Restriction equivalence conjecture for semi-infinite, Hilbert, and ordinary spectra

Suppose ${\mathcal{H}}$ is equipped with a fixed polarization, and fix a decomposition

$${\mathcal{H}}={\mathcal{H}}^-\oplus {\mathcal{H}}^+$$

in the polarization class. Let $Gr_{\textrm{res}}({\mathcal{H}})$ denote the Grassmannian of decompositions in the polarization class, let $Gr({\mathcal{H}}^+)$ denote the Grassmannian of finite-dimensional subspaces of ${\mathcal{H}}^+$, and let $Gr({\mathbb R}^{\infty})$ denote the Grassmannian of finite-dimensional subspaces of a countably infinite-dimensional dense subspace ${\mathbb R}^{\infty}\subset {\mathcal{H}}^+$. A semi-infinite spectrum is a $Gr_{\textrm{res}}({\mathcal{H}})$-spectrum, and a Hilbert spectrum is a $Gr({\mathcal{H}}^+)$-spectrum. The natural restriction maps are

$$Gr_{\textrm{res}}({\mathcal{H}})\text{-spectra}\longrightarrow Gr({\mathcal{H}}^+)\text{-spectra}\longrightarrow Gr({\mathbb R}^{\infty})\text{-spectra}.$$

Restriction equivalence conjecture. There is a notion of weak equivalence of semi-infinite spectra and a notion of weak equivalence of Hilbert spectra such that the restriction map from $Gr_{\textrm{res}}({\mathcal{H}})$-spectra to $Gr({\mathcal{H}}^+)$-spectra induces an equivalence of simplicial homotopy categories. Similarly, the restriction map from $Gr({\mathcal{H}}^+)$-spectra to $Gr({\mathbb R}^{\infty})$-spectra induces an equivalence of homotopy categories.

The conjecture would identify the homotopy theories of semi-infinite spectra, Hilbert spectra, and spectra indexed on finite-dimensional subspaces of ${\mathbb R}^{\infty}$. Establishing it requires constructing semi-infinite spectrification and sphere spectra, a semi-infinite notion of stable weak equivalence, and left adjoints to the restriction maps; the source leaves these developments for future work.