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.

$\DeclareMathOperator{\Ker}{\mathrm{Ker}}\DeclareMathOperator{\Mod}{\mathrm{Mod}}\DeclareMathOperator{\Der}{\mathrm{Der}}\DeclareMathOperator{\Hom}{\mathrm{Hom}}$While reading about Hochschild cohomology, I learned that we could define derivations in terms of the Hochschild complex: writing
\begin{align*}
M &\xrightarrow{d^1} \Hom_{\Mod_R}(S,M)\\
&\xrightarrow{d^2} \Hom_{\Mod_R}(S\otimes_RS,M)\\
&\xrightarrow{d^3}\Hom_{\Mod_R}(S\otimes_RS\otimes_RS,M)\\
&\xrightarrow{d^4}\cdots.
\end{align*}
for the Hochschild cochain complex of an $R$-algebra $S$ with coefficients in an $S$-bimodule $M$, we have
$$\Der_R(S,M)\cong\Ker(d^2).$$
Now, derivations play an important role in deformation theory, and we can build an universal object corepresenting them, the **module of differentials** $\Omega_{S/R}$ of $S$ over $R$, defined by
$$\Hom_S(\Omega_{S/R},M)\cong\Der_R(S,M).$$
Naturally, this leads one to wonder about whether we have a similar universal object for the module
$$\Der^{n}_R(S,M)\cong\Ker(d^{n+1})$$
of "$n$-order Hochschild derivations of $S$ into $M$". For example, here's what such a higher derivation looks like for $n=2$ and $n=3$ (where below we identify a map $D\colon S^{\otimes_R n}\to M$ with the unique $n$-multilinear map $D\colon S^{\times n}\to M$ it represents):

- A
**second order Hochschild derivation**is a map $D\colon S\otimes_R S\to M$ satisfying the equation $$D(ab,c)-D(a,bc)=aD(b,c)-D(a,b)c$$ for each $a,b,c\in S$. - A
**third order Hochschild derivation**is a map $D\colon S\otimes_RS\otimes_RS\to M$ satisfying the equation $$D(ab,c,d)-D(a,bc,d)+D(a,b,cd)=aD(b,c,d)+D(a,b,c)d.$$ for each $a,b,c,d\in S$.

Lastly, we could also work with the cyclic complex of $S$ with coefficients with $R$, defining "higher cyclic derivations" in a similar manner. These satisfy one extra equation: $$D(a_1,\ldots,a_n)=(-1)^{n-1}D(a_n,a_1,\ldots,a_{n-1}).$$ So again, in the low degree cases, we have $D(a,b)=-D(b,a)$ and $D(a,b,c)=D(c,a,b)=D(b,c,a)$.

Now, write $\Der^{\mathrm{cycl},n}_R(S,M)$ for the set of "$n$-order cyclic derivations", and note that given an $S$-module morphism $f\colon M\to N$ and an $n$-order (cyclic) derivation $D$, the composition $f\circ D$ is still an $n$-order (cyclic) derivation. This gives us functors $\Der^{n}_R(S,-)$ and $\Der^{\mathrm{cycl},n}_R(S,-)$.

Question.The above two functors are corepresentable by $\Omega_{S/R}$ when $n=1$. Are they also corepresentable for $n\geq2$ (in the commutative case)?

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