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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 2.6***: 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 2.7:***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 are satisfied- for any

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

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**Previous:**Terminal Object Cecilia Flori 2006-11-20