Glossary of Category Theory - Functors

Functors

A functor F is said to be:

• a constant provided that F maps every object in a category to the same object A and every morphism to the identity on A.
• faithful provided that F is injective when restricted to each hom-set.
• full provided that F is surjective when restricted to each hom-set.
• isomorphism-dense (sometimes called essentially surjective) provided that for every B there exists A such that F(A) is isomorphic to B.
• an equivalence provided that F is faithful, full and isomorphism-dense.
• amnestic provided that if k is an isomorphism and F(k) is an identity, then k is an identity.
• reflect identities provided that if F(k) is an identity then k is an identity as well.
• reflect isomorphisms provided that if F(k) is an isomorphism then k is an isomorphism as well.