Categories

[Jonathan Post]

*"Formalized Proof, Computation, and the Construction Problem in Algebraic
geometry", by Carlos Simpson.

[PDF]* arXiv:*math*.AG/0410224 v1 8 Oct
2004<http://arxiv.org/pdf/math/0410224> File
Format: PDF/Adobe Acrobat - View as
HTML<http://72.14.253.104/search?q=cache:8FWyl8lGqRwJ:arxiv.org/pdf/math/0410224+%22how+many+categories%22+math+theory&hl=en&gl=us&ct=clnk&cd=5>
finite integer N, *how many categories* are there with N morphisms? What *
...* Algebraic *theory* of machines, I. Trans. Amer. *Math*. Soc. 116
(1965), 450-464.



On 11/23/06, Jonathan Post <jvospost3 at gmail.com> wrote:
>
> Roland Bacher correctly refers to quivers as summarized in wikipedia at
>
> http://en.wikipedia.org/wiki/Quiver_%28mathematics%29
>
> On 11/22/06, Roland Bacher < Roland.Bacher at ujf-grenoble.fr> wrote:
> >
> >
> > This numbers can also be obtained as the numbers
> > of the following quotients of quivers:
> >
> > Associate to a category with n morphisms and k objects
> > the quiver with k vertices corresponding to the objects
> > and a(X,Y) arrows directed from the object X to the object
> > Y if there are a(X,Y) morphisms from X into Y.
> >
> > Given two morphism g:X-->Y, f:Y-->Z with composition
> > h=f o g:X-->Z, put the relation gf=h on the quiver algebra
> >
> > The resulting quotient algebra has dimension n and a basis
> > given by the (simple) arrows.
> >
> > This leads to the following "algorithm" for enumerating
> > all categories with n morphisms and k objects:
> >
> > (a) enumerate all quivers (directed graphs) with k vertices and
> > n oriented edges.
> >
> > (b) Associate the following "weight" to such a quiver as follows :
> >
> > (b1)given a triplet of vertices X,Y,Z,
> > set w(X,Y,Z)=(a(X,Z))^{a(X,Y) \cdot a(Y,Z)}
> > (this counts the number of ways which associate a morphism
> > X-->Z to a composition X-->Y-->Z) where a(U,V) denotes the
> > number of oriented arrows starting at U and ending at W.
> >
> > (b2) associate to a given quiver the weight
> > \prod_{X,Y,Z} w(X,Y,Z)
> > where the product is over all triplets of vertices.
> >
> > The total sum of such weighted quivers yields then the solution.
> >
> > Roland Bacher
> >
> >
> >
> >
> >
> >
> > On Wed, Nov 22, 2006 at 04:20:32PM -0500, 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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