Categories

[Jonathan Post]

First, John Baez agrees that an earlier post of his partially answered my
follow-up question about 2-categories (some assembly required):

http://golem.ph.utexas.edu/category/2006/10/klein_2geometry_vi.html#c005157

 Re: Klein 2-Geometry VI

Tim writes:

So I'll wrap up and say what I think I've discovered. Just as
finite-dimensional vector spaces and their subspaces are intimately related
to finite sets and their subsets, so finite-dimensional 2-vector spaces and
their 2-subspaces are intimately related to finite graphs and their
subgraphs. These are obviously much more complicated and I'm not at all
confident my analyses here are correct, particularly the identification of
the various 2-Grassmannians associated to various graph maps. The basic idea
seems sound though.

Yeah, that's a great idea!

The way I see it, your fundamental idea is this. There's a 2-functor from
the 2-category of directed graphs to the 2-category of vector 2-spaces
(≃2-term chain complexes). And, it goes like this:

   - Any directed graph gives a 2-term chain complex.
   - Any map between directed graphs gives a chain map between 2-term
   chain complexes.
   - And, there's also a kind of "homotopy between maps between directed
   graphs", which gives a chain homotopy between chain maps between 2-term
   chain complexes.

(Do graph theorists ever think about those homotopies? They should!)

All this generalizes further, to an (n+1)-functor from n-dimensional cell
complexes to (n+1)-term chain complexes.

I'm not sure how to get some amazing new insights into projective n-geometry
from this way of thinking - but that's not surprising, since I just read
your post 2 minutes ago. We should mull on it.

Thanks!
Posted by: John Baez <http://math.ucr.edu/home/baez/> on October 8, 2006
7:25 PM |

Second, I took Category theory in grad school 1976 or 1977, and forgot what
little I knew, and have relearned only some. So anyone in seqfans who
learned more and remebered more should jump in here, and don't worry about
making me look foolish, as I take the risk of exposing my ignorance every
time I post here.

If one asks: "how many categories are there with n morphisms and k objects?"
one is asking up to isomorphism. I'm making a first cut here off the top of
my head, which may be no more than an ill-informed guess.

The sagittal graph of a general morphism a.k.a. homomorphism (not restricted
to a monomorphism, epimorphism, isomorphism, endomorphism, or automorphism)
leaves us wondering what the structure of the objects are.  I think, by the
way that the question is asked, that they have no structure, i.e. that they
are just points. If they have structure, that affects everything.
unstructrured objects (points), then we still get different counts if the
vertices and arcs are labeled or unlabeled, and whether one allows loops or
not. if the edges are unlabeled, and loops are allowed, then any of the k
objects can be mapped to any of the k objects including itself, giving k^k,
if n = k.  If n>k then we can 't use more than k of the n because the
mapping must be functional, that is, only one arrow out of each point (but
multiple arrows to a point is allowed).

 If n<k then the first point can be mapped to any of the k points, and we
run out of arrows when we reach k^n.  Now we see why labeling matters, as if
they are unlabeled, we count only once in equivalence classes, but if
labeled each such mapping is distinct.  Each loop and arc is de facto
labeled by its source point and target point, as an ordered pair.  If you
want to allow other labels, you have to multiply by the proper combinational
multiplier.

So we're taking a summation of n=0 (just the k points and none map to any),
n=1 (one of k points maps to one of k), n=2 and so forth.

That starts the answer.  For k=n and any point can map to any, we have the
count of endomorphisms being the same as the count of endofunctions already
in OEIS.

Category theory has differing approaches depending on whether we have
digraphs, multgraphs,  reversible digraphs, reflexive digraphs, irreflexive
digraphs, and various tricks to represent undirected graphs with directed
graphs.

So exact definitions are necessary.

Sorry, just a first cut.  Not enough sleep.  Some time tomorrow before gests
and Thanksgiving dinner kicks in, but no promises.

Thanks for the dialog.

Best,

Jonathan Vos Post
<http://golem.ph.utexas.edu/cgi-bin/MT-3.0/sxp-comments.pl?entry_id=955;parent_id=5157>


On 11/22/06, Max A. <maxale at gmail.com> wrote:
>
> On 11/22/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
> > How many categories are there?
> >
> > First, how many categories are there with n morphisms and k objects?
>
> Does that correspond to transitively closed digraphs with k (labeled?)
> vertices and n edges?
>
> Max
>
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