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 $q\to 1$ 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 $k=\text{dim}W-\text{dim}V$. 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