Operads with general groups of equivariance, and some 2-categorical aspects of operads in Cat

Abstract

We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits. We then apply this notion to study the 2-category of algebras over an operad in Cat. We show that any operad is finitary, that an operad is cartesian if and only if the group actions are nearly free (in a precise fashion), and that the existence of a pseudo-commutative structure largely depends on the groups of equivariance. We conclude by showing that the operad for braided strict monoidal categories has two canonical pseudo-commutative structures.

K-theory for 2-categories

“…where the two vertical arrows are the S-algebra structures on C,D, satisfying two axioms. Using the structure of the operad S and following the calculations in [CG14b], we see that this amounts to the following data and axioms:…”

Operads, tensor products, and the categorical Borel construction

“…Proof. The category Λ-operads in M is the category of monoids for the composition product • M on [BΛ op , M ] constructed in [4]. Composition with F gives a functor…”

“…where both isomorphisms are induced by universal properties (see [4] for more details) and the unlabeled arrow is induced by the same argument as that for coproducts above but this time using coends. The arrow t is constructed in a similar fashion, and is the composite below.…”

“…There is a clear parallel between these definitions of different types of operads and the definitions of different kinds of monoidal category, with each given by some general schema in which varying an N-indexed collection of groups produced the types of operads or monoidal categories seen in nature. Building on the work in [4], the goal of this paper is to show that this parallel can be upgraded from an intuition to precise mathematics using the notion of action operad.…”

Categories of operators and actions of group operads

“…It will be established in the author's future work. was developed in [5] and [8] while they use different terminology "action operads." We give just sketches; for the details, we refer the reader to the literature above and the author's previous work [20].…”