Symmetric monoidal G-categories and their strictification

TL;DR

This paper develops a categorical framework for genuine symmetric monoidal G-categories, proving their classifying spaces are genuine E_ G-spaces, and extends equivariant infinite loop space theory to more general categorical inputs.

Contribution

It introduces an operadic definition and strictification theory for genuine symmetric monoidal G-categories within a broad categorical framework.

Findings

Classifying spaces of genuine symmetric monoidal G-categories are genuine E_ G-spaces.

The strictification process converts pseudoalgebras into strict genuine permutative G-categories.

The theory generalizes equivariant infinite loop space theory to category-level inputs.

Abstract

We give an operadic definition of a genuine symmetric monoidal G-category, and we prove that its classifying space is a genuine E_\infty G-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power, and Lack, to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal G-categories to genuine permutative G-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When G is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal G-categories as input to an equivariant infinite loop space machine that gives genuine G-spectra as output.