Generalized Operads and Their Inner Cohomomorphisms

In this paper we introduce a notion of 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. §0. Introduction 0.1. Inner cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A,A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A,A1) and B = (B,B1), define the category A ⇒ B by the following data. An object of A ⇒ B is a pair (F, u) where F is a k–algebra and u : A→ F ⊗B is a homomorphism of algebras such that u(A1) ⊂ F ⊗B1 (all tensor products being taken over k). A morphism (F, u) → (F , u) in A ⇒ B is a homomorphism of algebras v : F → F ′ such that u = (v ⊗ idB) ◦ u. The following result was proved in [Ma3] (see. Prop. 2.3 in Chapter 4): 0.1.1. Theorem. The category A ⇒ B has an initial object