In the mathematical field of category theory, an amnestic functor ''F'' : ''A'' → ''B'' is a

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

for which an ''A''-

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

''ƒ'' is an identity whenever ''Fƒ'' is an identity. An example of a functor which is ''not'' amnestic is the forgetful functor Met_c→Top from the category of

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

s with continuous functions for morphisms to the

category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...

. If

d_1

and

d_2

are equivalent metrics on a space

X

then

\operatorname\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

"Abstract and Concrete Categories. The Joy of Cats"
Jiri Adámek, Horst Herrlich, George E. Strecker. Functors {{cattheory-stub