  In this paper, we extend a result of Lafont and M{\'e}tayer and prove that
the polygraphic homology of a small category, defined in terms of polygraphic
resolutions in the category $\omega$Cat of strict $\omega$-categories, is
naturally isomorphic to the homology of its nerve. Along the way, we prove some
results on homotopy colimits with respect to the Folk model structure and
deduce a theorem which formally resembles Quillen's Theorem A.
