Categories

[Jonathan Post]

This actually relates to the enumeration results I've made on "prime"
endofunctions, namely endofunctions which are not unions of other prime
endofunctions nor categorical products of endofunctions. My partial results
here are in A125024, A124023, and a comment on A048888 that I submitted a
day or two ago about the enumeration of prime endofunctions in terms of
numbrals.

I have been working on enumerating not just endofunctions, but
(categorically) endomorphisms of endofunctions, (2-categorically)
endomorphisms of endomorphisms of endofunctions.  I've been posting partial
results on John Baez's "n-category cafe" and apparently just annoyed John
Baez, who didn't yet see that I was finding a new combinatorial result at
the base level (endofunctions) and working on the n-category hierarchy above
that, so those comments may already have been deleted from his blog.

That is, in counting all categories on n objects, most are trivially the
union of smaller disjoint categories (that is, the sagittal digraph is not
connected), or categorical products of smaller categories.  They key to
enumeration involves thus both a tree-enumeration, and the numbral part
which, per a recent paper in the Journal on Integer Sequences, relate to
balanced ordered trees.

Hard to summarize less cryptically, here, but I stand by in support of what
you're doing, and think my approach is genuinely related.

-- Jonathan Vos Post

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?
> 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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