ONT Re: Category Theory
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CAT. Note 16
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| 1.2. Categories (concl.)
|
| Large Categories. In addition to the metacategory of all sets --
| which is not a set -- we want an actual category Set, the category
| of all 'small' sets. We shall assume that there is a big enough set
| U, the "universe", then describe a set x as "small" if it is a member
| of the universe, and take Set to be the category whose set U of objects
| is the set of all small sets, with arrows all functions from one small set
| to another. With this device (details in Section 7 below) we construct other
| familiar large categories, as follows:
|
| Set. Objects, all small sets;
| arrows, all functions between them.
|
| Set_*. Pointed sets: Objects, small sets each with a selected base point;
| arrows, base-point-preserving functions.
|
| Ens. Category of all sets and functions within a (variable) set V.
|
| Cat. Objects, all small categories;
| arrows, all functors (Section 3).
|
| Mon. Objects, all small monoids;
| arrows, all morphisms of monoids.
|
| Grp. Objects, all small groups;
| arrows, all morphisms of groups.
|
| Ab. Objects, all small (additive) abelian groups,
| with morphisms of such.
|
| Rng. All small rings, with the ring homomorphisms
| (preserving units) between them.
|
| CRng. All small commutative rings and their morphisms.
|
| R-Mod. All small left modules over the ring R, with linear maps.
|
| Mod-R. Small right R-modules.
|
| K-Mod. Small modules over the commutative ring K.
|
| Top. Small topological spaces and continuous maps.
|
| Toph. Topological spaces, with arrows homotopy classes of maps.
|
| Top_*. Spaces with selected base point,
| base-point-preserving maps.
|
| Particular categories (like these) will always appear
| in bold-face type [not shown here]. Script capitals
| are used by many authors to denote categories.
|
| Mac Lane, 'Cat Work Math', p. 12.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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