Today I Learned

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

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.