# Functors

...whenever new abstract objects are constructed in a specified way out of given ones, it is advisable to regard the construction of the corresponding induced mappings on these objects as an integral part of their definition.

...every sufficiently good analogy is yearning to become a functor.

Before functoriality, people lived in caves.

# Definition

Maps between categories are called *functors*. Briefly, a functor between categories consists of maps of objects and arrows that preserve the categorical structure. In more detail:

Suppose **functor**

- For each object
, an object - For each arrow
in , an arrow in

with the following properties:

- (Compatibility with composition) For every pair of composable arrows
in , we must have - (Identity Preservation) For every object
, one has

You may run across the phrase "covariant functor" or "contravariant functor." Both are holdovers from the early days of category theory, when the foundations were still being established. Back then, many of the inspiring examples in algebraic geometry and algebraic topology involved functor-like maps that reversed the directions of arrows. We will even see examples of this in both module theory (see here) and category theory (see Yoneda's Lemma). In other words, we'll see maps between categories that send an arrow

Since such maps between categories arose naturally and behaved well in all other respects, they were called **contravariant functors**. By contrast, our usual (direction-preserving) maps were called **covariant functors**.^{[1]} Note that a contravariant functor "reverses" composition, i.e., satisfies

The more modern view has reverted to simply "functors", and has incorporated the contravariant functors via the notion of opposite categories: a contravariant functor

# Examples

## The power set functor

One of the first constructions you usually see in set theory is that of the *power set*. Indeed, it is usually part of the very axioms of set theory. This construction is the object map of a functor. The **power set functor**

- Objects: For each set
we assign its power set , i.e., the set of all subsets of - Arrows: For each set map
, we assign the set map defined by sending a subset to its image .

You should verify for yourself that these maps satisfy the properties of a functor.

## Forgetful functors

A functor that simply "forgets" some or all of the structure of an algebraic object is commonly called a **forgetful** (or **underlying**) functor. For example, the forgetful functor

A fair question. For now the answer is simply "Let's wait and see." Soon we will see that many (all?) constructions and "universal properties" depend crucially on the category in which one is working. As such, being able to be incredibly specific about which category is being considered will often prove critically important.

## Remember me not

Are there functors inverse to "forgetting?" Sadly, we will see that the answer to that question is generally no: that which is forgotten cannot be remembered. However, we will also see that there is a type of functor which can be regarded as complementary to forgetting (later to be codified in the idea of an adjoint).

For example, there is a functor **free group**

Note that these two functors are definitely not inverses. For instance, even if

## Abelianization

Suppose **commutator subgroup** and denoted

is always an abelian group; - Any group morphism
carries commutators to commutators, hence to - Every group morphism
to an abelian group factors uniquely through the projection - The assignment
is the object function of a functor , called the **abelianization**functor (or sometimes**factor-commutator**functor).

The abelianization functor is adjoint to the forgetful functor.

## A non-functor

Although nearly every common construction in algebra is the object function of a functor, there are a few notable exceptions. One such is the assignment to each group *not* have an induced morphism

## General linear group

For each commutative ring **general linear group**

## Unit groups

This example is closely related to the previous. For each ring

## Examples in topology

Singular homology in a given dimension

Similarly, the assignment of a homotopy group **fundamental groupoid**.

## Families of objects in a category

Suppose **family of objects in indexed by ** is, as it sounds, a collection of the form

For example, the set of objects of

In general, we have a (natural) bijection

This allows us to interchangeably think of families of objects in a given category

## Suggested next notes

Natural transformations

Universal Properties I - Inspiring Examples

Hence the origin for the name of this wiki! β©οΈ