nLab object of isomorphisms



In the same way that internal homs in a cartesian monoidal category behave as an internal version of hom-sets between two objects, the object of isomorphisms should behave as an internal version of the sub-hom-set of isomorphisms between two objects.

These generalize bijection sets in the category of sets Set.


Given a cartesian monoidal category CC and two objects ACA \in C and BCB \in C, there is a functor

XHom Core(C/X)(A×X,B×X)X \mapsto \mathrm{Hom}_{\mathrm{Core}(C/X)}(A \times X, B \times X)

which takes an object XCX \in C to a morphism between objects A×XA \times X and B×XB \times X in the core of the slice category C/XC/X. A representing object of the above functor, if it exists, is the object of isomorphisms between AA and BB and is denoted as [A,B] iso[A, B]_\mathrm{iso}.


If the category is a cartesian closed category, then there is a monomorphism i:Mono C([A,B] iso,[A,B])i:\mathrm{Mono}_C([A, B]_\mathrm{iso}, [A, B]) between the object of isomorphisms between AA and BB and the internal hom from AA to BB.

 See also

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 20:55:29. See the history of this page for a list of all contributions to it.