Today I Learned

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

Category Archives: category theory

89: Functors

In category theory, a functor is a map between categories \mathcal{C}, \mathcal{D} consisting of two components:

  1. a map f from the objects of \mathcal{C} to those of \mathcal{D}
  2. a map from each morphism g: X \rightarrow Y in \mathcal{C} to a morphism g': f(X) \rightarrow f(Y) in \mathcal{D}.

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

  1. f(1_X) = 1_{f(X)} for objects X in \mathcal{C}
  2. f(a \circ b) = f(a) \circ f(b)

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

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86: Forgetful Functors

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 U: \mathbf{Grp} \rightarrow \mathbf{Set}, which maps groups to their underlying sets and group homomorphisms to themselves. U forgets the group structure and that group homomorphisms are anything other than functions between sets.
  • The functor V: \mathbf{Ab} \rightarrow \mathbf{Grp} from the category of abelian groups to the category of groups. V simply maps groups and group homomorphisms to themselves, essentially forgetting that the source groups are abelian.

85: The Group as a Category

The group can be viewed as equivalent to a specific kind of category, specifically the category C 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 C, the group operation to \circ (composition), and the group identity to the identity morphism on the single element of C. Since all morphisms in C are isomorphisms, each ‘element’ has an inverse, and \circ is associative, analogous to the associativity of the group operation.

79: Hom-Set

In category theory, the hom-set between 2 objects X, Y in a category C, often denoted as

\textrm{hom}_C (X, Y)

or simply

\textrm{hom} (X, Y)

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