Defining Hochschild homology of non-commutative DG-algebras with animated rings with a circle action A cool construction of Hochschild homology (that I saw on B. Antieau's website here ) 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). 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.
 A: 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
