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
.
When
is a binary and quadratic operad, we show that a measuring of
–algebras
leads to maps on operadic homology. In general, for any operad
in vector spaces
over a field
,
we construct universal measuring coalgebras to show that the category of
–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
of an
–algebra
and the modules over it. Finally, we describe the Sweedler product
of a coalgebra
and an
–algebra
. The object
is universal among
–algebras that arise as
targets of
–measurings
starting from
.
