Category Theory

Basic Concepts

A category C consists of

  • a collection C0, whose elements are called the objects of C and are usually denoted by uppercase letters, X, Y, Z, …;
  • a collection C1, whose elements are called the morphisms, or arrows, of C and

are usually denoted by lowercase letters, f, g, h, …;

such that

  • each morphism is assigned two objects, called source and target, or domain and codomain. We denote the source and target of the morphism f by s(f) and t(f), respectively. If the morphism f has source S and target T, we also write \(f : X \rightarrow Y\), or, more graphically \(X \xrightarrow{f} Y\),
  • each object X has a distinguished morphism \(id_X: X \rightarrow X\), called identity morphism.
  • for each pair of morphisms f, g, with t(f) = s(g), there exists a specified morphism \(g \circ f\), called the composite morphism, such that \(s(g \circ f) = s(f)\) and \(t(g \circ f) = t(g)\).

These structures need to satisfy the following axioms:

  • Unitality: For every morphism \(f: X → Y\), the compositions \(f \circ id_X\) and \(id_Y \circ f\) are both equal to f.
  • Associativity: For \(f: X \rightarrow Y\), \(g: Y \rightarrow Z\) and \(h: Z \rightarrow W\), the compositions \(h \circ (g \circ f) = (h \circ g) \circ f\).

(Perrone 2024, 3)

References:

Perrone, Paolo. 2024. Starting Category Theory. World Scientific.