Mapping spaces between spherical polynomial (monoid) ring spectra

Determine the homotopy type of the mapping space Map_{CAlg(Sp)}(Sph[N^i], Sph[N^j]) for any integers i, j ≥ 1, where Sph[N^k] denotes the spherical monoid ring on the free commutative monoid N^k. This problem asks for a computation of maps between spherical polynomial ring spectra and addresses a fundamental unresolved case of understanding maps between spherical monoid rings beyond the group-ring setting.

Background

The paper proves rigidity results for maps between spherical group rings, showing that for finitely generated abelian groups A and B the space of E_infinity-ring maps Sph[A] → Sph[B] is discrete and corresponds to group homomorphisms A → B. It also gives p-adic analogues over spherical Witt vectors.

Beyond group rings, the authors raise a broader program: characterize when a widehat-delta ring R lifts to a commutative ring spectrum Sph_R and when mapping spaces between such lifts agree with morphisms of widehat-delta rings. While significant progress is achieved for group rings, fundamental monoid-ring cases remain unresolved.

In particular, even for polynomial spherical monoid rings Sph[Ni] and SphNj, the mapping spaces in the E_infinity category are not known in any nontrivial case, motivating a concrete computation of Map_{CAlg(Sp)}(Sph[Ni], Sph[Nj]).

References

Yet, some fundamental cases of \Cref{question} remain open: for instance, one can study spherical monoid rings, rather than group rings. In fact, to the knowledge of the authors, even the case of polynomial rings is open in all nontrivial cases:

\begin{question} What is the space of maps $\Map_{\CAlg(\Sp)}(\Sph[\mathbb{N}i],\Sph[\mathbb{N}j])$ for any $i,j\geq 1$? \end{question}

— Maps between spherical group rings (2405.06448 - Carmeli et al., 10 May 2024) in Introduction, Background and motivation (Question following the statement that polynomial cases are open)