The Witt group $\mathcal{W}$ of braided fusion categories (see also the sequel paper) can be defined over any field; I am happy to restrict to characteristic $0$ if it matters.

Is $\mathbb k \mapsto \mathcal W(\mathbb k)$ an (affine?) algebraic group scheme?

Assuming $\mathcal W$ is sufficiently scheme-like for the following question to make sense, what I really want to know is:

What is the etale homotopy type of $\mathcal{W}$?