nLab presetoid



Category theory

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


Graph theory



In set theory, a setoid is usually described as a set with a pseudo-equivalence relation. However, in homotopy type theory or more generally in intensional type theory, there are multiple different ways to define the concept of a “collection with a pseudo-equivalence relation”. Presetoids arise from the most general possible notion of “collection with a pseudo-equivalence relation” in homotopy type theory , in the same way that the notion of precategory and pregroupoid arise when defining categories and groupoids.


A pseudo-equivalence relation on a type AA is a family of sets Hom(a,b)Hom(a, b) of morphisms for each element a:Aa:A and b:Ab:A, which comes with the following additional structure

  • for each element a:Aa:A a family of morphisms

    id(a):Hom(a,a)\mathrm{id}(a):Hom(a, a)
  • for each object a:Aa:A and b:Ab:A a family of functions

    dagger(a,b):Hom(a,b)Hom(b,a)\mathrm{dagger}(a, b):Hom(a, b) \to Hom(b, a)
  • for each object a:Aa:A, b:Ab:A, and c:Ac:A, a family of functions

    comp(a,b,c):(Hom(a,b)×Hom(b,c))Hom(a,c)\mathrm{comp}(a, b, c):(Hom(a, b) \times Hom(b, c)) \to Hom(a, c)

A presetoid AA consists of a type Ob(A)Ob(A) of objects and a pseudo-equivalence relation (Hom,id,dagger,comp)(\mathrm{Hom}, \mathrm{id}, \mathrm{dagger}, \mathrm{comp}).

Dagger functors

The structure-preserving maps between two presetoids are called dagger-functors. A dagger-functor F:ABF:A \to B between two presetoids AA and BB consists of

  • a function f Ob:Ob(A)Ob(B)f_{Ob}:\mathrm{Ob}(A) \to \mathrm{Ob}(B)

  • a family of functions f hom(a,b):Hom A(a,b)Hom B(f Ob(a),f Ob(b))f_{hom}(a, b):\mathrm{Hom}_A(a, b) \to \mathrm{Hom}_B(f_{Ob}(a), f_{Ob}(b))

  • and identifications indicating that the identity morphism, dagger, and the composition operation are preserved:

idpreserve(a):f Hom(a,a)(id A(a))=id B(f Ob(a))\mathrm{idpreserve}(a):f_{Hom}(a, a)(\mathrm{id}_A(a)) = \mathrm{id}_B(f_{Ob}(a))
daggerpreserve(a,b,f):f Hom(a,b)(dagger A(a,b)(f))=dagger B(f Ob(a),f Ob(b))(f Hom(a,b)(f))\mathrm{daggerpreserve}(a, b, f):f_{Hom}(a, b)(\mathrm{dagger}_A(a, b)(f)) = \mathrm{dagger}_B(f_{Ob}(a), f_{Ob}(b))(f_{Hom}(a, b)(f))
comppreserve(a,b,c,f,g):f E(a,c)(comp A(f,g))=comp B(f Hom(a,b)(f),f Hom(b,c)(g)\mathrm{comppreserve}(a, b, c, f, g):f_E(a, c)(\mathrm{comp}_A(f,g)) = \mathrm{comp}_B(f_{Hom}(a, b)(f), f_{Hom}(b, c)(g)

An dagger functor between two presetoids is faithful if every morphism function f Hom(a,b):Hom A(a,b)Hom B(f Ob(a),f Ob(b))f_{Hom}(a, b):\mathrm{Hom}_A(a, b) \to \mathrm{Hom}_B(f_{Ob}(a), f_{Ob}(b)) is an injection. A dagger functor is embedding-on-objects if the object function f V:V AV Bf_V:V_A \to V_B is an embedding, and equivalent-on-objects if the object function is a equivalence of types.

Core of a presetoid

A subpresetoid GG of a presetoid AA is a presetoid GG with an faithful embedding-on-objects dagger functor f:GAf:G \to A. A subpresetoid GG of AA is a subpregroupoid if the pseudo-equivalence relation of GG additionally come with:

  • for every object a:Ob(G)a:Ob(G), b:Ob(G)b:Ob(G), c:Ob(G)c:Ob(G), and d:Ob(G)d:Ob(G) and morphism f:Hom G(a,b)f:Hom_G(a, b), g:Hom G(b,c)g:Hom_G(b, c), and h:Hom G(c,d)h:Hom_G(c, d), an identification

    assoc(a,b,c,d,fg,h):comp(a,b,d)(f,comp(b,c,d)(g,h))=comp(a,c,d)(comp(a,b,c)(f,g),h)\mathrm{assoc}(a, b, c, d, f g, h):\mathrm{comp}(a, b, d)(f, \mathrm{comp}(b, c, d)(g, h)) = \mathrm{comp}(a, c, d)(\mathrm{comp}(a, b, c)(f, g), h)
  • for every object a:Ob(G)a:Ob(G), b:Ob(G)b:Ob(G) and morphism f:Hom G(a,b)f:Hom_G(a, b), an identification

    runital(a,b,f):comp(a,b,b)(f,id(b))=f\mathrm{runital}(a, b, f):\mathrm{comp}(a, b, b)(f, \mathrm{id}(b)) = f
  • for every object a:Ob(G)a:Ob(G), b:Ob(G)b:Ob(G) and morphism f:Hom G(a,b)f:Hom_G(a, b), an identification

    lunital(a,b,f):comp(a,a,b)(id(a),f)=f\mathrm{lunital}(a, b, f):\mathrm{comp}(a, a, b)(\mathrm{id}(a), f) = f
  • for every object a:Ob(G)a:Ob(G), b:Ob(G)b:Ob(G) and morphism f:Hom G(a,b)f:Hom_G(a, b), an identification

    sec(a,b,f):comp(a,b,a)(f,dagger(a,b)(f))=id(a)\mathrm{sec}(a, b, f):\mathrm{comp}(a, b, a)(f, \mathrm{dagger}(a, b)(f)) = \mathrm{id}(a)
  • for every object a:Ob(G)a:Ob(G), b:Ob(G)b:Ob(G) and morphism f:Hom G(a,b)f:Hom_G(a, b), an identification

    ret(a,b,f):comp(b,a,b)(dagger(a,b)(f),f)=id(b)\mathrm{ret}(a, b, f):\mathrm{comp}(b, a, b)(\mathrm{dagger}(a, b)(f), f) = \mathrm{id}(b)

A subpregroupoid GG of a presetoid AA is a maximal subpregroupoid if the dagger functor f:GAf:G \to A is equivalent-on-objects, and additionally if, for every other subgroupoid HH of AA with a faithful embedding-on-objects dagger functor g:HAg:H \to A, the type of faithful embedding-on-objects dagger functors h:HGh:H \to G with an identification up(f,g,h):hf=g\mathrm{up}(f, g, h):h \circ f = g is contractible.

The core pregroupoid Core(A)Core(A) of a setoid AA is the maximal subpregroupoid of AA. Since the core is a pregroupoid, every morphism in the core is an isomorphism, and the type of morphisms between two objects aa and bb in the core of AA is denoted as a Aba \cong_A b.

Univalent setoids

There are two ways to show that two objects of a presetoid are identified with each other: by way of the identity type, and by way of the type of isomorphisms in the core pregroupoid of a presetoid. Univalent setoids are precisely the presetoids which the two notions of identifying objects with each other coincide.

For each object a:Ob(A)a:Ob(A) and b:Ob(A)b:Ob(A) of a presetoid AA, there is a function

idtoiso(a,b):(a= Ab)(a Ab)\mathrm{idtoiso}(a, b):(a =_A b) \to (a \cong_A b)

from the identity type to the type of isomorphisms in the core pregroupoid Core(A)\mathrm{Core}(A) of AA. A presetoid is a univalent setoid if for each object a:Ob(A)a:Ob(A) and b:Ob(A)b:Ob(A) the function

idtoiso(a,b):(a= Ab)(a Ab)\mathrm{idtoiso}(a, b):(a =_A b) \simeq (a \cong_A b)

is an equivalence of types.

Since every type of morphisms in the core of the presetoid is a h-set, this means that every univalent setoid is an h-groupoid.

See also

Last revised on September 22, 2022 at 11:47:18. See the history of this page for a list of all contributions to it.