#+title: category
- tags :: [[file:20200708122545-category_theory.org][category theory]]
- source :: [[file:literature/20210116145100-act4e_session_1_transmutation.org][ACT4E - Session 1 - Transmutation]]

A *category* is specified by four characteristics:

1. Objects, $X$
2. Morphisms, $X \to Y$. For every pair of objects in a category there exists a
   set whose elements map X to Y (this set could be called $Hom(X,Y)$)
3. Identitiy morphisms, a morphism $X \to X$
4. Composition. Given morphisms f and g, there exists a morphism h such that $h
   = f \circ g$

Additionally, categories adhere to the following conditions:

1. Unitality: for any morphism in the category $X \to Y$, $id_x \circ f = f = f \circ
   id_y#$
2. Associativity: given morphisms f, g, and h in the category, $(f \circ g) \circ h = f \circ (g \circ h)$