Suppose $X$ is a finite $n$-connected CW complex, then the function spectrum $F(\Sigma^\infty_+ X,S^0)$ is an $E_\infty$-ring spectrum, induced by the diagonal of $X$. I believe it is known that if we consider $F(\Sigma^\infty_+ X,S^0)$ as an $E_n$-ring spectrum, its Koszul dual is the $E_n$-ring spectrum $\Sigma^\infty_+ \Omega^n X$. Let me sketch an argument:
The Koszul dual $E_n$-ring spectrum is computed as the Spanier-Whitehead dual of the factorization homology $\int_{(\mathbb{R}^n)^+}(-)$. We know factorization homology commutes with taking suspension spectra, so
$\int_{(\mathbb{R}^n)^+} \Sigma^\infty_+ \Omega^n X \simeq \Sigma^\infty_+ \int_{(\mathbb{R}^n)^+} \Omega^n X$. By work of Ayala-Francis, the latter is known to be equivalent to $\Sigma^\infty_+ B^n\Omega^n X\simeq \Sigma^\infty_+ X$.
The tricky part is then to figure out the $E_n$-coalgebra structure on the result. The $E_n$-coalgebra structure is obtained from the functoriality of factorization homology with respect to pinch maps. Hence, the $E_n$-coalgebra structure can be computed before taking suspension spectra. From this question, I learned that in a cartesian category like $(\mathrm{Top}_*,\times)$, all $E_n$-coalgebras come from the diagonal, so the $E_n$-coalgebra structure of $X$ induced from factorization homology has to be coming from the diagonal. The dual of this is then the original $E_n$-algebra structure on $F(\Sigma^\infty_+ X,S^0)$.
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Is there a usual reference for this fact?
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Is the supplied proof correct?
Best Answer
I am not sure if this gives what you want, but maybe it is: I went in the other direction in a paper The McCord model for the tensor product of a space and a commutative ring spectrum, in Progress in Math. 215 (2003).
Let $D(Y) = F(Y,S)$ denote the $S$-dual of a spectrum $Y$. Given a space $X$, $D(\Sigma^{\infty}_+X)$ - let's write this as $D(X_+)$ - is a commutative augmented $S$-algebra using the diagonal on $X$. The category of such things is tensored over based spaces, and I consider the augmented commutative algebra $S^n \otimes D(X_+)$. (This can surely be viewed as factorization homology.)
This has an $E_n$-coalgebra structure using the naturality in the $S^n$ variable, and so the $S$--dual, $D(S^n \otimes D(X_+))$ is an $E_n$-algebra. In my paper, I observe that when $X$ is finite and $n$-connected, this $E_n$-algebra identifies with the $E_n$-algebra $\Sigma^{\infty}_+ \Omega^n X$. (I prove a more general result, and compare with a paper of Greg Arone. I show that certain Goodwillie towers he wrote down are obtained by taking the $S$--dual of filtered objects that I construct.)