2. Functors

There are many examples in mathematics where from one object $A$ one can construct another kind of object $F(A)$. For example, with a Boolean algebra there is associated a topological space (its prime ideal space), and with a topological space there is associated its homology group. These constructions extend to maps between the objects and have reasonable properties for maps. However, upon closer examination it can be seen that there are really two kinds:

Definition. For categories ${\cal C}$ and ${\cal D}$, a covariant functor $F$ from ${\cal C}$ to ${\cal D}$ consists of

(i) for each object $A \in {\cal C}$, an object $F(A) \in {\cal D}$, and

(ii) for each $A,B \in {\cal C}$ and morphism $\phi \in$   Morph$(A,B)$, a morphism $F(\phi) \in$   Morph$(F(A), F(B))$,

with the properties

(a) $F( \phi \circ \psi ) = F( \phi ) \circ F( \psi )$ (whenever $\circ$ makes sense) and

(b) $F($   1$_ A ) =$   1$_ {F(A)}$ for each $A \in {\cal C}$.

An example is the fundamental-group functor $\pi _ 1$. Here ${\cal D}$ is the category of groups; the objects of ${\cal C}$ are pairs $(X,p)$ consisting of a topological space $X$ and a designated point in $X$, and the morphisms of ${\cal C}$ are continuous functions that take designated points to designated points. $F(X,p)$ is the fundamental group $\pi _ 1 (X,p)$ consisting of equivalence classes of loops at $p$ in $X$.

A contravariant functor is defined similarly, except that in (ii) we have $F(\phi) \in$   Morph$( F(B), F(A) )$ and in (a) $F (\phi \circ \psi) = F(\psi) \circ F(\phi)$. An example is the prime-ideal-space functor from Boolean algebras to topological spaces.

For a functor $F$ from ${\cal C}$ to ${\cal D}$ and a functor $G$ from ${\cal D}$ to ${\cal E}$, the result of following $F$ by $G$ is again a functor, which will be covariant if both $F$ and $G$ are covariant or both $F$ and $G$ are contravariant.