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I added to Hochschild homology in the section Function algebra on derived loop space a statement and proof of the theorem that “the function complex $C(\mathcal{L}X)$ on the derived loop space $C(\mathcal{L}C)$ is the Hochschild homology complex of $C(X)$”.
There is a curious aspect to this: we are to compute the corresponding pushout in $\infty$-algebras. But in the literature on Hochschild homology, the pushout is of course taken not in algebras, but in modules
$HH_k(C(X)) := Tor_k(C(X), C(X))_{C(X) \otimes C(X)} \,.$So how is that $HH$-complex actually an derived algebra?
The solution of this little conundrum is remarkably trivial using Badzioch-Berger-Lurie’s result on homotopy T-algebras .This tells us that we may model the $\infty$-algebras as simplicial copresheaves on our syntactic category $T$, using the left Bousfield localizatoin of the injective model structure at maps that enforce the algebra property.
But since we are computing a pushout and since the traditional bar complex provides a cofibrant resolution of our pushout diagram already in the unlocalized structure, and since left Bousfield localization does not affect the cofibrations, due to all these reasons we may (or actually: have to) compute the pushout of $\infty$-algebras as just a pushout in simplicial copresheaves.
In particular it follows that the pushout of our product-preserving coproseheaves is not actually product-presrving itself. Instread, it is (the simplicial set underlying) the standard Hochschild complex. So everything comes together. We know that if we wanted to find the actual $\infty$-algebra structure on this, we’d have to form the fibrant replacement in the localized model structure. That would make a bit of machinery kick in and actually produce the $\infty$-algebra structure on the Hochschild complex for us.
But if we don’t feel like doing that, we don’t have to. The homotopy groups of our simplicial copresheaf won’t change by that replacement.
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