ONT Re: Category Theory
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CAT. Note 10
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| 1.1. Axioms for Categories (concl.)
|
| Since the objects of a metacategory correspond exactly to
| its identity arrows, it is technically possible to dispense
| altogether with the objects and deal only with arrows. The
| data for an 'arrows-only metacategory' C consist of arrows,
| certain ordered pairs <g, f>, called the composable pairs of
| arrows, and an operation assigning to each composable pair
| <g, f> an arrow g o f, called their composite. We say
| "g o f" is defined" for "<g, f> is a composable pair".
|
| With these data one 'defines' an identity of C to be an arrow u
| such that f o u = f whenever the composite f o u is defined and
| u o g = g whenever u o g is defined. The data are then required
| to satisfy the following three axioms:
|
| 1. The composite (k o g) o f is defined if and only if
| the composite k o (g o f) is defined. When either is
| defined, they are equal (and this 'triple composite' is
| written as k o g o f).
|
| 2. The triple composite k o g o f is defined
| whenever both composites k o g and g o f
| are defined.
|
| 3. For each arrow g of C there exist identity arrows
| u and u' of C such that u' o g and g o u are defined.
|
| In view of the explicit definition given above for
| identity arrows, the last axiom is a quite powerful
| one; it implies that u' and u are unique in (3), and
| it gives for each arrow g a codomain u' and a domain u.
| These axioms are equivalent to the preceding ones. More
| explicitly, given a metacategory of objects and arrows,
| its arrows, with the given composition, satisfy the
| "arrows-only" axioms; conversely, an arrows-only
| metacategory satisfies the objects-and-arrows
| axioms when the identity arrows, defined as
| above, are taken as the objects (Proof as
| exercise).
|
| Mac Lane, 'Cat Work Math', p. 9.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
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