(Edit: the following should be a comment following Philipp's "Reid, what does that mean?", not an Answer. Can't see how to change it now, though.though. Further edit: I've appended a guess at an answer.)

As to for your questionsmain question, I've been trying for hours with no successthere's weak evidence that the answer is 1. Theorem 1.6 of Here's the argument.

It's a theorem (in the paper tells you what cite) that whenever you have an adjunction between a pair of finite categories, and both their Euler characteristics are defined, then their Euler characteristics are equal. It's also a theorem that whenever two categories are equivalent, the inverse matrix EC of one is defined iff the EC of the other is, but in such and then they're equal. Finally, it's a theorem that if a category has a terminal object then its EC, if defined, is 1.

Now, your category is equivalent to the category C of nontrivial finite-dimensional vector spaces over Fq. The usual adjunction between vector spaces and sets restricts to an indirect manner adjunction between C and the category S of nonempty finite sets. Moreover, S has a terminal object. So if the three theorems described extend (in some sense) to infinite categories, then the EC of your category is 1.

I'm not at all sure that this is the right answer, and it might contradicts Qiaochu Yuan's suggestion. If the EC (or more precisely the limit that you mention) is notbe any use 1, then that's interesting, because it would imply some difference in behaviour between finite and infinite categories.

Finally, you ask 'How can one interpret the answer?' I don't think I can say much about interpreting this particular case. But in general you can think of the Euler characteristic of a category C as the same thing as the Euler characteristic of its nerve NC (a simplicial set), or its classifying space BC (the geometric realization of NC). Of course, not every topological space has a well-defined Euler characteristic, so theorems of this kind are subject to some hypotheses. The result on adjunctions mentioned above is of the flavour 'if spaces are homotopy equivalent then their Euler characteristics are equal'.

2 clarification of purpose; deleted 9 characters in body

(Edit: the following should be a comment following Philipp's "Reid, what does that mean?", not an Answer. Can't see how to change it now, though.)

Baez and Dolan have a notion of the cardinality of a groupoid. You pick one object ai from each isomorphism class, and define the cardinality to be

\sumi 1/order(Aut(ai)).

For example, if E is the groupoid of finite sets and bijections then you have to choose one i-element set ai for each natural number i; then Aut(ai) = Sn and the cardinality is e. See their paper "From finite sets to Feynman diagrams".

The cardinality of a finite groupoid is the same as its size/Euler characteristic in my sense. (And you can try your luck at extending to the infinite case.) I guess Reid was pointing out that given an arbitrary category C, not necessarily a groupoid, there are two numbers associated to it: (i) the Euler characteristic of C itself, (ii) the cardinality of the underlying groupoid of C. These are in general different (e.g. consider the two-object category consisting of a single arrow). I guess the reason why Reid made his comment was your second-to-last question: "Is it equal or related to a sum of 1/Gl(Vi)?"

As to your questions, I've been trying for hours with no success. Theorem 1.6 of the paper tells you what the inverse matrix is, but in such an indirect manner that it might not be any use.

1

Baez and Dolan have a notion of the cardinality of a groupoid. You pick one object ai from each isomorphism class, and define the cardinality to be

\sumi 1/order(Aut(ai)).

For example, if E is the groupoid of finite sets and bijections then you have to choose one i-element set ai for each natural number i; then Aut(ai) = Sn and the cardinality is e. See their paper "From finite sets to Feynman diagrams".

The cardinality of a finite groupoid is the same as its size/Euler characteristic in my sense. (And you can try your luck at extending to the infinite case.) I guess Reid was pointing out that given an arbitrary category C, not necessarily a groupoid, there are two numbers associated to it: (i) the Euler characteristic of C itself, (ii) the cardinality of the underlying groupoid of C. These are in general different (e.g. consider the two-object category consisting of a single arrow). I guess the reason why Reid made his comment was your second-to-last question: "Is it equal or related to a sum of 1/Gl(Vi)?"

As to your questions, I've been trying for hours with no success. Theorem 1.6 of the paper tells you what the inverse matrix is, but in such an indirect manner that it might not be any use.