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Given a $\mathbb{C}$-linear category $\mathrm{C}$, let’s understand $\mathbf{HH}(\mathrm{C})$, the Hochschild homology of $\mathrm{C}$ as natural transformation. Then for any $A\in \mathbf{HH}(\mathrm{C})$,$E\in \mathrm{C}$, and $f, g\in \mathbf{End}(E,E)$ we can then deform (in a naive way) the composition of morphism $f\circ g$ as $A\circ(f\circ g)$.

However, when $f=g=\mathbf{Id}$, this deformation is not necessary preserving $\mathbf{Id}$, I wonder is there any other kind of deformation given by Hochschild homology that can preserve the $\mathbf{Id}$(probably in a non-naive way)?

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  • $\begingroup$ Deformation using $A \in \mathbf{HH}_\bullet(\mathrm{C})$ modifies $f \circ g$ to $A \circ (f \circ g)$, but typically fails to preserve the identity morphism, as $A \circ \mathbf{Id}_E \neq \mathbf{Id}_E$. $\endgroup$
    – Furdzik Zbignew
    Commented Nov 24 at 18:58

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