Bilinear pairings on two-dimensional cobordisms and generalizations of the Deligne category

The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. It is semisimple for generic values of the parameter $t$ while producing categories of representations of the symmetric group when modded out by the ideal of negligible morphisms when $t$ is a nonnegative integer. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. The Deligne category and its semisimple quotients admit similar interpretations. This viewpoint coupled to the universal construction of two-dimensional topological theories leads to multi-parameter monoidal generalizations of the partition and the Deligne categories, one for each rational function in one variable.