#### Volume 22, issue 3 (2022)

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On measurings of algebras over operads and homology theories

### Abhishek Banerjee and Surjeet Kour

Algebraic & Geometric Topology 22 (2022) 1113–1158
##### Abstract

The notion of a measuring coalgebra, introduced by Sweedler, induces generalized maps between algebras. We begin by studying maps on Hochschild homology induced by measuring coalgebras. We then develop a notion of measuring coalgebra 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 develop the notion of measuring coalgebras for algebras over any operad $\mathsc{𝒪}$. When $\mathsc{𝒪}$ is a binary and quadratic operad, we show that a measuring of $\mathsc{𝒪}$–algebras leads to maps on operadic homology. In general, for any operad $\mathsc{𝒪}$ in vector spaces over a field $K$, we construct universal measuring coalgebras to show that the category of $\mathsc{𝒪}$–algebras is enriched over $K$–coalgebras. We develop measuring comodules and universal measuring comodules for this theory. We also relate these to measurings of the universal enveloping algebra ${U}_{\mathsc{𝒪}}\left(\mathsc{𝒜}\right)$ of an $\mathsc{𝒪}$–algebra $\mathsc{𝒜}$ and the modules over it. Finally, we describe the Sweedler product $C▹\mathsc{𝒜}$ of a coalgebra $C$ and an $\mathsc{𝒪}$–algebra $\mathsc{𝒜}$. The object $C▹\mathsc{𝒜}$ is universal among $\mathsc{𝒪}$–algebras that arise as targets of $C$–measurings starting from $\mathsc{𝒜}$.

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##### Keywords
measuring coalgebras, algebras over an operad
##### Mathematical Subject Classification 2010
Primary: 16T15, 18D50
##### Publication
Received: 12 January 2020
Revised: 1 March 2021
Accepted: 29 March 2021
Published: 25 August 2022
##### Authors
 Abhishek Banerjee Department of Mathematics Indian Institute of Science Bengaluru India Surjeet Kour Department of Mathematics Indian Institute of Technology, Delhi Delhi India