ONT Re: Category Theory
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CAT. Note 11
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
| 1.2. Categories
|
| A category (as distinguished from a metacategory) will
| mean any interpretation of the category axioms within
| set theory. Here are the details. A 'directed graph'
| (also called a "diagram scheme") is a set O of objects,
| a set A of arrows, and two functions:
|
| dom
| ------->
| A O (1)
| ------->
| cod
|
| In this graph, the set of composable pairs of arrows is the set:
|
| A x_O A = {<g, f> : g, f in A and dom g = cod f},
|
| called the "product over O".
|
| A 'category' is a graph with two additional functions:
|
| id o
| 1. O ------> A, 2. A x_O A -----> A,
| (2)
| c ~~~~~~> id_c, <g, f> ~~~~~> g o f,
|
| called identity and composition,
| [the latter] also written as g f,
| such that:
|
| dom(id_a) = a = cod(id_a),
|
| dom(g o f) = dom f,
|
| cod(g o f) = cod g, (3)
|
| for all objects a in O and all composable pairs
| of arrows <g, f> in A x_O A, and such that the
| associativity and unit axioms (1.1) and (1.2)
| hold. In treating a category C, we usually
| drop the letters A and O, and write:
|
| c <epsilon> C, f in C (4)
|
| for "c is an object of C" and "f is an arrow of C",
| respectively. We also write:
|
| hom(b, c) = {f : f in C, dom f = b, cod f = c} (5)
|
| for the set of arrows from b to c. Categories can
| be defined directly in terms of composition acting
| on these "hom-sets" (Section 8 below); we do not
| follow this custom because we put the emphasis
| not on sets (a rather special category), but
| on axioms, arrows, and diagrams of arrows.
| We will later observe that our definition
| of a category amounts to saying that a
| category is a monoid for the product
| x_O, in the general sense described
| in the introduction.
|
| Mac Lane, 'Cat Work Math', p. 10.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o