Functoriality

Functoriality

Functor

A (covariant) functor between categories and is a mapping of objects to objects and morphisms to morphisms such that for all -objects and :

A contravariant functor is a similar mapping , but reverses the direction of morphisms:

• ,
• ,
• .

Equivalently, a contravariant functor is a (covariant) functor from the opposite category to .

e.g.

• Every category has an identity functor .
• The mapping sends any vector space to its dual space , and any linear map to its dual map (i.e. the pullback) is a contravariant functor from the category of vector spaces to itself.
• The functor that sends each finite set to its power set and each function to the corresponding function on the power sets is a contravariant functor.
  

The Category of Small Categories

We now have another example of a category, namely , the category of all small categories and corresponding functors.

Injective & Surjective on Objects

A functor is said to be injective on objects if the object part is injective, it is surjective on objects if is surjective. Similarly, is injective or surjective on morphisms if the morphism part is injective or surjective.

Faithful & Full Functor

Suppose , are locally-small categories. A functor is faithful if for all , the map is injective. It is full if is surjective.

Embedding

A functor is called an embedding if it is full, faithful, and injective on objects.

Full Subcategory

A full subcategory consists of some objects of and all of the morphisms between them.

Representable Structure

Representable Functor

Let be a locally-small category, we have the representable functors: for all objects .

Generator

A generator for category is an object has the property that for any objects and and morphisms , if then there is an arrow such that . That is, the arrows in the category are distinguished by their effect on generalized elements based at .

Contravariant Representable Functors

Let be a locally small category, we have the contravariant representable functors: taking to by for .

Equivalence of Categories

Equivalent Categories

An equivalence of categories consists of a pair of functors and and a pair of natural isomorphisms
In this situation, the functor is called a pseudo-inverse of . The categories and are then said to be equivalent, written .

Proposition

The following conditions on a functor are equivalent:

• is part of an equivalence of categories.
• is full and faithful and “essentially surjective” on objects: for every there is some such that .