$omega$-Operads of coendomorphisms and fractal $omega$-operads for higher structures

Abstract

In this article we introduce the notion of textit{Fractal $omega$-operad} emerging from a natural $omega$-operad associated to any coglobular object in the category of higher operads in Batanin's sense, which in fact is a coendomorphism $omega$-operads. We have in mind coglobular object of higher operads which algebras are kind of higher transformations. It follows that this natural $omega$-operad acts on the globular object associated to these higher transformations. To construct the natural $omega$-operad we introduce some general technology and give meaning to saying an $omega$-operad possesses the textit{fractal property}. If an $omega$-operad $B^{0}_{P}$ has this property then one can define a globular object of all higher $B^{0}_{P}$-transformations and show that the globular object has a $B^{0}_{P}$-algebra structure.