Representations of infinite species

We consider species, consisting of a possibly infinite set of rings, and bimodules between them.

Simson realised the category of representations as a functor category, which we prove is hereditary when each of the rings is semisimple.

We use purity to provide sufficient conditions, in order for a representation to decompose into indecomposables with local endomorphism rings.

For any bifunctor valued in bimodules, we functorially construct species equipped with commutativity conditions.

This generates examples coming from a range of topics, such as subobject lattices in abelian length categories, the field choice problem in persistent homology, and topological field theories with defects.