Polynomial functors as affine spaces

Polynomial functors are like spaces of objects (e.g. k-way tensors) without fixed size and come with an action of (products of) general linear groups. The aim of this talk is to answer the following question: what happens when you replace vector spaces by polynomial functors when defining affine spaces?

I will define polynomial functors, the maps between them and their Zariski-closed subsets and give examples of these things. Then, I will discuss how to extend some of the basic results from affine algebraic geometry to this setting. This is joint work with Jan Draisma, Rob Eggermont and Andrew Snowden.