category
tags
: [[category theory]]
source
: [[ACT4E - Session 1 - Transmutation]]
A category is specified by four characteristics:
- Objects, \(X\)
- 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 \(Hom(X,Y)\))
- Identitiy morphisms, a morphism \(X \to X\)
- Composition. Given morphisms f and g, there exists a morphism h such that \(h = f \circ g\)
Additionally, categories adhere to the following conditions:
- Unitality: for any morphism in the category \(X \to Y\), \(id_x \circ f = f = f \circ id_y#\)
- Associativity: given morphisms f, g, and h in the category, \((f \circ g) \circ h = f \circ (g \circ h)\)