The size of a finite skeletal category C in the sense of Leinster is defined as follows: Label the objects of C by integers 1,2,...,n and let a_{ij} be the number of morphisms from i to j (for i and j between 1 and n). The **size (or Euler characteristic) of C** is defined as the sum of the entries of the inverse of the nxn matrix A=(a_{ij}), if the inverse exists.

Let F_{q} be a finite field with q elements. For every natural number i, there is up to isomorphism exactly one F_{q}-vector space V_{i} of dimension i. The number of linear maps from V_{i} to V_{j} is equal to q^{ij}. We ignore the zero dimensional vector space V_{0}. Consider the infinite matrix

Q=(q^{ij})

where rows and and columns are indexed by positive integers 1,2,3,... From now on let us treat q as a formal parameter, don't care about convergence issues, and set v=q^{-1}.

**Is there a notion of an inverse of Q?** (The entries will probably be formal power series in v.) **If the answer is yes, what is a closed form for the sum of the entries of the inverse (as a formal power series in v), i.e. the size of the category of finite dimensional F _{q}-vector spaces?**

At least every truncation Q_{n} of Q to an upper left nxn corner has an inverse for every positive integer n, since Q_{n} is a Vandermonde matrix. What is the limit of the sum of the entries of Q_{n}^{-1} as n goes to infinity? I believe the answer is a power series in v. Is there an explicit form?

How can you interpret the answer? Is it the Euler characteristic of some moduli space? Is it equal or related to a sum of 1/Gl(V_{i})? Does something interesting happen at q=1?