Some remarks on higher Morita categories

TL;DR

This paper revisits the construction of higher Morita categories, providing simplified proofs and comparisons, and confirms their correct deloopings for $E_{n}$-algebras, advancing the theoretical understanding of these structures.

Contribution

It offers a simplified proof of the Segal condition, compares different constructions of higher Morita categories, and demonstrates correct deloopings for $E_{n}$-algebras.

Findings

Simplified proof of the Segal condition

Comparison between different constructions of higher Morita categories

Confirmation of correct deloopings for $E_{n}$-algebras

Abstract

We take another look at the construction of double $\infty$-categories of algebras and bimodules and prove a few supplemental results about these, including a simpler proof of the Segal condition and a comparison between our construction and that of Lurie. We then take a more streamlined, inductive approach to the higher Morita categories of $E_{n}$-algebras and show that these deloop correctly.