# at.algebraic topology – Is there a “spectral exterior algebra” construction in higher algebra?

Given a ring spectrum $$R$$ and an $$R$$-module $$E$$, we have the spectral symmetric algebra $$mathrm{Sym}_R(E)$$ of $$E$$ over $$R$$, defined by
begin{align*} mathrm{Sym}_R(E) &overset{mathrm{def}}{=} mathrm{colim}_{mathbb{F}}(Delta_{E})\ &cong bigoplus_{ninmathbb{N}}E^{otimes_mathbb{S}n}_{mathsf{h}Sigma_{n}}, end{align*}
where $$mathbb{F}overset{mathrm{def}}{=}mathsf{FinSets}^simeq$$ is the groupoid of finite sets and permutations. As A Rock and a Hard Place showed here, the $$mathbb{E}_infty$$-ring $$mathrm{Sym}_R(E)$$ comes with a natural grading by the sphere spectrum, inducing a $$mathbb{Z}$$-grading on $$pi_0(mathrm{Sym}_R(E))congmathrm{Sym}_{pi_0(R)}(pi_0(E))$$. So e.g. picking $$R=E=mathbb{S}$$, gives
begin{align*} pi_0(mathrm{Sym}_{mathbb{S}}(mathbb{S})) &overset{mathrm{def}}{=} pi_0(mathbb{S}{t})\ &cong mathbb{Z}(t), end{align*}
which carries the natural $$mathbb{Z}$$-grading.

However, the $$pi_0$$ of an $$mathbb{S}$$-graded ring can be more complicated than just a commutative $$mathbb{Z}$$-grading, and for instance allows for the multiplication on the $$pi_0$$ to be supercommutative, satisfying $$ab=(-1)^{deg(a)deg(b)}ba$$. This led me to the following pair of questions:

1. Is there an “spectral exterior algebra” construction $$bigwedge_RE$$, whose $$pi_0$$ is the $$mathbb{Z}$$-graded supercommutative exterior algebra $$bigwedge_{pi_0(R)}pi_0(E)$$? If so, does it come with an $$mathbb{S}$$-grading?
2. One of the more homotopy-theoretic points of view on symmetric and exterior algebras is that the passage from the former to the latter corresponds to considering a larger portion of the sphere spectrum. More generally, do we have an $$mathbb{N}$$-indexed sequence of “higher exterior algebra” constructions $$mathrm{Sym}_R(E)$$, $$bigwedge_R(E)$$, $$bigwedge^{mathbf{2}}_R(E)$$, $$ldots$$?
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