Categories

[Jonathan Post]

One should also count loops, i.e. an arrow from an object to itself.

Then there are 2 categories on one object, with and without the loop:
Notating V(C) = the vertices in category C (which we label in this
illustrartion but enumerate up to isomorphism, i.e. on forgetting labels;
and L(C) is the set of loops of C, and A(C) is the set of (proper) arcs.
That is, I've separated the loops out from their being a mere subset of
arcs, to aid in enumeration. Then each category is, for this enumeration, a
triple: C= V(C) U L(C) U A(C).

C_1,1 = (1,null,null);
C_1,2 = (1, 1->1,null).

There are 6 categories on 2 objects:
C_2,1 = ({1,2},null,null);
C_2,2 = ({1,2},null,(1->1))
note that the above is isomorphic to C_2,2 = ({1,2},null,(2->2))
C_2,3 = ({1,2},{(1->1),(2->2)},null);
C_2,4 = ({1,2},null,(1->2)) which is isomorphic to C_2,4 =
({1,2},null,(2->1));
C_2,5 = ({1,2},(1>1),(1->2)) which is isomorphic to ({1,2},(2>2),(2->1));
C_2,6 = ({1,2},null,(1>1),(2->1)).

There do seem to be 35 categories on 3 objects, which are essentially the
same as the 7 endomorphisms on 3 objects, with each loop either left in, or
thrown away.

The combinatorics are now straightforward.  We have the endofunction
enumeration, and binary choices on each loop.

I have many, many pages of drawings of these, but not wanting to either scan
them (not having a scanner) nor write out the explicit sets as above.

Is this clear enough, though, to proceed?

-- Jonathan
C_2,2 = ({1,2},(1->1),null)


On 11/24/06, Edwin Clark <eclark at math.usf.edu> wrote:
>
> On Wed, 22 Nov 2006 franktaw at netscape.net wrote:
>
> > How many categories are there?
> >
> > First, how many categories are there with n morphisms and k objects?
> > This table starts:
> >
> >  1
> >  2  1
> >  7  3 1
> > 35 16 3 1
> >
> > The first column is A058129, the number of monoids; the main diagonal
> > is all 1's.  I am not
> > 100% certain of the 16 in the final row.
> >
>
> As you say, the number of one object categories on n objects is M(n)=the
> number of monoids with n elements. But this is known only up to n = 7
> according to the OEIS. This seems to make the number of TWO object
> categories with n morphisms a very difficult problem...Let alone THREE
> object categories...
>
> MathSciNet gives only one hit on "two object categor*":
> ------------------------------------------------------------------
> Rosick\'y, Ji\v rí
> Codensity and binding categories.
> Comment. Math. Univ. Carolinae 16 (1975), no. 3, 515--529.
>
> If there is no measurable cardinal, the author constructs a two-object
> category all of whose well-powered complete extensions are binding.
> -------------------------------------------------------------------
> which is clearly not related to the enumeration problem.
>
> A search on "categor* with two objects" give a few more hits
> but as far as I can see still no papers related to enumeration..
>
>
>
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