equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
identity type, equivalence of types, definitional isomorphism
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
A natural isomorphism between two functors and
is equivalently
a natural transformation with a two-sided inverse;
a natural transformation each of whose components for all is an isomorphism in ;
an isomorphism in the functor category .
In this case, we say that and are naturally isomorphic. (Synonym: and are isomorphic functors; the naturality is understood when one says that two functors are isomorphic.)
Notably, especially in expositions and lectures, despite the third bullet point above, one does not need to define the concept of a functor category in order to define isomorphic functors. Authors sometimes make cautionary remarks about the category of all functors from to (cf. e.g. Auslander (1971, p.9))
If you want to speak of ‘the’ functor satisfying certain conditions, then it should be unique up to unique natural isomorphism.
A natural isomorphism from a functor to itself is also called a natural automorphism.
A fundamental use of the concept of isomorphic functors is the usual definition of equivalent categories which involves functors isomorphic to identity functors.
The Yoneda lemma implies that in any category , and for any objects and of , the following are equivalent:
and are isomorphic in ,
the representable presheaves and are isomorphic functors,
the representable copresheaves and are isomorphic functors.
Queen Mary College Mathematics Notes (1971) Republished in: Selected works of Maurice Auslander. American Mathematical Society (1999)
Last revised on December 7, 2023 at 03:50:08. See the history of this page for a list of all contributions to it.