$\omega$-Operads of Coendomorphisms for Higher Structures

Abstract

It is well known that strict $\omega$-categories, strict $\omega$-functors, strict natural $\omega$-transformations, and so on, form a strict $\omega$-category. A similar property for weak $\omega$-categories is one of the main hypotheses in higher category theory in the globular setting. In this paper we show that there is a natural globular $\omega$-operad which acts on the globular set of weak $\omega$-categories, weak $\omega$-functors, weak natural $\omega$-transformations, and so on. Thus to prove the hypothesis it remains to prove that this $\omega$-operad is contractible in Batanin's sense. To construct such an $\omega$-operad we introduce more general technology and suggest a definition of $\omega$-operad with the \textit{fractal property}. If an $\omega$-operad $B^{0}_{P}$ has this property then one can define a globular set of all higher $B^{0}_{P}$-transformations and, moreover, this globular set has a $B^{0}_{P}$-algebra structure.