Next question is thus: "How many 2-categories (bicategories) are there whose objects are the categories on n objects?"<br>
<br>
See:<br>
Basic Bicategories<br>
Tom Leinster (1998-10-04) arXiv.org:math/9810017
<div> 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
</div>
<br>
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.<br>
<br>
<font size="-1"><span class="a"><a href="http://www.ima.umn.edu/categories/abstracts.html">www.ima.umn.edu/categories/abstracts.html</a> <br>
</span></font><font size="-1">Abstracts for the IMA 2004 Summer Program: n-Categories.<br>
Ross Street has proposed two related definitions of n-category, one in 1987 and one in 2002. <br>
<br>
</font><span class="authors"></span>See also: <br>
Tom Leinster (2003-05-02) arXiv.org:math/0305049
<div> 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
</div>
<br>
<font color="black" face="" size="-1"><a href="http://www.lepp.cornell.edu/spr/2002-04/msg0041086.html"><font color="blue">http://www.lepp.cornell.edu/spr/2002-04/msg0041086.html</font></a> </font><br>
<pre id="line194"><font size="4">In the beginning, there was nothing.<br>But with nothing came nothingness, the vacuity, which was something.<br>So now there was nothingness and somethingness,<br>vacuity and triviality, falsehood and truth,
<br>which were 2 things.<br>So now there was a set of nothingness and somethingness,<br>and the whole realm of sets of elements<br>sprang up out of the set of nothingness and somethingness,<br>and sprouted functions between them
<br>to relate them back to the set of nothingness and somethingness.<br>So now there was a category of sets and functions,<br>and the whole realm of categories of objects and morphisms<br>sprang up out of the category of sets and functions,
<br>and sprouted functors between them,<br>to relate them back to the category of sets and functions,<br>which sprouted natural transformations between *them*,<br>to relate the relationships.<br>So now there was ...<br><br>
<br>-- Toby<br> <a href="mailto:toby@math.ucr.edu">toby@math.ucr.edu</a></font></pre>
<font size="-1">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.<br>
<br>
Happy Thanksgiving,<br>
<br>
Jonathan Vos Post<br>
</font><br><div><span class="gmail_quote">On 11/22/06, <b class="gmail_sendername">Jonathan Post</b> <<a href="mailto:jvospost3@gmail.com">jvospost3@gmail.com</a>> wrote:</span><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
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.<br>
<br>
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.<br>
<br>
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.<br>
<br>
Hard to summarize less cryptically, here, but I stand by in support of
what you're doing, and think my approach is genuinely related.<br>
<br>
-- Jonathan Vos Post<div><span class="e" id="q_10f11e1bf7a16489_1"><br><br><div><span class="gmail_quote">On 11/22/06, <b class="gmail_sendername"><a href="mailto:franktaw@netscape.net" target="_blank" onclick="return top.js.OpenExtLink(window,event,this)">
franktaw@netscape.net</a></b> <<a href="mailto:franktaw@netscape.net" target="_blank" onclick="return top.js.OpenExtLink(window,event,this)">franktaw@netscape.net
</a>> wrote:</span><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">How many categories are there?<br><br>First, how many categories are there with n morphisms and k objects?
<br>This table starts:<br><br> 1<br> 2 1<br> 7 3 1<br>35 16 3 1<br><br>The first column is A058129, the number of monoids; the main diagonal<br>is all 1's. I am not<br>100% certain of the 16 in the final row.<br><br>Taking the row sums, we get:
<br><br>1,3,11,55<br><br>the number of categories with n morphisms. This is probably not in the<br>OEIS (only<br>A001776 is possible - other matches become less than A058129). The<br>inverse Euler<br>transform,<br><br>
1,2,8,41
<br><br>is the number of connected categories with n morphisms; this is<br>likewise probably not<br>in the OEIS (only A052447 is possible).<br><br>Can somebody generate more data?<br><br>Franklin T. Adams-Watters<br><br>
A category is a collection of objects and morphisms; each morphism is
<br>from one object<br>to another (not necessarily different) object. Where the destination<br>of one morphism<br>is the source of a second, their composition is defined; composition is<br>associative where<br>it is defined. Each object has an identity morphism, which connects it
<br>to itself; this<br>is an identity when composed with morphisms coming in and with<br>morphisms going<br>out.<br>________________________________________________________________________<br>Check Out the new free AIM(R) Mail -- 2 GB of storage and
<br>industry-leading spam and email virus protection.<br><br></blockquote></div><br>
</span></div></blockquote></div><br>