On measurings of algebras over operads and homology theories

TL;DR

This paper develops a comprehensive theory of coalgebra measurings between algebras over operads, exploring their impact on homology theories, universal constructions, and module categories, extending Sweedler's notion to a broad operadic context.

Contribution

It introduces and studies coalgebra measurings for algebras over any operad, establishing universal constructions and their relations to homology and modules.

Findings

Maps on Hochschild and Lie algebra homology induced by measurings

Construction of universal measuring coalgebras for operad algebras

Development of measuring comodules and the Sweedler product

Abstract

The notion of a coalgebra measuring, introduced by Sweedler, is a kind of generalized ring map between algebras. We begin by studying maps on Hochschild homology induced by coalgebra measurings. We then introduce a notion of coalgebra measuring between Lie algebras and use it to obtain maps on Lie algebra homology. Further, these measurings between Lie algebras satisfy nice adjoint like properties with respect to universal enveloping algebras. More generally, we introduce and undertake a detailed study of the notion of coalgebra measuring between algebras over any operad $\mathcal O$. In case $\mathcal O$ is a binary and quadratic operad, we show that a measuring of $\mathcal O$-algebras leads to maps on operadic homology. In general, for any operad $\mathcal O$, we construct universal measuring coalgebras to show that the category of $\mathcal O$-algebras is enriched over coalgebras. We develop measuring comodules and universal measuring comodules for this theory. We also relate these to measurings of the universal enveloping algebra $U_{\mathcal O}(\mathscr A)$ of an $\mathcal O$-algebra $\mathscr A$ and the modules over it. Finally, we construct the Sweedler product $C\rhd \mathscr A$ of a coalgebra $C$ and an $\mathcal O$-algebra $\mathscr A$. The object $C\rhd \mathscr A$ is universal among $\mathcal O$-algebras that arise as targets of $C$-measurings starting from $\mathscr A$.