Introduction to Category Theory/Functors

Functor

A structure-preserving map between categories is called functor. A (covariant) functor F from category ${\mathcal {C}}$ to category ${\mathcal {D}}$ satisfies

$F$ sends objects of ${\mathcal {C}}$ to objects of ${\mathcal {D}}$ .
$F$ sends arrows of ${\mathcal {C}}$ to arrows of ${\mathcal {D}}$ .
If $m$ is an arrow from $A$ to $B$ in ${\mathcal {C}}$ , then $F(m)$ is an arrow from $F(A)$ to $F(B)$ in ${\mathcal {D}}$ .
$F$ sends identity arrows to identity arrows: $F(1_{A})=1_{F(A)}\;$ .
$F$ preserves compositions: $F(g\circ f)=F(g)\circ F(f)$ .

A contravariant functor reverses arrows:

If $m$ is an arrow from $A$ to $B$ in ${\mathcal {C}}$ , then $F(m)$ is an arrow from $F(B)$ to $F(A)$ in ${\mathcal {D}}$ .
$F$ preserves compositions: $F(g\circ f)=F(f)\circ F(g)$ .

Natural Transformations

If F and G are covariant functors between the categories ${\mathcal {C}}$ and ${\mathcal {D}}$ , then a natural transformation $\eta$ from F to G associates to every object X in ${\mathcal {C}}$ a morphism $\eta _{X}:F(X)\to G(X)$ in ${\mathcal {D}}$ called the component of $\eta$ at X, such that for every morphism $f:X\to Y$ in ${\mathcal {C}}$ we have $\eta _{Y}\circ F(f)=G(f)\circ \eta _{X}$ . This equation can conveniently be expressed by the commutative diagram

If both F and G are contravariant, the horizontal arrows in this diagram are reversed. If $\eta$ is a natural transformation from F to G, we also write $\eta :F\to G$ . This is also expressed by saying the family of morphisms $\eta _{X}:F(X)\to G(X)$ is natural in X.

If, for every object X in C, the morphism $\eta _{X}$ is an isomorphism in ${\mathcal {D}}$ , then $\eta$ is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors). Two functors F and G are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from F to G.

Natural transformations are usually far more natural than the definition above.

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