Centers of categorified endomorphism rings

TL;DR

This paper proves that for a broad class of cocomplete categories, the weak and strong Drinfeld centers of the monoidal category of cocontinuous endofunctors are the same, generalizing previous results limited to module categories.

Contribution

It extends the equivalence of weak and strong Drinfeld centers to a wider class of cocomplete categories, beyond module categories over a ring.

Findings

Weak and strong Drinfeld centers coincide for many cocomplete categories

Generalizes known results from module categories to broader settings

Provides a unifying framework for centers in categorified algebra

Abstract

We prove that for a large class of well-behaved cocomplete categories $\mathcal C$ the weak and strong Drinfeld centers of the monoidal category $\mathcal{E}$ of cocontinuous endofunctors of $\mathcal{C}$ coincide. This generalizes similar results in the literature, where $\mathcal{C}$ is the category of modules over a ring $A$ and hence $\mathcal{E}$ is the category of $A$-bimodules.