S. D. V. Stephens

notes / Category Theory / chapter 2: Functors and Natural Transformations

2.1 Functors

In progress: these notes stop partway.

Definition 2.1.1 (Functor). Let and be categories. A functor consists of a function , written , and for each a function , written , which satisfy the following axioms:

  1. whenever in ;
  2. whenever .

Remark. The definition of a functor is such that, from each string of maps in (with ), it is possible to construct exactly one map in . For example, given maps in , we can construct maps in ; by the axioms these are implied to be equal.