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[Jonathan Post]

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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