Left Adjoints for Generalized Multicategories
A. D. Elmendorf

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 -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.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Algebraic structures and combinatorial models
