Categories#
Definition: Category
A category \(\mathcal{C}\) consists of the following:
- a class \(\mathop{\operatorname{ob}}(\mathcal{C})\) whose elements are called objects;
- a class \(\hom(\mathcal{C})\) whose elements are called morphisms / arrows, where each morphism \(f\) has a source object \(a \in \mathop{\operatorname{ob}}(\mathcal{C})\) and a target object \(b \mathop{\operatorname{ob}}(\mathcal{C})\). We say that \(f\) is a morphism from \(a\) to \(b\).
- a compo
Notation
To expresses that \(f\) is a morphism from \(a\) to \(b\), we use the following notation:
\[ f: a \to b \]
The class of all morphisms from \(a\) to \(b\) is denoted in one of the following ways:
\[ \hom (a, b) \qquad \hom_{\mathcal{C}} (a, b) \qquad \mathop{\operatorname{mor}}(a,b) \qquad \mathcal{C}(a, b) \]
They satisfy the following conditions:
- If \(f\) is a morphism from \(a\) to \(b\) and \(g\) is a