Functors

A functor is a structure-preserving map between categories. A functor F: C → D assigns to each object A in C an object F(A) in D, and to each morphism f: A → B in C a morphism F(f): F(A) → F(B) in D, preserving identity morphisms and composition: F(id_A) = id_{F(A)} and F(g ∘ f) = F(g) ∘ F(f).

Functors make precise the notion of a 'forgetful' or 'free' construction: the forgetful functor from groups to sets discards the group structure; its left adjoint — the free group functor — reconstructs structure from raw sets. This free/forgetful adjunction is one of the most common patterns in mathematics, and functors are the language in which it is stated.

A functor is covariant if it preserves the direction of morphisms, contravariant if it reverses them. Contravariant functors appear naturally in geometry: the operation that sends a topological space X to its ring of continuous functions C(X) is contravariant, since a continuous map f: X → Y induces a ring map f*: C(Y) → C(X) in the opposite direction. This reversal — topology to algebra with arrows flipped — is the structural signature of duality throughout mathematics.