definition. category [kostecki2011introduction, 1.1] [tt-0002]

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A category \({\cal C}\) consists of:

1. objects \(\operatorname {Ob}({\cal C})\): \(O, X, Y, \dots \)
2. arrows \(\operatorname {Arr}({\cal C})\): \(f, g, h, \dots \), where for each arrow \(f\),
   • a pair of operations \(\operatorname {dom}\) and \(\operatorname {cod}\) assign a domain object \(X=\operatorname {dom}(f)\) and a codomain object \(Y=\operatorname {cod}(f)\) to \(f\),
   • thus f can be denoted by \[f : X \to Y\] or \[X \xrightarrow {f} Y\]
3. compositions: a composite arrow of any pair of arrows \(f\) and \(g\), denoted \(g \circ f\) or \( f \mathbin {\bullet } g \), makes the diagram commute (we say that \(f \mathbin {\bullet } g\) factors through \(Y\)),
4. a identity arrow for each object \(O\), denoted \(\mathit {1}_O : O \to O\)

1. associativity of composition: the diagram commutes,
2. identity law: the diagram commutes.