ONT Re: Category Theory
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CAT. Note 15
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| 1.2. Categories (cont.)
|
| Ordinal Numbers. We regard each ordinal number n as the linearly ordered
| set of all the preceding ordinals n = {0, 1, ..., n-1}; in particular, 0
| is the empty set, while the first infinite ordinal is !w! = {0, 1, 2, ...}.
| Each ordinal n is linearly ordered, and hence is a category (a preorder).
| For example, the categories $1$, $2$, and $3$ listed above are the preorders
| belonging to the (linearly ordered) ordinal numbers 1, 2, and 3. Another
| example is the linear order !w! [omega]. As a category, it consists of
| the arrows:
|
| 0 -> 1 -> 2 -> 3 -> ...,
|
| all their composites, and the identity arrows for each object.
|
| !D! is the category with objects all finite ordinals and arrows
| f : m -> n all order-preserving functions (i =< j in m implies
| f_i =< f_j in n). This category !D! [Delta], sometimes called
| the 'simplicial category', plays a central role (Chapter 7).
|
| Finord = Set_!w! is the category with objects all finite ordinals n
| and arrows f : m -> n all functions from m to n. This is essentially
| the category of all finite sets, using just one finite set n for each
| finite cardinal number n.
|
| Mac Lane, 'Cat Work Math', pp. 11-12.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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