ONT Re: Category Theory
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CAT. Note 17
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| 1.3. Functors
|
| A 'functor' is a morphism of categories. In detail, for
| categories C and B a functor T : C -> B with domain C and
| codomain B consists of two suitably related functions: The
| 'object function' T, which assigns to each object c of C an
| object Tc of B and the 'arrow function' (also written T) which
| assigns to each arrow f : c -> c' of C an arrow Tf : Tc -> Tc'
| of B, in such a way that:
|
| T(1_c) = 1_Tc, T(g o f) = Tg o Tf, (1)
|
| the latter whenever the composite g o f is defined in C. A functor,
| like a category, can be described in the "arrows-only" fashion: It
| is a function T from arrows f of C to arrows Tf of B, carrying each
| identity of C to an identity of B and each composable pair <g, f>
| in C to a composable pair <Tg, Tf> in B, with Tg o Tf = T(g o f).
|
| A simple example is the power set functor $P$ : Set -> Set. Its object
| function assigns to each set X the usual power set $P$X, with elements
| all subsets S c X; its arrow function assigns to each f : X -> Y that
| map $P$f : $P$X -> $P$Y which sends each S c X to its image fS c Y.
| Since both $P$(1_X) = 1_$P$X and $P$(g o f) = $P$g o $P$f, this
| clearly defines a functor $P$ : Set -> Set.
|
| Mac Lane, 'Cat Work Math', p. 13.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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