ONT Re: Category Theory
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CAT. Note 9
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| 1.1. Axioms for Categories (cont.)
|
| A metacategory is to be any interpretation which satisfies all these axioms.
| An example is the 'metacategory of sets', which has objects all sets and
| arrows all functions, with the usual identity functions and the usual
| composition of functions. Here "function" means a function with
| specified domain and specified codomain. Thus a function
| f : X -> Y consists of a set X, its domain, a set Y,
| its codomain, and a rule x ~> fx (i.e., a suitable
| set of ordered pairs <x, fx>) which assigns, to
| each element x in X, an element fx in Y. These
| values will be written as fx, f_x, or f(x), as
| may be convenient. For example, for any set S,
| the assignment s ~> s for all s in S describes
| the 'identity function' 1_S : S -> S; if S is a
| subset of Y, the assignment s ~> s also describes
| the 'inclusion' or 'insertion function' S -> Y;
| these functions are 'different' unless S = Y.
| Given functions f : X -> Y and g : Y -> Z,
| the 'composite' function g o f : X -> Z is
| defined by (g o f)x = g(fx) for all x in X.
| Observe that g o f will mean first apply f,
| then g -- in keeping with the practice of
| writing each function f to the left of its
| argument. Note, however, that many authors
| use the opposite convention.
|
| To summarize, the metacatgory of all sets has as
| objects, all sets, as arrows, all functions with the
| usual composition. The metacategory of all groups is
| described similarly: Objects are all groups G, H, K;
| arrows are all those functions f from the set G to
| the set H for which f : G -> H is a homomorphism
| of groups. There are many other metacategories:
| All topological spaces with continuous functions
| as arrows; all compact Hausdorff spaces with the
| same arrows; all ringed spaces with their morphisms,
| etc. The arrows of any metacategory are often called
| its 'morphisms'.
|
| Mac Lane, 'Cat Work Math', pp. 8-9.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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