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Category Theory

Category Theory is a branch of mathematics which seek to abstract away as much underlying structure as possible. As such, many theorems from across mathematics are corollaries of results from Category Theory.

The Category

A [[Category]] \(C\) consists of the following information:

  1. A collection \(\text{ob}(C)\) of objects.
  2. A collection \(\text{mor}(C)\) of morphisms, each of which has a source object and a target object.
  3. A binary operation \(\circ\) on morphisms via which morphisms can be composed.

The Functor

A [[Functor]] \(F: C \to D\) is a mapping from one category \(C\) to another category \(D\), such that

  1. Every object \(c \in C\) is mapped to an object \(Fc \in D\).
  2. Every morphism \(f: c \to c'\) in \(C\) is mapped to a morphism \(Ff: Fc \to Fc'\) in \(D\).
  3. The functoriality axiom \(Ff \circ Fg = F(f \circ g)\) for \(f, g \in \text{mor}(C)\) is satisfied.