On Wed, Apr 11, 2012 at 7:49 PM, Marshall Lochbaum <mwlochb...@gmail.com> wrote: > In an attempt to clarify things, here are the collected axioms for a > category: > A category is a collection of objects and a collection of arrows. > Each arrow has a source object and a destination object (which may be the > same object). > Each object has an identity arrow from itself to itself, which is unique in > the sense that it is the only identity arrow. Other arrows may go from that > object to itself, but are not the identity. > For any two arrows A->B and B->C (i.e. the destination of one is the source > of the other), there is a unique arrow which is the composition of these > arrows. > The operation of composition on arrows is associative.
If that is the complete set of axioms then we need additional axioms before category theory can say anything. For example, I could have a "red arrow" and a "green arrow" which represent the same function, with the above set of axioms. Only the identity arrow for an object cannot have multiple instances... And I do not see any significant constraints on what objects are. > These axioms completely "define" objects and arrows. The category of sets > is a representation of sets and functions in these terms; we can only know > category-theoretical things like which arrows lead where in this category. > We cannot ask what set a given object represents within the confines of > category theory. When we say that set theory can be a basis for all of mathematics, we mean that a set of axioms which includes the set axioms is sufficient to represent all of mathematics. (There's also an implication that those axioms are expressed in terms of sets, but that's not much of a constraint -- a formulation equivalent to a set is "a sequence of bits which is infinite in length" -- here, membership in a set corresponds to a bit being set, and the cardinality of the set is the sum of the bits (using the J convention of 1: true, 0: false) -- most of the information and interesting stuff happens in what the bits stand for.) Anyways, what you seem to be saying, here, is that we need additional axioms to make category theory useful. And I guess I have no problems accepting that. > Nonetheless, we can find familiar landmarks. The empty set > is distinguished by having a unique arrow to every other object (the one > corresponding to the trivial function). Two sets A and B are isomorphic if > there are arrows f:A->B and g:B->A such that f o g is the identity on B and > g o f is the identity on A. And, given the original constraints on arrows, there can be multiple g and multiple f, but since the composition is the identity when they compose to an identity they must be considered the same arrow. That tells us something of minor interest. > Using this, we can find the one-element set up > to isomorphism. Any set which has a unique arrow from any set to it has > only one element, and all of these sets are isomorphic. I do not see how there can be a unique arrow between different sets in the complete category of functions of sets. Each set can apparently have an infinite number of arrows leading from the set to itself. The only uniqueness constraints are on the identity arrow (which is unique to an object) and the composition of two arrows (which is unique to those two specific arrows). So it seems to me that if the category is complete that one element sets can have multiple arrows between them (my red arrow and green arrow example from earlier can be an illustration of this issue). Unless, of course, we introduce an additional axiom to prevent this. (Which does like seem a reasonable course of action.) Alternatively, we can consider certain arrows to be unique in some sense (like being the identity arrow, or being a complete arrow, or some such thing) but that would not guarantee that the set has only one element. > One way to find how many elements are in a set is to check > how many arrows there are from a one-element set to it, > since each identifies a single element in the set. However, > the strength of category theory is not that it allows us to > rebuild the sets in this way, but that it allows us to see > exactly how much precision functions give us in discussing > sets. Specifically, there is no way to determine the elements > of sets--sets are distinguished only up to isomorphism. Unless we introduce an axiom that lets us do that... > If you want to learn the formal workings of category theory, the J forums > aren't the best place to do it. I would recommend getting a book on the > subject; unfortunately I've picked up category theory from ad-hoc work in > other disciplines and don't have a good reference. It shouldn't be too hard > to find a source online, though. It's the best I have though. -- Raul ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
