I don't know whether or not this satisfies the criterion of "categorification" (what that?), but the equation (1 - t)(1 + t + t^{2} + ...) = 1 and its relation to vector spaces is well-known to differential topologists and geometers. We use it all the time in K-theory and index theory where it becomes the identity Λ_{-1}V ⊗ S_{1}V = ℂ. Here, Λ_{-1}V denotes the alternating sum of the exterior powers of V whilst S_{1}V is the sum of the symmetric powers.

Of course, the sum of the symmetric powers isn't a class in K-theory as it is an infinite sum. To get round this, we work in K[[t]] and allow parameters, whereupon the equation becomes Λ_{-t}V ⊗ S_{t}V = ℂ. Here, the t means formally multiply the kth exterior or symmetric power by t^{k}.

Where this breaks out of mere formalism and becomes very powerful is in equivariant K-theory. Then the parameter becomes a way of measuring the action of the group, which is usually S^{1} or a finite cyclic group for index theory calculations. In particular, in Witten's original adaptation of index theory to loop spaces, we end up with positive energy representations of S^{1} which become vector bundles over the original (finite dimensional) manifold with circle actions preserving the fibres. One can decompose these according to the circle action whereupon one has a vector bundle for each k ∈ ℤ. The positive energy criterion means that these are trivial below a certain integer and are always finite dimensional. However, as there are an infinite number of them then the total dimension can be infinite dimensional. Then the identity Λ_{-t}V ⊗ S_{t}V = ℂ has real meaning as the power of the t parameter indicates how the circle acts on that component of the vector bundle. That is, if the circle action on V is the standard action then it is multiplication by t^{k} on Λ^{k} V and on S^{k} V.