Generalized operads and their inner cohomomorphisms

TL;DR

This paper introduces generalized operads encompassing various operad-like structures, constructs their inner cohomomorphisms, and explores their applications in non-commutative geometry and categorical contexts.

Contribution

It provides a unified axiomatic framework for generalized operads and extends inner cohomomorphism constructions to broader categorical settings.

Findings

Unified axiomatic treatment of generalized operads.

Construction of inner cohomomorphisms in operad categories.

Application to symmetry and moduli in non-commutative geometry.

Abstract

In this paper we introduce a notion of {\it generalized operad} containing as special cases various kinds of operad--like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provide an approach to symmetry and moduli objects in non-commutative geometries based upon these "ring--like" structures. We give a unified axiomatic treatment of generalized operads as functors on categories of abstract labeled graphs. Finally, we extend inner cohomomorphism constructions to more general categorical contexts. This version differs from the previous ones by several local changes (including the title) and two extra references.