this is a sub-entry of A Survey of Elliptic Cohomology, see there for background and context.
-dimensional Riemanninan field theories are symmetric monoidal functors from -dimensional Riemannian bordisms to topological vector spaces.
A field theory is very similar to a representation of a group. Only where a representation of a group is a functor from the delooping of to Vect, an FQFT is a representation of a more complicated domain category.
how does topology enter?
of Riemannian bordisms equipped with a continuous map to .
Notice that does depend covariantly on . This means that is contravariant in .
When special structure is around, however, we also have a push-forward of such functors along morphisms.
Example: push-forward to the point: for as above and the unique map to the point heuristically we want a map
heuristically the pushforward
acts on field theories over
by the assignment
for instance when then . This is clearly reminiscent of the pushforward of a sheaf along a continuous functions and suggests that should be looked at as a sheaf on . It is however not so, since if then an object in (i.e., a -dimensional closed manifold with a map to ) cannot in general be reconstructed from and . On the other hand, such a reconstruction is possible if one allows objects in to have -dimensional boundaries. This point of view leads to extended topological quantum field theory.