Categories

[Jonathan Post]

Next question is thus: "How many 2-categories (bicategories) are there whose
objects are the categories on n objects?"

See:
Basic Bicategories
Tom Leinster (1998-10-04)  arXiv.org:math/9810017 A concise guide to very
basic bicategory theory, from the definition of a bicategory to the
coherence theorem. Comment: 11 pages; LaTeX 2e with Paul Taylor's diagram
macros

This is even harder to answer, because there seem to be at least 7 different
definitions of n-categories, of which categories = 1-categories, and I've
just raised the enumeration probem for n=2.

www.ima.umn.edu/categories/abstracts.html
Abstracts for the IMA 2004 Summer Program: n-Categories.
Ross Street has proposed two related definitions of n-category, one in 1987
and one in 2002.

See also:
Tom Leinster (2003-05-02) arXiv.org:math/0305049 Higher-dimensional category
theory is the study of n-categories, operads, braided monoidal categories,
and other such exotic structures. It draws its inspiration from areas as
diverse as topology, quantum algebra, mathematical physics, logic, and
theoretical computer science. This is the first book on ... Comment: Book,
410 pages

http://www.lepp.cornell.edu/spr/2002-04/msg0041086.html

In the beginning, there was nothing.
But with nothing came nothingness, the vacuity, which was something.
So now there was nothingness and somethingness,
vacuity and triviality, falsehood and truth,
which were 2 things.
So now there was a set of nothingness and somethingness,
and the whole realm of sets of elements
sprang up out of the set of nothingness and somethingness,
and sprouted functions between them
to relate them back to the set of nothingness and somethingness.
So now there was a category of sets and functions,
and the whole realm of categories of objects and morphisms
sprang up out of the category of sets and functions,
and sprouted functors between them,
to relate them back to the category of sets and functions,
which sprouted natural transformations between *them*,
to relate the relationships.
So now there was ...


-- Toby
   toby at math.ucr.edu

But your question is fundamental, and I am eager to see an answer.  I hope
it is not rude for me to be anticipating follow-ups.

Happy Thanksgiving,

Jonathan Vos Post

On 11/22/06, Jonathan Post <jvospost3 at gmail.com> wrote:
>
> 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.
> > ________________________________________________________________________
> > Check Out the new free AIM(R) Mail -- 2 GB of storage and
> > industry-leading spam and email virus protection.
> >
> >
>
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