Amnestic functor

☆ Save On Wikipedia ↗

In the mathematical field of category theory, an amnestic functor F : A → B is a functor for which an A-isomorphism ƒ is an identity whenever Fƒ is an identity.

An example of a functor which is not amnestic is the forgetful functor Metc→Top from the category of metric spaces with continuous functions for morphisms to the category of topological spaces. If d 1 {\displaystyle d_{1}} {\displaystyle d_{1}} and d 2 {\displaystyle d_{2}} {\displaystyle d_{2}} are equivalent metrics on a space X {\displaystyle X} {\displaystyle X} then id : ( X , d 1 ) → ( X , d 2 ) {\displaystyle \operatorname {id} \colon (X,d_{1})\to (X,d_{2})} {\displaystyle \operatorname {id} \colon (X,d_{1})\to (X,d_{2})} is an isomorphism that covers the identity, but is not an identity morphism (its domain and codomain are not equal).

References