Categories

[Edwin Clark]

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..