Convolution over homology

**Abstract**

Many representation theoretic gadgets can be constructed
from geometry by considering convolution products on the homology of
certain spaces. This is in itself some sort of categorification.
However, it seems that there should often be monoidal categories,
whose objects are spans, that categorify the algebraic gadgets by
replacing convolution by pullback. I am interested in exploring some
of these situations to see why and when the categories of spans
replace the defining relations with isomorphisms. I will discuss some
examples.