20 research outputs found
Operads and Jet modules
Let be an algebra over an operad in a cocomplete closed symmetric
monoidal category. We study the category of -modules. We define certain
symmetric product functors of such modules generalising the tensor product of
modules over commutative algebras, which we use to define the notion of a jet
module. This in turn generalises the notion of a jet module over a module over
a classical commutative algebra. We are able to define Atiyah classes (i.e.
obstructions to the existence of connections) in this generalised context. We
use certain model structures on the category of -modules to study the
properties of these Atiyah classes. The purpose of the paper is not to present
any really deep theorem. It is more about the right concepts when dealing with
modules over an algebra that is defined over an arbitrary operad, i.e. the aim
is to show how to generalise various classical constructions, including modules
of jets, the Atiyah class and the curvature, to the operadic context. For
convenience of the reader and for the purpose of defining the notations, the
basic definitions of the theory of operads and model categories are included.Comment: 43 page