ONT Re: Category Theory
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CAT. Note 21
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| 1.3. Functors (cont.)
|
| An 'isomorphism' T : C -> B of categories is a functor
| T from C to B which is a bijection, both on objects and
| on arrows. Alternatively, but equivalently, a functor
| T : C -> B is an isomorphism if and only if there is a
| functor S : B -> C for which both composites S o T and
| T o S are identity functors; then S is the 'two-sided
| inverse' S = T^(-1).
|
| Certain properties much weaker than isomorphism will be useful.
|
| A functor T : C -> B is 'full' when to every pair c, c' of objects of C
| and to every arrow g : Tc -> Tc' of B, there is an arrow f : c -> c' of C
| with g = Tf. Clearly the composite of two full functors is a full functor.
|
| A functor T : C -> B is 'faithful' (or an embedding) when to every pair
| c, c' of objects of C and to every pair f_1, f_2 : c -> c' of parallel
| arrows of C the equality Tf_1 = Tf_2 : Tc -> Tc' implies f_1 = f_2.
| Again, composites of faithful functors are faithful. For example,
| the forgetful functor Grp -> Set is faithful but not full and
| not a bijection on objects.
|
| These two properties may be visualized in terms of hom-sets (see (2.5)).
| Given a pair of objects c, c' in C, the arrow function of T : C -> B
| assigns to each f : c -> c' an arrow Tf : Tc -> Tc' and so defines
| a function:
|
| T_c,c' : hom(c, c') -> hom(Tc, Tc'), f ~> Tf.
|
| Then T is full when every such function is surjective, and faithful
| when every such function is injective. For a functor which is both
| full and faithful (i.e., "fully faithful"), every such function is
| a bijection, but this need not mean that the functor itself is an
| isomorphism of categories, for there may be objects of B not in
| the image of T.
|
| Mac Lane, 'Cat Work Math', pp. 14-15.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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