Categories

A category, C, is an identified arrow world equipped with an associative composition, oC. It is conventional, when it is unambiguous, to omit the C from o.

For example, a transform world is, in fact, a category under its composition (but only an interesting arrow world under conjoining). Furthermore, if the arrow worlds on which its arrows are transformations are, in fact, categories then composition and conjoinability satisfy an interesting interchange law: given conjoinable (T, S) and (U, V) for which (T, U) and (S, V) are composable, (T · S, U · V) is composable and (T o U, S o V) is conjoinable with the composite of the former equal to the conjoint of the latter. The picture which demonstrates this is instructive !

It is possible to recast the above into a form in which the arrow world under conjoining is a category; we replace the collection of transformations with a collection of triplets (F, T, G) wherein T is the transformation replaced, F is a functor which ends it and G is a functor which begins it; however, it is reasonable to require that every such triplet be inserted in place of T (ie one for each pair (F, G) of functors (ending, beginning) T). Composition is defined point-wise and conjoinability becomes: ((F, T, G), (H, S, I)) is conjoinable precisely if G = H and (T, S) is conjoinable; in which case the conjoint is (F, T · S, I). However, this is less elegant and unnecessary; all the interesting properties can be described in terms of a category on which there is a second operation for which it is an arrow world with given properties, namely those of a transform domain.

In a category, zero arrows are unique (within any given Hom(,)). Proof: suppose f, x are initial and g, y are terminal with (f, g) and (x, y) composable and f o g parallel to x o y; then pick the unique d in Hom(ω(g), α(x)) and e in Hom(ω(y), α(f)) to see that x o d is in Hom(α(f), ω(f)) = {f} as f is initial, so x o d = f, e o y is in Hom(α(g), ω(g)) = {g} as g is terminal, so e o y = g, and d o e is in Hom(α(x), ω(y)) which consists, simply, of a single identity arrow [as x is initial, y is terminal and α(x) = ω(y)], so that d o e = α(x). This then implies f o g = (x o d) o (e o y) = (x o (d o e)) o y = (x o α(x)) o y = x o y. Thus parallel zero arrows are equal; hence zero arrows are unique.

Examples

A great variety of mathematical constructions can be expressed in terms of categories (and many of the rest as arrow worlds). This should be no surprise: category theory may fairly be described as a (roughly general) theory of abstraction.

It may fairly be said that the category Set (whose definition I originally included here) inspired mathematicians first to invent the abstraction of categories (below the foundations of which I here burrow), then to search for a better category. Set has many virtues, but it is inescapably tied to logical premises (among which the law of the excluded middle stands pre-eminent) which I mistrust. It does, however, provide an excellent model (probably because of the very severe certainty it inherits from reductio ad absurdum) within which to search out and comprehend the properties, of whatever system, that will illuminate any discussion.

Written by Eddy.
\$Id: category.html,v 1.2 2001-10-17 17:11:30 eddy Exp \$