May I recommend this introduction to Category Theory:
http://www.ling.ohio-state.edu/~plummer/courses/winter09/ling681/barrwells.pdf
The first ten pages should suffice to get a bit of clarity with the
terminology. (Section 2.1.11 defines the "category of sets".)
Robby
On Thu, Apr 12, 2012 at 10:49 PM, Raul Miller <rauldmil...@gmail.com> wrote:
> On Thu, Apr 12, 2012 at 3:46 PM, Jordan Tirrell
> <jordantirr...@gmail.com> > We are using what we know about functions
> on sets simply to define the
> > category *Set* (where objects are all sets and arrows are all functions
> on
> > sets).
>
> That seems to be ambiguous, though, and I am trying to understand
> what's really meant.
>
> The issues which are significant, to me, are:
>
> 1. The difference(s) between a "Function" and an "Arrow" which in some
> way represents a function.
>
> 2. The difference between a member of a domain and the domain itself.
>
> 3. What exactly is being represented by the composition of arrows.
>
> > We then usually try to capture ideas about functions in categorical
> > language in *Set*. Marshall gave a great example about how we can look at
> > the category *Set* alone and identify which object corresponds to the
> empty
> > set, and which objects correspond to one-element sets. Specifically, the
> > empty set is the object in our category which has a unique arrow to every
> > object (in general this is called an initial object).
>
> And, this uniqueness works with a variety of different concepts for
> how Arrows relate to Functions.
>
> > The dual notion of terminal object identifies the singleton sets.
> > The power of category theory is right here: we have left behind
> > our normal definitions of empty set and singleton set and discovered
> > that they correspond to the categorical definitions of initial and
> > terminal object (within this specific category *Set*).
>
> No.
>
> We have not yet left behind our normal definition of the -- that
> definition was a part of the definition of this *Set*. We also
> have not yet fully defined *Set*.
>
> > Now a set theorist can sit down with, say, a group theorist, and even if
> > both are ignorant of each other's field, they can have a meaningful
> > conversation: Does the category you study have an initial/terminal
> object?
> > One or many? Of course this is a toy example, but I think its a good
> > glimpse into the usefulness of category theory as a unifying theme in
> math.
>
> I have no problems believing that a notation can be useful.
>
> That is why I am interested in the first place.
>
> But before I can go there, there's a little matter of making sure
> I understand what the notation means.
>
> >> 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.
> >
> > As far as I know, set theory axioms would not allow a definition like
> "the
> > set of all sets which are not contained in any categories".
>
> All I need, to be able to do this, is to have a formal definition
> of "category" in my system.
>
> > There are some very interesting issues with the use of sets
> > vs collections when we discuss categories, which is why
> > restricting to categories whose objects and arrows both form
> > sets (called a small category) is common. *Set* is not small, so
> > we do have to be careful. Let's not get into set theory though,
> > since I will quickly be unable to offer good answers to questions,
> > and it is another step removed from J programming.
>
> Yes... I wanted to work with a category where the objects represented
> the domain of the values 0 and 1 and where arrows represented functions
> on that domain. Another domain I liked was where objects represented
> the domain of the values 0, 1, 2, 3 or 4.
>
> >> 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.
> >
> > This is a category, it is like *Set* but restricted to onto functions.
>
> Then I need a more complete definition of the category *Set*.
>
> But I think you are saying here that the function with the
> domain {0, 1} and the image {0} with codomain {0} is a different
> function from the function with the domain {0, 1} and the image
> {0} with the codomain {0, 1}.
>
> And, while I can accept that these are different arrows, it
> seems to me that these are the same function.
>
> --
> Raul
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>
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