|There are more categories in the world than just general theories. |
- Any group can be considered a category: group elements are arrows over one single object. Id is the group's neutral element. Composition is multiplication.
- A partially ordered set can be represented as a category. The set's elements are objects. Add a single arrow a → b for each pair a, b such that a < b, and unit arrow a → a for each a.
For each pair of objects there's no more than one arrow, and since partial order is transitive, we have composition (a<b, b<c => a<c), and there is no need to worry about its associativity.
- As a special case of the previous example, a segment of integers, [N..M] can be thought of as a category.
- Take any oriented graph. We can turn it into a category by treating its paths as arrows. An empty path is a unit arrow; path composition is concatenation.
- Natural numbers as objects, matrices as arrows. Any N×M matrix is an arrow N → M.
Matrix multiplication plays the role of composition; a unit N×N matrix is the unit arrow N → N.
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