A cool construction of Hochschild homology (that I saw on B. Antieau's website [here](https://antieau.github.io/2021/01/15/filtered2020.html) ) is the following:


Let $k$ be a commutative ring, then denote by $\mathfrak{a}\text{CAlg}\_k$ the category of animated commutative rings. For any $x\in BS^1$, we have a natural forgetful functor
$$x^\*:\mathfrak{a}\text{CAlg}\_k^{BS^1}\to \mathfrak{a}\text{CAlg}\_k.$$
Then Hochschild homology is the left adjoint $x\_!:\mathfrak{a}\text{CAlg}\_k\to \mathfrak{a}\text{CAlg}\_k^{BS^1}$. I've not found a proof that this does agree with the "usual" Hochschild homology of commutive rings, is there a good reference for this?


My main question is in the non-commutative setting. Hochschild homology has a natural definition for non-commutative DG-algebras [(see for instance here)](https://mathoverflow.net/questions/263429/topological-hochschild-homology-and-hochschild-homology-of-dg-algebras). DG-algebras are, at least in characteristic zero, equivalent to simplicial commutative algebras, so we can (probably) use $x\_!$ as a definition for Hochschild homology of commutative DG-algebras (even if I'm not sure if this is backwards-compatible). My question is if we can use $x\_!$ to define Hochschild homology for non-commutative DG-algebras. I think non-commutative DG-algebras are equivalent to simplicial non-commutative rings, and so we can consider the category of animated non-commutative rings $\mathfrak{a}\text{Alg}\_k$ and the forgetful functor
$$x^\*:\mathfrak{a}\text{Alg}\_k^{BS^1}\to \mathfrak{a}\text{Alg}\_k$$
and the left adjoint $x\_!$ thereof.


**Edit:** Maybe animated non-commutative rings is not the right way of going about it, but what I'm probably most interested in is having a **universal property** for non-commutative Hochschild homology, which is maybe a more amenable quesion.



Defining Hochschild homology of non-commutative DG-algebras with animated rings with a circle action
No, this does not work in the non-commutative case. In general we have $HH(A)=A\otimes\_{A\otimes A^{\mathrm{op}}} A$, and this is only a $k$-module, not an algebra. If $A$ is commutative, the tensor product happens to compute coproducts/pushouts of commutative $k$-algebras, and we have $HH(A)=\operatorname{colim}\_{S^1}A=x\_!(A)$ since $S^1=\*\coprod\_{\*\sqcup \*}\*$, but in the non-commutative case the tensor product does not have such an interpretation.


To see that the group $S^1$ acts on $HH(A)$ in general, one can use the formalism of factorization homology, $HH(A)=\int\_{S^1} A$, which makes the functoriality on $BS^1$ apparent. A more classical approach is to use the fact that the usual "Hochschild complex" extends to a cyclic $k$-module (a functor on Connes' cyclic category $\Lambda$), whose geometric realization acquires an action of $S^1$ due to the $\infty$-groupoid completion of $\Lambda$ being $BS^1$. A reference for the latter approach is Appendix B of the article by Nikolaus and Scholze: <https://arxiv.org/pdf/1707.01799.pdf>


