nLab bijection set



Given sets AA and BB, the bijection set Bij(A,B)\mathrm{Bij}(A, B) is the set of bijections between the set AA and BB. In the foundations of mathematics, the existence of such a set may be taken to follow from the existence of power sets, from the axiom of subset collection, from the existence of function sets, from the existence of injection sets, or as an axiom (the axiom of bijection sets) in its own right.


If the set theory has function sets, then the bijection set between AA and BB is a subset of the function set between AA and BB: Bij(A,B)B A\mathrm{Bij}(A, B) \subseteq B^A. Similarly, if the set theory has injection sets, then the bijection set between AA and BB is a subset of the injection set between AA and BB: Bij(A,B)Inj(A,B)\mathrm{Bij}(A, B) \subseteq \mathrm{Inj}(A, B)

The bijection set between AA and itself is the symmetric group on AA, Sym(A)Bij(A,A)\mathrm{Sym}(A) \coloneqq \mathrm{Bij}(A, A).

In the context of the axiom of choice or stronger axioms such as a choice operator satisfying the axiom schemata of existence, having bijection sets implies having power sets in the foundations, since the axiom of choice implies that for every set AA, there is a bijection 𝒫(A)Sym(A)\mathcal{P}(A) \cong \mathrm{Sym}(A) between the power set of AA and the symmetric group on AA. Thus, bijection sets are a form of impredicativity from the point of view of mathematics with the axiom of choice.

The universal property of the bijection set states that given a function XBij(A,B)X \to \mathrm{Bij}(A, B) from a set XX to the bijection set between sets AA and BB, there exists a bijection A×XB×XA \times X \cong B \times X in the slice category Set/X\mathrm{Set}/X.


Thinking of Set as a locally small category, this is a special case of the hom-set of the core of Set.

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

Last revised on January 10, 2023 at 20:54:43. See the history of this page for a list of all contributions to it.