Categories

[franktaw at netscape.net]

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.

Taking the row sums, we get:

1,3,11,55

the number of categories with n morphisms.  This is probably not in the 
OEIS (only
A001776 is possible - other matches become less than A058129).  The 
inverse Euler
transform,

1,2,8,41

is the number of connected categories with n morphisms; this is 
likewise probably not
in the OEIS (only A052447 is possible).

Can somebody generate more data?

Franklin T. Adams-Watters

A category is a collection of objects and morphisms; each morphism is 
from one object
to another (not necessarily different) object.  Where the destination 
of one morphism
is the source of a second, their composition is defined; composition is 
associative where
it is defined.  Each object has an identity morphism, which connects it 
to itself; this
is an identity when composed with morphisms coming in and with 
morphisms going
out.
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