojakova W-suspensions

The second notion of graph leads to the following higher inducive type defined in Sojakova15:

Given types AA and BB, a type family R(x)R(x) indexed by x:Bx:B, and functions s:BAs:B \to A and t:BAt:B \to A, one could construct the W-suspension Wsus s,tx:BR(x)\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x) which is generated by the graph in the following sense: there are point constructors

points:AWsus s,tx:BR(x)\mathrm{points}:A \to \underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)

and for each endpoint configuration x:Bx:B, path constructors

paths(x):R(x)(points(s(x))= Wsus s,tx:BR(x)points(t(x)))\mathrm{paths}(x):R(x) \to \left(\mathrm{points}(s(x)) =_{\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)} \mathrm{points}(t(x))\right)

The recursion principle of Wsus s,tx:BR(x)\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x) says that given a type CC together with

  • a function c points:ACc_\mathrm{points}:A \to C
  • a dependent function c paths: x:BR(x)(c points(s(x))= Cc points(t(x)))c_\mathrm{paths}:\prod_{x:B} R(x) \to (c_\mathrm{points}(s(x)) =_C c_\mathrm{points}(t(x)))

there exists a function c:(Wsus s,tx:BR(x))Cc:\left(\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)\right) \to C such that

  • for all x:Ax:A, c(points(x))c points(x)c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x), and
  • for all x:Bx:B, and r:R(x)r:R(x), ap c(paths(s,r))c paths(s,r)\mathrm{ap}_c(\mathrm{paths}(s,r)) \equiv c_\mathrm{paths}(s, r)

Similarly, the induction principle of Wsus s,tx:BR(x)\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x) says that given a type family C(z)C(z) indexed by z:Wsus s,tx:BR(x)z:\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x) together with

  • a dependent function c points: x:AC(points(x))c_\mathrm{points}:\prod_{x:A} C(\mathrm{points}(x))
  • a dependent function c paths: x:B r:R(x)c points(s(x))= C paths(x,r)c points(t(x))c_\mathrm{paths}:\prod_{x:B} \prod_{r:R(x)} c_\mathrm{points}(s(x)) =_C^{\mathrm{paths}(x,r)} c_\mathrm{points}(t(x))

there exists a dependent function c: z:Wsus s,tx:BR(x)C(z)c:\prod_{z:\underset{x:B}{\mathrm{Wsus}^{s, t}} R(x)} C(z) such that

  • for all x:Ax:A, c(points(x))c points(x)c(\mathrm{points}(x)) \equiv c_\mathrm{points}(x), and
  • for all x:Bx:B, and r:R(x)r:R(x), apd c(paths(x,r))c paths(x,r)\mathrm{apd}_c(\mathrm{paths}(x,r)) \equiv c_\mathrm{paths}(x, r)

W-suspensions of parallel functions

For the final notion of graph, the resulting definition of W-suspension is just the coequalizer type of s:RAs:R \to A and t:RAt:R \to A.

Wsus R,A(s,t)coeq R,A(s,t)\mathrm{Wsus}_{R,A}(s, t) \coloneqq \mathrm{coeq}_{R,A}(s, t)

Relation between the definitions

W-suspensions in the first sense could be constructed as the coequalizer of the left and middle dependent pair projections π A\pi_{\sum}^A and π Aπ y:AR(,y)\pi_{\sum}^A \circ \pi_{\sum}^{\sum_{y:A} R(-, y)} of the dependent pair type x:A y:AR(x,y)\sum_{x:A} \sum_{y:A} R(x, y), both of which have domain x:A y:AR(x,y)\sum_{x:A} \sum_{y:A} R(x, y) and codomain AA:

Wsusx:A,y:AR(x,y)coeq x:A y:AR(x,y),A(π A,π Aπ y:AR(,y))\underset{x:A,y:A}{\mathrm{Wsus}} R(x, y) \coloneqq \mathrm{coeq}_{\sum_{x:A} \sum_{y:A} R(x, y),A}(\pi_{\sum}^A, \pi_{\sum}^A \circ \pi_{\sum}^{\sum_{y:A} R(-, y)})

Alternatively, given two parallel functions s:BAs:B \to A and t:BAt:B \to A, the coequalizer of ss and tt could be defined as the W-suspension (in the first sense) generated by the graph defined by the family of dependent sum types z:B(s(z)= Ax)×(t(z)= Ay)\sum_{z:B} (s(z) =_A x) \times (t(z) =_A y)

Wsusx:A,y:A z:B(s(z)= Ax)×(t(z)= Ay)\underset{x:A,y:A}{\mathrm{Wsus}} \sum_{z:B} (s(z) =_A x) \times (t(z) =_A y)

 References

Last revised on February 1, 2024 at 21:04:43. See the history of this page for a list of all contributions to it.