Set and Function

A function is, set-theoretically, a subset that is total, meaning every has a value, and single-valued, meaning . Single-valuedness is what justifies the notation . The profile is the pair of domain and codomain. Since the codomain is part of the datum, the same rule on different domains is a different function.

The vocabulary of maps

is injective, also called an embedding, if ; surjective if it hits all of ; an isomorphism if it has a two-sided inverse with and . A self-map, one with , is an endomorphism, and an endomorphism that is also an isomorphism is an automorphism.

is the prototype category

Sets and functions form the category : composition is the usual one, identities the identity functions, and the isomorphisms are exactly the bijections. Most structured categories are with extra data on the objects and a compatibility condition on the morphisms; a $\mathbf{Set}$-valued functor is how a category “acts on” plain sets.