On Wed, Apr 11, 2012 at 9:01 PM, Raul Miller <rauldmil...@gmail.com> wrote: > 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).
I wanted to expand on this line of thinking, after having slept on it. When we say "The category of functions of sets", we know we have inherited all axioms about functions, and all axioms about sets (there are different axiom systems which describe each of these, but lets ignore that for now). But what we have not defined is how these relate to categories. First off, as a side note, let us consider "the set of all sets which are not contained in any categories". Without further axioms we do not know if this set is empty or non-empty. (Example axioms: "The set of all sets which are not contained in any categories is a non-empty set", or "The set of all sets which are not contained in any categories is an empty set".) So right off, we know that involving sets introduces all of the problems that sets have. Anyways, we could say that an arrow corresponds to a set and an object corresponds to a function (and arrow composition could be set intersection, for example). But that is probably not what you meant, so let's ignore that possibility. (Also, in this system, we know we have to have some property of sets that lets us form unique ordered pairs between functions, which may be a problem.) We could say that an arrow corresponds to a function and an object corresponds to the domain of a function (and the object that an arrow leads to corresponds to the image of the function). Here, identifying an arrow uniquely identifies a function. We could say that an arrow corresponds to a function, and an object corresponds to an element of the function's domain or an element of the function's range. Here, we have many arrows for each function. We could say that the objects are sets of functions and an arrow leads from each set that contains a function to each object that contains that function. Here, arrows represent a subset/superset relationship. Anyways, none of these choices are specified in the axioms you gave, that apparently define category theory. -- Raul ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
