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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