  The present article exploits the fact that permutads (aka shuffle algebras)
are algebras over a terminal operad in a certain operadic category Per. In the
first, classical part we formulate and prove a claim envisaged by Loday and
Ronco that the cellular chains of the permutohedra form the minimal model of
the terminal permutad which is moreover, in the sense we define, self-dual and
Koszul. In the second part we study Koszulity of Per-operads. Among other
things we prove that the terminal Per-operad is Koszul self-dual. We then
describe strongly homotopy permutads as algebras of its minimal model. Our
paper shall advertise analogous future results valid in general operadic
categories, and the prominent role of operadic (op)fibrations in the related
theory.

Category Theory;Algebraic Topology;Rings and Algebras