Invertibility in Category Representations
Sanjeevi Krishnan (The Ohio State University)
It is often desirable to equip a representation of a poset or more general small category with inner products on the relevant vector spaces so that the linear maps are partial isometries, maps which restrict to isometries on orthogonal complements of kernels. This sort of inner product structure can be used, for example, to simply representations of interest in multidimensional persistence, circuit design, and network coding. The existence of suitable inner product structure is much more difficult to ascertain in the general categorical setting than in the group setting. However, we can characterize the existence of a slightly weaker inner product structure as factorizability of the representation through a special dagger category called an inverse category. This factorizability admits a coordinate-free, numerical characterization that is decidable for finite categories. We give some concrete applications in circuit design. Time-permitting, we will discuss some connections between this work and a nascent theory (by others) of principle S-bundles for S an inverse semigroup. This talk is joint work with Crichton Ogle.