# category

tags
: [[category theory]]

source
: [[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)\\)