A category is an abstract collection of objects

## constituents

- collection of objects, where if \(X\) is an object of \(C\) we write \(X \in C\)
- for a pair of objects \(X, Y \in C\), a set of morphisms acting upon the objects which we call the homset

## additional information

## requirements

- there exists the identity morphism; that is, \(\forall X \in C, \exists I_{X}: X\to X\)
- morphisms are always composable: given \(f: X\to Y\), and \(g: Y\to Z\), exists \(gf: X \to Z\)
- the identity morphism can compose in either direction: given \(f: X \to Y\), then \(f I_{x} = f = I_{y} f\)
- morphism composition is associative: \((hg)f=h(gf)\)