  We introduce two applications of polygraphs to categorification problems. We
compute first, from a coherent presentation of an $n$-category, a coherent
presentation of its Karoubi envelope. For this, we extend the construction of
Karoubi envelope to $n$-polygraphs and linear $(n,n-1)$-polygraphs. The second
problem treated in this paper is the construction of Grothendieck
decategorifications for $(n,n-1)$-polygraphs. This construction yields a
rewriting system presenting for example algebras categorified by a linear
monoidal category. We finally link quasi-convergence of such rewriting systems
to the uniqueness of direct sum decompositions for linear
$(n-1,n-1)$-categories.
