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
