On Fri, Apr 13, 2012 at 4:06 PM, Jordan Tirrell <jordantirr...@gmail.com> wrote:
> On Fri, Apr 13, 2012 at 3:15 PM, Raul Miller <rauldmil...@gmail.com> wrote:
>> On Fri, Apr 13, 2012 at 2:38 PM, Marshall Lochbaum <mwlochb...@gmail.com>
>> wrote:
>> > You're making a mistake in your treatment of functions. A function is
>> not a
>> > formula. It is an association of a value in the codomain to each value in
>> > the domain, and if two functions have all of the same associations (which
>> > forces them to have the same domain), then they are identical. Therefore
>> >:
>> > and 1: with domain {0} are the same function.
>>
>> The reference I was reading
>> http://en.wikibooks.org/wiki/Haskell/Category_theory begins with "a
>> directed multigraph with loops".My statement here was wrong. The reference I was reading was http://www.ling.ohio-state.edu/~plummer/courses/winter09/ling681/barrwells.pdf >> That means that we can have multiple distinct arrows leading from 0 and 1. >> >> You are telling me that we can only have one arrow leading from 0 to >> 1, which makes their explicit statement pointless. But I see no >> statements in their exposition to support this constraint. > > > Two objects in an arbitrary category may or may not have many arrows > between them. Agreed. > Categories in general are not about sets and functions. Nevertheless, sets and functions are useful in the context of discussions involving mathematics. > If you restrict yourself to two objects that represent singleton > sets and you want arrows that represent functions between > them, then certainly there is only one (well, one in each direction > one for each identity). Any object can be also be thought of as a singleton set which contains that object. So the above simplifies to: ] If you restrict yourself to two objects and you want arrows ] that represent functions between them, then certainly there ] is only one (well, one in each direction one for each identity). Furthermore, the arrows were originally edges in a directed multigraph with loops, and you can model the connection between the two nodes as a function and the distinction between different arrows connecting the same node as a label. If this meaning of "function" is the "representation" you are describing here, then the above simplifies to: ) If you restrict yourself to two objects and you want arrows ) between them, then certainly there is only one (well, one in ) each direction one for each identity). But it seems to me that this statement is not true for arrows, in general. > Numbers are used to label vertices (objects) of directed graph examples in > the link Robert sent out. That's an example, and not a constraint. Also, consider Godel numbering. > Examples there might call objects 0 and 1, but these are just names. > It does not mean that they represent the sets {0} and {1} nor that > arrows between them represent functions on those sets. Of course it does not mean that. Nevertheless an edge of a directed graph can be thought of as a constant function with a one element domain. And an edge in a directed multigraph can be thought of the same way, with the inclusion of some additional information which is not a part of the function, such that multiple edges might exist between the same node pair. > Be very careful not to confuse categories in general with > categories that represent sets and functions. I do not think I have done that. -- Raul ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
