## Functors

Next: Heyting algebra Up: Appendix Previous: Sets

# Functors

I will now briefly explain the concept of a functor.
Generally speaking a functor is a transformation from one category $\mathscr{C}$ to another category $\mathscr{D}$, such that the categorical structure of the domain $\mathscr{C}$ is preserved i.e. gets mapped onto $\mathscr{D}$.
There are two types of functors:
1. Covariant Functor
2. Contravariant Functor
1. Definition 3.3   : A covariant functor from a category $\mathscr{C}$ to a category $\mathscr{D}$ is a map $F:\mathscr{C}\rightarrow\mathscr{D}$ that assigns to each $\mathscr{C}$-object a $\mathscr{D}$-object F(a) and to each $\mathscr{C}$-arrow $f:a\rightarrow b$ a $\mathscr{D}$-arrow $F(f):F(a)\rightarrow F(b)$ such that the following are satisfied:
1. $F(1_a)=1_{F(a)}$
2. $F(f o g)=F(f) o F(g)$ for any $g:c\rightarrow a$

It is clear from the above that a covariant functor is a transformation that preserves both:
• the domain's and the codomain's identities

• the composites of functions i.e. it preserves the direction of the arrows
This can be easily seen with the aid of the following diagram;

$\displaystyle \xymatrix{ a\ar[rr]^f\ar[rrdd]_h&&b\ar[dd]^g\\ &&\\ &&c\\ } \xymatrix{\ar@{=>}[rr]^F&&} \xymatrix{ F(a)\ar[rr]^{F(f)}\ar[rrdd]_{F(h)}&&F(b)\ar[dd]^{F(g)}\\ &&\\ &&F(c)\\ }$

2. Definition 3.4

A contravariant functor from a category $\mathscr{C}$ to a category $\mathscr{D}$ is a map $X:\mathscr{C}\rightarrow\mathscr{D}$ that assigns to each $\mathscr{C}$-object a a $\mathscr{D}$-object X(a) and to each $\mathscr{C}$-arrow $f:a\rightarrow b$ a $\mathscr{D}$-arrow $X(f):X(b)\rightarrow X(a)$ such that the following conditions are satisfied

1. $X(1_a)=1_{X(a)}$
2. $X(f o g)=X(g) o X(f)$ for any $g:c\rightarrow a$

A diagrammatic representation of a contravariant functor is the following:

$\displaystyle \xymatrix{ a\ar@{>}[rr]^f\ar[rrdd]_h&&b\ar[dd]^g\\ &&\\ &&c\\ } \xymatrix{\ar@{=>}[rr]^X&&} \xymatrix{ F(a)&&X(b)\ar[ll]^{X(f)}\\ &&\\ &&X(c)\ar[lluu]^{X(h)}\ar[uu]^{X(g)}\\ }$

As we can see from the above diagram, a contravariant functor in mapping arrows from one category to the next reverses the directions of the arrows, by mapping domains to codomains and vice versa.

Next: Heyting algebra Up: Appendix Previous: Sets
Cecilia Flori 2007-01-02