Higher Strictification Data

Strict Elements

Definition The anima of $n$-(strict-)elements of a spectrum $X$ is given by $nElem(X):=Ma{p}_{Al{g}^{(n)}(S)}(\mathbb{Z},{\Omega }^{\infty }X)$

Remark: Koszul-dual description The right hand side can be rewritten using toposic Koszul duality, namely that

$Ma{p}_{Al{g}^n(S)}(\mathbb{Z},{\Omega }^n{\Omega }^{\infty }{\Sigma }^n(X))\simeq Ma{p}_S(B^n\mathbb{Z},{\Omega }^{\infty }{\Sigma }^{n}X)$ Example: 1 and 2 elements There are two instances of this calculation that will be important

• $1Elem(X)\simeq Ma{p}_S(B\mathbb{Z},{\Omega }^{\infty }\Sigma X)\simeq Ma{p}_{Sp}(\Sigma \mathbb{S},\Sigma X)\simeq {\Omega }^{\infty }X$
• $2Elem(X)\simeq Ma{p}_S(B^2\mathbb{Z},{\Omega }^{\infty }{\Sigma }^{2}X)\simeq Ma{p}_{Sp}({\Sigma }_{+}^{\infty }\mathbb{C}{P}^{\infty },{\Sigma }^{2}X)$

That is, they’re the spectra respectively encoding $X$-cohomology at $\ast$ and at $\mathbb{C}{P}^{\infty }$.

Remark: forgetful maps We have functors

$U_n:nElem(X)\to (n-1)Elem(X)$ giving by forgetting $E_n$ maps to $E_{n-1}$.

Definition A $E_2$ strictification of $x\in 1Elem(X)$ is a lift along $U_2$ to $\tilde{x}\in 2Elem(X)$.

Remark: where this is going Combing with the example calculation, the main result of the paper of Grossman-Naples is that the map $U_1:X^{{\Omega }^2\mathbb{C}{P}^{\infty }}\to X$ is the one induced by restriction along $\mathbb{C}{P}^1\to \mathbb{C}{P}^{\infty }$.

Remark Doing this in families, we can get $E_n$-stacks that are moduli of $n$-elements. The local geometry on these stacks is “strict deformation theory”.

Moduli

Proposition The functor

$CAlg(Sp)\stackrel{U}{\to }Sp\stackrel{{\Omega }^n}{\to }Sp\stackrel{nElem}{\to }S$ is corepresented by the spherical group ring $\mathbb{S}[B^n\mathbb{Z}]\in CAlg(Sp)$. That is

$Spec(\mathbb{S}[B^n\mathbb{Z}])\simeq nElem({\Omega }^n\underline{(-)})$ Proof: by unpacking the adjunctions.

It turns out these are loop stacks of the flat affine line

Proposition $Spec(\mathbb{S}[B^n\mathbb{Z}])\simeq {\Omega }_0^{n}Spec(\mathbb{S}[\mathbb{N}]):={\Omega }^n\mathbb{A}$

Example Let $V^{\otimes }$ be a symmetric monoidal stable $\infty$-category $Elem(Pic(V))\simeq Elem(\Omega Br(V))\simeq {\Omega }^n\mathbb{A}(Br(V))$ That is, the strict picard group is the anima of points $\{x:Spec(Br(V))\to {\Omega }^n\mathbb{A}\}$.

Example the element $1\in {\Omega }^{\infty }MU$ (this discussion could explain the appearance of Wilson spaces)