Spherical Ring Stacks from moduli of n-elements

More on anima of strict elements

Let $X$ be a spectrum, we have

$$Ma{p}_{kGp}(\mathbb{Z},{\Omega }^{\infty }X)\simeq Ma{p}_{S_{\ast }}(B^k\mathbb{Z},{\Omega }^{\infty -k}X)\simeq Ma{p}_{Sp}({\Sigma }^{\infty }{B}^k\mathbb{Z},{\Sigma }^{k}X)$$

hence, we have a representability result:

Lemma The functor $kElem:Sp\to S$ is co-represented by the spectrum ${\Omega }^k{\Sigma }^{\infty }{B}^k\mathbb{Z}$ Examples: $n=1$ and $\infty$

• $1Elem$ is corepresented by $\Omega {\Sigma }^{\infty }{S}^1\simeq \Omega \Sigma \mathbb{S}\simeq \mathbb{S}$. Indeed, $1Elem$ is “weak”/“ordinary” elements.
• $\infty Elem$ is corepresented by $B^{\infty }Z\simeq H\mathbb{Z}$, this is “strict”-elements. For example, $\infty Elem(Pic(V^{\otimes }))$ is the groupoid of strict picard elements.

Hence, $kElem$ is a sequence functors interpolation between strict and weak elements.

Example: n= 2

For example, $k=2$ case is representable by $2Elem(-)\simeq Ma{p}_{Sp}({\Omega }^2{\Sigma }^{\infty }\mathbb{C}{P}^{\infty },-)$ Hence, for a spectrum $X$, $2Elem(X)$ is an anima with homotopy groups ${\pi }_q(2Elem(X))\simeq X_{red}^{q+2}(\mathbb{C}{P}^{\infty })$ where on the right hand side is the reduced $X$ cohomology groups for pointed spaces: $X_{red}^{\bullet }(pt\stackrel{z}{\to }Z):={\pi }_{\bullet }Ma{p}_{Sp}({\Sigma }^{\infty }Z,X)\simeq {\pi }_{\bullet }Ma{p}_{S_{\ast }}(Z,{\Omega }^{\infty }X)$

As explained in Higher Strictification Data, there is a tower of forgetful maps

$\infty Elem=\lim (\cdots \to 2Elem\to 1Elem)$

Moduli Theory

Using the construction from from Higher Strictification Data (which we encountered first in the context of Complex Orientations), we can construct moduli of elements for algebraic geometry over any nice $\infty$-operad $\mathcal{O}$.

Namely, giving an $\mathcal{O}$-algebra $A$, we can associate to it the anima of $k$-elements $kElem(A)$. This is given by the composite $Al{g}_{\mathcal{O}}(S{p}^{\otimes })\stackrel{U}{\to }Sp\stackrel{kElem}{\to }S$ Which one can view as a prestack on $Af{f}_{\mathcal{O}}$.

Structure in the $n=1$ case

In the case $n=1$, we have several structures present

Representability 1-strict elements are ordinary/weak elements. We can see $1Elem(U(A))\simeq Ma{p}_{Sp}(\mathbb{S},U(A))\simeq Ma{p}_{Al{g}_{\mathcal{O}}}(F_{\mathcal{O}}(\mathbb{S}),A)$ That is, $1Elem$ is an affine scheme in $\mathcal{O}$-geometry corresponding free $\mathcal{O}$-algebra on one generator in degree $0$. $1Elem(U_{\mathcal{O}}(-))\simeq Spec(F_{\mathcal{O}}(\mathbb{S}))$

Additive structure Furthermore, this lifts to an abelian group object in anima. Because $1Elem(X)\simeq Ma{p}_{Sp}(\mathbb{S},X)\simeq {\Omega }^{\infty }X$ lifts to a $E_{\infty }$ group object, classified by the connective cover $X_{\ge 0}$.

Multiplicative structure

$1Elem\simeq {\Omega }^{\infty }:Sp\to S$ is lax-symmetric monoidal, hence, it induces a functor $Al{g}_{\mathcal{O}}(Sp)\stackrel{{\Omega }^{\infty }}{\to }Al{g}_{\mathcal{O}}(S)$ that lives over, via the forgetful functors $U_{\mathcal{O}}$, the functor $1Elem$. That is $U^S({\Omega }^{\infty }R)\simeq {\Omega }^{\infty }(U^{Sp}(R))$ Hence, $1Elem$ is a $\mathcal{O}$ group stack.

We’d like to ask what happens to these structures at $n=2$ and beyond. This will be used to study a theory of transmutations, as in the geometric approach to prismatic cohomology.