categorification

The BPS sector of the (bigraded) Hilbert space of a twisted supersymmetric theory may give a cohomology group, whose Euler characteristic is exactly the studied invariant.
eg. 4d Seiberg-Witten theory and the 3d one, Khovanov homology & (colored) Jones polynomials, bigraded BPS Hilbert space $T[M_3]$ and half index ${\hat{Z}}_a(q)$ & superconformal index & WRT invariant (3-manifold, SU(2) Chern-Simons theory; suggested by 3d-3d correspondence, string theory), Q-cohomology& Witten index, the same as singular homology & Euler characteristic

Conversely, decategorification is the dimensional reduction, where the partition function of the TQFT gives a graded trace by compactification (periodic) and is then twisted into a homotopical invariant (when there's a $\mathbb{Z}\times \mathbb{Z}$ graded symmetry) of the corresponding (n-1)-manifold boundary.
eg. Witten index~periodic boundary condition, $\hat{F}$ & $\hat{H}$

ref: arxiv 1101.3216 by Witten; arxiv 1701.06567 BPS spectra and 3-manifold invariant; Knot Invariants and Categorification
brane