Polynomials in $\mathbb Z[t]$ are categorified by considering Euler characteristics of complexes of finite-dimensional graded vector spaces. Now, given a rational function that has a power series expansion with integer coefficients, it seems natural to consider complexes of (locally finite-dimensional) graded vector spaces.


Are there nice examples of this in nature?



Can we categorify the equation (1 - t)(1 + t + t^2 + ...) = 1?
Yes, the particular equation you wrote is categorified by the free resolution of k as module over k[x] by the complex $k[x] \overset{x}\longrightarrow k[x]$ given by multiplication by x. It also appears in the numerical criterion for Koszulity of k[x] (see the paper of [Beilinson, Ginzburg and Soergel](http://mathaware.org/jams/1996-9-02/S0894-0347-96-00192-0/S0894-0347-96-00192-0.pdf)).


