Functors


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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