Category Theory

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)