Introduction to Category Theory/Functors

Functor

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

• $F$ sends objects of $𝒞$ to objects of $𝒟$.
• $F$ sends arrows of $𝒞$ to arrows of $𝒟$.
• If $m$ is an arrow from $A$ to $B$ in $𝒞$, then $F(m)$ is an arrow from $F(A)$ to $F(B)$ in $𝒟$.
• $F$ 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 $𝒞$, then $F(m)$ is an arrow from $F(B)$ to $F(A)$ in $𝒟$.
• $F$ preserves compositions: $F(g\circ f)=F(f)\circ F(g)$.

Natural Transformations

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

diagram defining natural transformations

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 $𝒟$, 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.

Related resources

• w:Functor
• w:Natural transformation