The coalgebraic enrichment of algebras in higher categories

TL;DR

This paper establishes a higher categorical framework showing that algebras in certain monoidal infinity-categories are enriched over coalgebras, extending classical results to the realm of higher categories and operads.

Contribution

It introduces a coalgebraic enrichment of algebras in higher categories, generalizing the universal measuring coalgebra to the infinity-categorical setting.

Findings

Enrichment, tensoring, and cotensoring of algebra categories over coalgebra categories in infinity-categories.

Higher categorical analogue of the universal measuring coalgebra.

Extension of classical results to the setting of presentably symmetric monoidal infinity-categories.

Abstract

We prove that given $\mathcal{C}$ a presentably symmetric monoidal $\infty$-category, and any essentially small $\infty$-operad $\mathcal{O}$, the $\infty$-category of $\mathcal{O}$-algebras in $\mathcal{C}$ is enriched, tensored and cotensored over the presentably symmetric monoidal $\infty$-category of $\mathcal{O}$-coalgebras in $\mathcal{C}$. We provide a higher categorical analogue of the universal measuring coalgebra. For categories in the usual sense, the result was proved by Hyland, L\'{o}pez Franco, and Vasilakopoulou.