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.