## M Theory Lesson 92

In the 1960s Gian-Carlo Rota wrote a series of papers on the foundations of combinatorics, looking in particular at incidence algebras for locally finite posets. This is an algebra of functions $f(a,b)$ defined on closed intervals of the poset. Multiplication is given by convolution

$(f * g)(a,b) = \sum_{x \in [a,b]} f(a,x) g(x,b)$

This algebra includes the zeta functions $\zeta (a,b) = 1$ and their inverses the Mobius functions, as discussed in Leinster’s paper on Euler characteristics for categories.

In the fourth paper in the series  Rota considered the analogue for vector spaces over finite fields, which are of some interest here since a category of vector spaces may be seen as a simple quantum analogue to the topos Set. The paper begins with q-analogues of binomial coefficients, namely the Gaussian coefficients, which count the rank $k$ subspaces of an $n$ dimensional vector space over the field of order $q$. It is then observed that by allowing $q$ to take a wider range of values, the limit of $q \rightarrow 1$ reduces q-identities to the classical case. The Mobius function $\mu (V,W)$ for elements of the lattice of vector subspaces is given by

$\mu (V,W) = (-1)^{k} q^{(k;2)}$

where $(k;2)$ is the binomial coefficient and $k = \textrm{dim} W – \textrm{dim} V$. The zeta function is defined by $\zeta (V,W) = 1$ when $V$ is contained in $W$, and $0$ otherwise. It is the inverse of the Mobius function in the incidence algebra for the $n$ dimensional vector space.

The close analogy with poset incidence algebras suggests an extension of Leinster’s weightings for a category, which are defined in terms of the Mobius function on the object set incidence algebra with $\zeta (A,B)$ counting the arrows in Hom($A,B$) (a simple extension of the single arrow counted for posets). For a category enriched in Vect the Hom space is a vector space, and it is natural to extend $\zeta (V,W)$ to the dimension of the Hom space Hom($V,W$). Euler characteristics that count dimensions of linear spaces, rather than elements of a set, hopefully bring us a little closer to the knot invariants in their categorified homological guise.

 J. Goldman and G-C. Rota, Stud. Appl. Math. 49 (1970) 239-258

## 3 Responses so far »

1. 1 ### Doug said,

Hi Kea,

I am not very familiar with the Rota work, although I recall that you posted about this once before.

You appear to be investigating probability in applied mathematics. Consider these two resources:

1 – SIAM classic with the most concise, cogent synopsis that I have read on this topic:
Appendix B: Some Notions of Probability Theory
Basar and Olsder, Dynamic noncoopertative game theory.

2 – IMU 2006 Gauss Prize to Kiyoshi Itô for stochastic analysis.
http://www.mathunion.org/Prizes/2006/
gauss_eng_long.pdf

2. 2 ### kneemo said,

Szabo came out with a new paper in which he uses KK-theory and categories to describe D-branes. See definitions 4.1 and 4.2 for the meaning of D-brane and D-brane charge in this new setting.

3. 3 ### Kea said,

Thanks for the link, kneemo. Yes, those guys are great mathematicians and I really wish I better understood the KK point of view, but it isn’t easy!