Left Adjoints for Generalized Multicategories

TL;DR

This paper develops a framework for constructing generalized multicategories from $ ext{Cat}$-operads, extending classical symmetric multicategory theory and establishing an adjunction between operad algebras and these multicategories.

Contribution

It introduces a new construction of generalized multicategories for any $ ext{Cat}$-operad that is $ ext{Sigma}$-free, generalizing known cases and enabling equivariant extensions.

Findings

Established an adjoint pair between operad algebras and generalized multicategories.

Extended classical symmetric multicategory theory to a broader operadic context.

Provided a flexible construction allowing for equivariant generalizations.

Abstract

We construct generalized multicategories associated to an arbitrary operad in Cat that is $\Sigma$-free. The construction generalizes the passage to symmetric multicategories from permutative categories, which is the case when the operad is the categorical version of the Barratt-Eccles operad. The main theorem is that there is an adjoint pair relating algebras over the operad to this sort of generalized multicategory. The construction is flexible enough to allow for equivariant generalizations.