# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## Category Archives: category theory

## 89: Functors

January 10, 2017

Posted by on In category theory, a **functor** is a map between categories consisting of two components:

- a map from the objects of to those of
- a map from each morphism in to a morphism in .

This map must preserve the identity morphisms and composition. That is, it must satisfy the following properties:

- for objects in

Functors can be thought of as kinds of homomorphisms between categories. With functors as arrows, categories can then form a category called , the category of categories.

## 86: Forgetful Functors

January 7, 2017

Posted by on **Forgetful functors** are, as the name implies, functors between categories that *forget* something about the structure or properties of the objects and arrows in the source category of the functor.

Examples:

- The functor , which maps groups to their underlying sets and group homomorphisms to themselves.
*forgets*the group structure and that group homomorphisms are anything other than functions between sets. - The functor from the category of abelian groups to the category of groups. simply maps groups and group homomorphisms to themselves, essentially
*forgetting*that the source groups are abelian.

## 85: The Group as a Category

January 5, 2017

Posted by on The group can be viewed as equivalent to a specific kind of category, specifically the category with only a single element and all of whose morphisms are isomorphisms. In this equivalence, the elements of the group correspond to the morphisms of , the group operation to (composition), and the group identity to the identity morphism on the single element of . Since all morphisms in are isomorphisms, each ‘element’ has an inverse, and is associative, analogous to the associativity of the group operation.

## 79: Hom-Set

December 30, 2016

Posted by on In category theory, the **hom-set** between 2 objects in a category , often denoted as

or simply

is the collection of arrows (morphisms) in from to . Note that despite the name, the hom-set is not a set in general.

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