ONT Re: Category Theory
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CAT. Note 14
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| 1.2. Categories (cont.)
|
| Preorders. By a preorder we mean a category P in which, given objects
| p and p', there is at most one arrow p -> p'. In any preorder P, define
| a binary relation =< on the objects of P with p =< p' if and only if there
| is an arrow p -> p' in P. This binary relation is reflexive (because there
| is an identity arrow p -> p for each p) and transitive (because arrows can be
| composed). Hence a preorder is a set (of objects) equipped with a reflexive
| and transitive binary relation. Conversely, any set P with such a relation
| determines a preorder, in which the arrows p -> p' are exactly those ordered
| pairs <p, p'> for which p =< p'. Since the relation is transitive, there is
| a unique way of composing these arrows; since it is reflexive, there are the
| necessary identity arrows.
|
| Preorders include 'partial orders' (preorders with the added axiom that
| p =< p' and p' =< p imply p = p') and 'linear orders' (partial orders
| such that, given p and p', either p =< p' or p' =< p).
|
| Mac Lane, 'Cat Work Math', p. 11.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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