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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 to another category , such that the categorical structure of the domain is preserved i.e. gets mapped onto .

There are two types of functors:

**Covariant Functor****Contravariant Functor**

- It is clear from the above that a covariant functor is a transformation that preserves both:
**Definition 3.3***: A***covariant functor**from a category to a category is a map that assigns to each -object a -object F(a) and to each -arrow a -arrow such that the following are satisfied:- for any

- the domain's and the codomain's identities

- the composites of functions i.e. it preserves the direction of the arrows

- A diagrammatic representation of a contravariant functor is the following:
**Definition 3.4***A***contravariant functor**from a category to a category is a map that assigns to each -object a a -object X(a) and to each -arrow a -arrow such that the following conditions are satisfied- for any

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