Topological–simplicial comodule–contramodule correspondence conjecture

Let $C$ be a simplicial set. The categories $S^F$, $S_T$, $K^F$, and $K_T$, equipped with the displayed horizontal dashed Quillen adjunctions and the vertical functors induced by geometric realization, the singular complex, and induction, are arranged in a commutative (in an appropriate sense) square:

\begin{tikzcd}[row sep=2.5em] \mathcal{S}^F \arrow[d,shift \right,swap,"|-|"] \arrow[rr,shift \left, dashrightarrow, "L"] & & \mathcal{S}_T\arrow[ll,shift \left, dashrightarrow,"R"] \arrow[d,shift \right,swap,"|-|"] \\ \mathcal{K}^F \arrow[u,shift \right,swap,"\mathcal{I}\!\mathpzc{n}\mathpzc{d}_{\bullet}^F \circ{SC}"] \arrow[rr,shift \left, dashrightarrow, "L"] & & \mathcal{K}_T\arrow[ll,shift \left, dashrightarrow, "R"] \arrow[u,shift \right,swap,"\mathcal{I}\!\mathpzc{n}\mathpzc{d}\!\downarrow\circ{SC}"] \end{tikzcd}

Topological–simplicial correspondence conjecture. The left adjoint functors are either on top or on the left, and the vertical solid arrows are Quillen equivalences.

This conjectural worldview relates the topological and simplicial comodule–contramodule correspondences. It proposes compatibility between the corresponding comodule and contramodule categories under geometric realization and singular-complex constructions, together with Quillen equivalences in the vertical directions. The source does not provide a resolution status.