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 [1] 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.

[1] 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.

    I do intend to read more about Rota.

    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.

  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!

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