bijection set (changes) in nLab

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Definition

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.

 Properties

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 hasinjection 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 X→Bij(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×X≅B×XA \times X \cong B \times X in the slice category Set/X\mathrm{Set}/X.

Generalisations

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

• equivalence type

• object of isomorphisms

• symmetric group

• function set, injection set

computational trinitarianism = \linebreak propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language