The coalgebraic enrichment of algebras in higher 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, López Franco, and Vasilakopoulou.

arXiv comment12 pages. To appear in JPAA. Added results on cofree coalgebras (Corollary 2.11, Proposition 3.10 and Corollary 3.11). Added an example in Remark 3.16