Let X be an affine variety over ℂ. Consider X(ℂ) with the classical topology, and create the topologists loop space ΩX(ℂ) of maps from the circle into X(ℂ). One can also construct the ind-variety X((t)), whose R-points are given by X(R((t))) for any ℂ-algebra R. Take the ℂ-points of this ind-variety, and give them the usual topology. Is the topological space X((t))(ℂ) thus defined homotopy equivalent to ΩX(ℂ)?
Edit: David Ben-Zvi's comment regarding using unbased loops instead of based loops is pertinent. We should be considering unbased loops (L not Ω). This checks out in the case where $X=\mathbb{G}_m$. The further commentary remark below applies to the affine Grassmannian case of based loopsalso provides positive evidence.
Commentary (based on comments): Note that the space X((t)) is not the base change of X to ℂ((t)). It isn't the restriction of scalars either, since $R\otimes \mathbb{C}((t))\neq R((t))$ in general. I'll try to come back with more details, an example or a reference in due time, but if someone has one ready to go, let me know.
Further commentary: Based on Regarding putting the comment classical topology of Stephen Griffeth, if loops on G is going to X((t))(ℂ), one should not be homotopic to the affine Grassmannian, and scared of the affine Grassmannian is ind-scheminess. ℂ((t)) has a homogenous space for the algebraic loop group, with the stabaliser natural structure of a point being topologically non-trivialtopological ring, then I guess and hence we topologise X(ℂ((t))) in the answer to usual manner, taking the original question as I asked it is nosubspace topology using a closed embedding into affine n-space for some n.
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