PROP Profile of Poisson Geometry

It is shown that some classical local geometries are of infinity origin, i.e. their smooth formal germs are (homotopy) representations of cofibrant (di) operads in spaces concentrated in degree zero. In particular, they admit natural infinity generalizations when one considers homotopy representations of the (di) operads in generic differential graded spaces. Poisson geometry provides us with a simplest manifestation of this phenomenon.