**This is not an answer, but rather a comment to A Rock and a Hard Place's answer.**

Here's another idle musing: what about replacing the relation $ab=(-1)^{\deg(a)\deg(b)}ba$ with coherent homotopies, having an analogue of $\mathbb{E}_{\infty}$ for graded-commutativity?

I'm not sure what's the best way to do this, but I see a possible (probably not totally satisfying) one. The basic idea is to generalise from the characterisation of exterior algebras as free graded commutative algebras.

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**1. The case of classical exterior algebras.**

Recall that:
 1. A $\mathbb{Z}$-graded $R$-algebra $A_\bullet$ is **$\mathbb{Z}$-graded commutative** if we have $$ab=(-1)^{\deg(a)\deg(b)}ba$$ for each $a,b\in A_\bullet$.
 2. The exterior algebra $\bigwedge_RM$ on an $R$-module $M$ is the **free $\mathbb{Z}$-graded commutative algebra on $M$**: the assignment $M\mapsto\bigwedge_RM$ defines a functor
 $$\textstyle\bigwedge_R\colon\mathsf{Alg}_R\to\mathsf{CommGr}_{\mathbb{Z}}\mathsf{Alg}_R$$
 that is left adjoint to the forgetful functor $\mathsf{CommGr}_{\mathbb{Z}}\mathsf{Alg}_R\hookrightarrow\mathsf{Alg}_R$.
 3. The category of $\mathbb{Z}$-graded commutative algebras embeds fully faithfully into that of [$\tau_{\leq1}\mathbb{S}$-graded commutative algebras](https://mathoverflow.net/q/403217), which are lax symmetric monoidal functors $(\tau_{\leq1}\mathbb{S},\otimes,0)\to(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$.

**Question**: is $\bigwedge_RM$ not only the free $\mathbb{Z}$-graded commutative algebra on $M$, but also the free $\tau_{\leq1}\mathbb{S}$-graded commutative algebra on $M$?

**2. The graded commutativity condition.**

As [I mentioned in my other question](https://mathoverflow.net/q/403217), a $\tau_{\leq1}\mathbb{S}$-graded $R$-algebra is a lax monoidal functor from the $1$-truncation of the sphere spectrum to $(\mathsf{Mod}_R,\otimes_{R},R)$, and it is $\tau_{\leq1}\mathbb{S}$-graded _commutative_ when this functor is _symmetric_ lax monoidal.

Now, for a functor $(\tau_{\leq1}\mathbb{S},\otimes,0)\to(\mathsf{Mod}_R,\otimes_{R},R)$ to be (_symmetric_) lax monoidal [is the same](https://mathoverflow.net/questions/130616) as for it to be a (_commutative_) monoid under the Day convolution monoidal structure on $\mathbf{Fun}(\tau_{\leq1}\mathbb{S},\mathsf{Mod}_R)$. So, given an $R$-module $M$, we can consider the constant functor
$$\Delta_M\colon\tau_{\leq1}\mathbb{S}\to\mathsf{Mod}_R$$ on $M$ and consider the free commutative $\otimes_{\mathsf{Day}}$-monoid on $\Delta_M$. If the answer to the question above is yes, then this is the exterior algebra of $M$.

**3. Spectral exterior algebras.**

Now we can repeat the same strategy for module spectra: given a ring spectrum $R$ and an $R$-module $M$, pick the constant functor
$$\Delta_M\colon\tau_{\leq k}\mathbb{S}\to\mathsf{ModSp}_R$$
and apply the free $\mathbb{E}_{\infty}$-monoid functor to $\Delta_M$ with respect to the Day convolution monoidal structure on $\mathsf{Fun}(\tau_{\leq k}\mathbb{S},\mathsf{ModSp}_R)$. The result is then a possible candidate for the spectral "$k$th higher" exterior algebra of $M$ over $R$.

**4. The catch.**

I'm not sure if this really gives a desirable result at all. For instance, does it recover spectral symmetric algebras when $k=0$?