"The category of sets" is defined to be the category of sets and the functions between them. It is an unambiguous term. If there is another category whose elements are sets (i.e. the category of sets containing a "base" element and the functions between them fixing that element), it will be properly delineated.
An arrow is only associated with one source and one destination. Thus there is no arrow which leads from 0 to 0 and from 1 to 1. If we are in the category of sets, this means each function has precisely one domain and codomain. Marshall On Tue, Apr 3, 2012 at 9:06 AM, Raul Miller <rauldmil...@gmail.com> wrote: > On Mon, Apr 2, 2012 at 9:29 PM, Marshall Lochbaum <mwlochb...@gmail.com> > wrote: > > "The category of sets" means the category whose objects are sets and > whose > > arrows are functions. I suppose it's more precise to call it "the > category > > of sets and functions," but no one does that. > > My problem is this: > > On Mon, Apr 2, 2012 at 6:30 PM, Marshall Lochbaum <mwlochb...@gmail.com> > wrote: > > A category is the collection of arrows AND objects. The objects alone do > > not define the category, so there can be many categories with the same > set > > of objects. > > If we can have two categories of sets, with the same > objects but different arrows, then it does not make > sense to call one of them "*The* category of sets". > How am I supposed to know which category we are talking > about? > > > Since each arrow has a start and end point, the identity arrow for one > > element can't be the same as the identity for another, because they have > > different endpoints. > > But, if we have a category with objects 0 and 1, there are several > arrows that could be identity arrows. > > Note that this is based on my current understanding of arrows, which > is that an arrow can represent an arbitrary function: > > First, there's the arrow which leads from 0 to 0 but does not lead > from 1 to 1. This is an identity arrow. > > Second, there's the arrow which leads from 1 to 1 but does not lead > from 0 to 0. This is also an identity arrow. > > Third, there's the arrow which leads from 0 to 0 and which also leads > from 1 to 1. We might call this the "complete" identity arrow, since > all objects in the category are in its domain. > > Finally, there's the arrow which connects no objects. This is the > trivial arrow, and it satisfies the requirements of an identity arrow. > > My question here is: which of these arrows are required by the axiom > that a category must have identity arrows for all objects? We > probably do not need the trivial arrow but I am not clear about the > other three. > > -- > Raul > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
