ONT Re: Category Theory
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CAT. Note 22
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| 1.3. Functors (concl.)
|
| A 'subcategory' S of a category C is a collection of
| some of the objects and some of the arrows of C, which
| includes with each arrow f both the object dom f and the
| object cod f, with each object s its identity arrow 1_s,
| and with each pair of composable arrows s -> s' -> s"
| their composite. These conditions ensure that these
| collections of objects and arrows themselves constitute
| a category S. Moreover, the injection (inclusion) map
| S -> C which sends each object and each arrow of S to
| itself (in C) is a functor, the 'inclusion functor'.
| This inclusion functor is automatically faithful.
|
| We say that S is a 'full subcategory' of C when the inclusion functor
| S -> C is full. A full subcategory, given C, is thus determined by
| giving just the set of its objects, since the arrows between any two
| of these objects s, s' are all morphisms s -> s' in C. For example,
| the category Set_f of all finite sets is a full subcategory of the
| category Set.
|
| Mac Lane, 'Cat Work Math', p. 15.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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