On measurings of algebras over operads and homology theories

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