nLab monoidal symmetric proset



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


Category theory


Monoid theory



In set theory, algebraic structures such as monoids are defined as a set with additional structure and equational axioms. However, in homotopy type theory, there are two possible notions of what it means for a type to be a set, and what it means that a set has equational axioms. Traditionally in homotopy type theory, a set is defined using the identity type, where every identity type between two elements a:Aa:A and b:Ab:A of a type AA is propositionally truncated. However, equality in set theory is usually defined as an equivalence relation, and one could directly translate the equivalence relation of equality into homotopy type theory, resulting in a symmetric proset. The equational axioms of algebraic structures could similarly be defined using the equivalence relation of the symmetric proset rather than the identity type. In the context of defining monoids, this results in the notion of a monoidal symmetric proset.


In homotopy type theory, a monoidal symmetric proset is a symmetric proset (M, M)(M, \equiv_M) with a binary function ()():M×MM(-)\cdot(-):M \times M \to M, an element 1:M1:M, and witnesses of associativity, left unitality, right unitality, and extensionality

a:M,b:M,c:Massoc(a,b,c):(ab)c Ma(bc)a:M, b:M, c:M \vdash \mathrm{assoc}(a, b, c):(a \cdot b) \cdot c \equiv_M a \cdot (b \cdot c)
a:Mlunit(a):1a Maa:M \vdash \mathrm{lunit}(a):1 \cdot a \equiv_M a
a:Mrunit(a):a1 Maa:M \vdash \mathrm{runit}(a):a \cdot 1 \equiv_M a
a:M,b:M,c:M,d:Mext(a,b,c,d):(a Mb)×(c Md)(ac) M(bd)a:M, b:M, c:M, d:M \vdash \mathrm{ext}(a, b, c, d):(a \equiv_M b) \times (c \equiv_M d) \to (a \cdot c) \equiv_M (b \cdot d)

A monoidal symmetric proset is univalent or a monoid if the canonical function

idtoequiv(a,b):(a= Mb)(a Mb)idtoequiv(a, b):(a =_M b) \to (a \equiv_M b)

is an equivalence of types for all elements a:Ma:M and b:Mb:M.


Every monoidal symmetric proset is (the delooping of) a (2, 1)-preorder? with only 1 object.

See also

Last revised on September 22, 2022 at 19:09:09. See the history of this page for a list of all contributions to it.