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Is there an analogue of the module of differentials for "higher order derivations" in the Hochschild/cyclic senses?
Note added by YC: the definition below of the cyclic sub-complex is incorrect; and the "higher order derivations" referred to here are traditionally known (since the 1940s) as n-cocycles.
$\...
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Categorical Kähler differentials and the Leibniz rule
From nlab, the module of Kähler differentials over some category $\mathcal{C}$ is the free functor:
$$\Omega: \mathcal{C} \to \mathsf{Mod_{\mathcal{C}}}$$
left-adjoint to the (forgetful) embedding:
$$...
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