nLab coequalizer 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




The instantiation of a coequalizer as a type in dependent type theory.


In dependent type theory, given types AA and BB and functions f:ABf:A \to B and g:ABg:A \to B, the coequalizer type or coequaliser type of ff and gg is a higher inductive type coeq A,B(f,g)\mathrm{coeq}_{A,B}(f, g). There are two different way to define the coequalizer type. The first way states that coeq A,B(f,g)\mathrm{coeq}_{A,B}(f, g) is generated by the constructors

in:Bcoeq A,B(f,g)\mathrm{in} \colon B \to \mathrm{coeq}_{A,B}(f, g)
glue: x:Ain(f(x))= coeq A,B(f,g)in(g(x))\mathrm{glue} \colon \prod_{x:A}in(f(x)) =_{\mathrm{coeq}_{A,B}(f, g)} in(g(x))

The second way states that coeq A,B(f,g)\mathrm{coeq}_{A,B}(f, g) is generated by

in:Bcoeq A,B(f,g)\mathrm{in} \colon B \to \mathrm{coeq}_{A,B}(f, g)
glue: x:B y:B z:A(f(z)= Bx)(g(z)= By)(in(x)= coeq A,B(f,g)in(y))\mathrm{glue} \colon \prod_{x:B} \prod_{y:B} \prod_{z:A} (f(z) =_B x) \to (g(z) =_B y) \to (\mathrm{in}(x) =_{\mathrm{coeq}_{A,B}(f, g)} \mathrm{in}(y))

Induction and recursion principles

The recursion principle for coequalizer types says that given a type CC together with

  • a function c in:BCc_\mathrm{in}:B \to C
  • a dependent function c glue: x:B y:B z:A(f(z)= Bx)(g(z)= By)(c in(x)= Cc in(y))c_\mathrm{glue}:\prod_{x:B} \prod_{y:B} \prod_{z:A} (f(z) =_B x) \to (g(z) =_B y) \to (c_\mathrm{in}(x) =_C c_\mathrm{in}(y))

there exists a function c:coeq A,B(f,g)Cc:\mathrm{coeq}_{A, B}(f, g) \to C such that

  • for all x:Bx:B, c(in(x))c in(x)c(\mathrm{in}(x)) \equiv c_\mathrm{in}(x), and
  • for all x:Bx:B, y:By:B, z:Az:A, p:f(z)= Bxp:f(z) =_B x, and q:g(z)= Byq:g(z) =_B y,
    ap c(glue(x,y,z,p,q))c glue(x,y,z,p,q)\mathrm{ap}_c(\mathrm{glue}(x,y,z,p,q)) \equiv c_\mathrm{glue}(x, y, z, p, q)

Similarly, the induction principle says that given a type family C(w)C(w) indexed by w:coeq A,B(f,g)w:\mathrm{coeq}_{A, B}(f, g) together with

  • a dependent function c in: x:BC(in(x))c_\mathrm{in}:\prod_{x:B} C(\mathrm{in}(x))
  • a dependent function c glue: x:B y:B z:A p:f(z)= Bx q:g(z)= Byc in(x)= C glue(x,y,z,p,q)c in(y)c_\mathrm{glue}:\prod_{x:B} \prod_{y:B} \prod_{z:A} \prod_{p:f(z) =_B x} \prod_{q:g(z) =_B y} c_\mathrm{in}(x) =_C^{\mathrm{glue}(x,y,z,p,q)} c_\mathrm{in}(y)

there exists a dependent function c: z:coeq A,B(f,g)C(z)c:\prod_{z:\mathrm{coeq}_{A, B}(f, g)} C(z) such that

  • for all x:Bx:B, c(in(x))c points(x)c(\mathrm{in}(x)) \equiv c_\mathrm{points}(x), and
  • for all x:Bx:B, y:By:B, z:Az:A, p:s(z)= Bxp:s(z) =_B x, and q:g(z)= Byq:g(z) =_B y,
    apd c(glue(x,y,z,p,q))c glue(x,y,z,p,q)\mathrm{apd}_c(\mathrm{glue}(x,y,z,p,q)) \equiv c_\mathrm{glue}(x,y,z,p,q)


Relation to W-suspensions

W-suspensions could be constructed as the coequalizer type 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 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)


  • The sum type of a type TT with itself is the coequalizer of two copies of the unique function rec T\mathrm{rec}_\emptyset^T from the empty type to TT.
    T+Tcoeq ,T(rec T,rec T)T + T \coloneqq \mathrm{coeq}_{\emptyset, T}(\mathrm{rec}_\emptyset^T, \mathrm{rec}_\emptyset^T)
  • The circle type is the coequalizer of any two endofunctions f:CCf:C \to C and g:CCg:C \to C on a contractible type CC.
    S 1coeq C,C(f,g)givenp:isContr(C)S^1 \coloneqq \mathrm{coeq}_{C, C}(f, g) \quad \mathrm{given} \quad p:\mathrm{isContr}(C)
  • The interval type is the coequalizer of the constant functions with value false\mathrm{false} and true\mathrm{true} from any contractible type CC to the boolean domain.
    𝕀coeq C,bool(const C,bool(false).const C,bool(true))givenp:isContr(C)\mathbb{I} \coloneqq \mathrm{coeq}_{C, \mathrm{bool}}(\mathrm{const}_{C, \mathrm{bool}}(\mathrm{false}). \mathrm{const}_{C, \mathrm{bool}}(\mathrm{true})) \quad \mathrm{given} \quad p:\mathrm{isContr}(C)
  • More generally, the suspension of a type TT is the coequalizer of the constant functions with value false\mathrm{false} and true\mathrm{true} from TT to the boolean domain.
    ΣTcoeq T,bool(const T,bool(false).const T,bool(true))\Sigma T \coloneqq \mathrm{coeq}_{T, \mathrm{bool}}(\mathrm{const}_{T, \mathrm{bool}}(\mathrm{false}). \mathrm{const}_{T, \mathrm{bool}}(\mathrm{true}))
  • The cyclic group of order n+1n + 1 for natural number n:n:\mathbb{N} is the coequalizer of the constant functions with value 00 and n+1n + 1 from any contractible type CC to the natural numbers type.
    n+1coeq C,(const C,(0),const C,(n+1))givenp:isContr(C)\mathbb{Z}_{n + 1} \coloneqq \mathrm{coeq}_{C, \mathbb{N}}(\mathrm{const}_{C, \mathbb{N}}(0), \mathrm{const}_{C, \mathbb{N}}(n + 1)) \quad \mathrm{given} \quad p:\mathrm{isContr}(C)
  • The pushout of a span f:CAf:C \to A and g:CBg:C \to B is the coequalizer of the composite functions in A+B Af\mathrm{in}_{A + B}^A \circ f and in A+B Bg\mathrm{in}_{A + B}^B \circ g, each with domain CC and codomain the sum type A+BA + B:
    A f,g CBcoeq C,A+B(in A+B Af,in A+B Bg)A \sqcup_{f,g}^C B \coloneqq \mathrm{coeq}_{C, A + B}(\mathrm{in}_{A + B}^A \circ f, \mathrm{in}_{A + B}^B \circ g)

Propositional truncation

The propositional truncation of a h-set AA is the set truncation of the coequalizer of the two product projection functions π 1:A×AA\pi_1:A \times A \to A and π 2:A×AA\pi_2:A \times A \to A of the product type A×AA \times A

A 1coeq A×A,A(π 1,π 2) 0\Vert A \Vert_{-1} \coloneqq \Vert \mathrm{coeq}_{A \times A,A}(\pi_1, \pi_2) \Vert_0

See also


Last revised on December 29, 2023 at 22:56:12. See the history of this page for a list of all contributions to it.