ONT Re: Category Theory
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CAT. Note 20
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| 1.3. Functors (cont.)
|
| Functors may be composed. Explicitly, given functors:
|
| T S
| C ---> B ---> A
|
| between categories A, B, and C, the composite functions:
|
| c ~> S(Tc), f ~> S(Tf)
|
| on objects c and arrows f of C define a functor S o T : C -> A, called the
| 'composite' (in that order) of S with T. This composition is associative.
| For each category B there is an identity functor I_B : B -> B, which acts as
| an identity for this composition. Thus we may consider the metacategory of
| all categories: its objects are all categories, its arrows are all functors
| with the composition above. Similarly, we may form the category Cat of all
| small categories -- but not the category of all categories.
|
| Mac Lane, 'Cat Work Math', p. 14.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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