Some of the things I've learned every day since Oct 10, 2016
Category Archives: category theory
January 10, 2017Posted 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.
January 7, 2017Posted 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.
- 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.
January 5, 2017Posted 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.
December 30, 2016Posted by on
In category theory, the hom-set between 2 objects in a category , often denoted as
is the collection of arrows (morphisms) in from to . Note that despite the name, the hom-set is not a set in general.