On Thu, Apr 12, 2012 at 7:52 AM, Raul Miller <rauldmil...@gmail.com> wrote:
>
> I wanted to expand on this line of thinking, after having slept on it.
>
> When we say "The category of functions of sets", we know we have
> inherited all axioms about functions, and all axioms about sets (there
> are different axiom systems which describe each of these, but lets
> ignore that for now). But what we have not defined is how these
> relate to categories.
>
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). 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). 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*).
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.
> 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". 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.
Anyways, we could say that an arrow corresponds to a set and an object
> corresponds to a function (and arrow composition could be set
> intersection, for example). But that is probably not what you meant,
> so let's ignore that possibility. (Also, in this system, we know we
> have to have some property of sets that lets us form unique ordered
> pairs between functions, which may be a problem.)
>
Yes I think this won't work as a meaningful category construction.
> 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.
> We could say that an arrow corresponds to a function, and an object
> corresponds to an element of the function's domain or an element of
> the function's range. Here, we have many arrows for each function.
>
Yes this would be a category. Objects are all "elements" of functions and
we simply have a unique arrow between every pair of objects.
> We could say that the objects are sets of functions and an arrow leads
> from each set that contains a function to each object that contains
> that function. Here, arrows represent a subset/superset relationship.
>
Sets with arrows representing nonempty intersections do not form a
category. You need an identity arrow on the empty set and also you would be
unable to compose some of these arrows, for example {0} -> {0,1} -> {1}.
However, sets with arrows that represent *either *subset *or* superset (not
strictly since we need identities) do form a category.
> Anyways, none of these choices are specified in the axioms you gave,
> that apparently define category theory.
Yes categories arise in many different ways. Most of our discussion about
category theory and functions has been about the example *Set* but this is
just one example of a category.
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