Yes, the particular equation you wrote is categorified by the free resolution of k as module over k[x] by the complex k[x] -> k[x] $k[x] \overset{x}\longrightarrow k[x]$ 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).