Let $G$ be a compact connected semisimple Lie group. The algebro-geometric definition of the affine Grassmannian is the coset space $$\mathcal{G}r=G_{\mathbb{C}}(\mathcal{\mathbb{C}((t))}/G_{\mathbb{C}}(\mathbb{C}[[t]]).$$ Technically, $\mathcal{G}r$ is an ind-scheme over $\mathbb{C}$, realized as an inductive limit of schemes.

On the other hand, we have a differential-geometric version of the affine Grassmannian described in Chapters 7 and 8 of Pressley-Segal. In particular, they construct $\mathcal{G}r_0^{\mathfrak{g}}$. This is homotopy-equivalent to $\Omega G$.

I am looking for a reference that explicitly relates these two notions of the affine Grassmannian. In particular, I am seeking answers to some of the questions below.

In what categories is it reasonable to compare these versions of the affine Grassmannian?

In these categories, are the two versions isomorphic?