FQFT and cohomology
Types of quantum field thories
If one regards a quantum field theory from the point of view of FQFT as a representation of a category of cobordisms with structure, then it is often helpful to decompose this information into two pieces:
the local geometric information: the value of the field theory on cobordisms of trivial topology (punctured spheres), depending thus only on the geometry (say Riemannian geometry) of these cobordisms. For instance in the case of conformal field theory this is encoded in a vertex operator algebra (as discussed there).
the global topological information: to obtain a representation of all cobordisms it must be possible to consistently glue the local data assigned to cobordisms of trivial topology to obtain the data assigned to more complicated cobordisms: it must be possible to “sew” together small cobordisms to large ones. The condition on the local data induced this way is called the sewing constraint.