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.

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