Let us consider the functors from dg algebras to graded spaces given by Hochshcild homology and cyclic homology. Then it is well known that in degree $0$, both functors are given by $A\longmapsto A/[A,A]$. However, if $A$ is a usual associative algebra over a field of characeristic $0$, and if we pick a free model $B=(TV,d)\longrightarrow A$, the homology of $B/[B,B]$ is $\mathrm{HC}_*(A)$. In this sense, HC is the right Taylor expansion for the abelianization functor in characteristic zero. At some point I naïvely thought that this would give HH instead. This motivates the following questions:
Q1. How should I have known that taking the left derived functor of abelianization in the model category $\mathsf{Alg}$ would give HC and not HH, even though at $t=0$ both theories coincide?
Q2. What other 'real life' examples are there of this phenomenon? (If the example is in $\mathsf{Alg}$, better!)
Q3. Can one describe the functors that coincide with HC and HH at $t=0$ in the sense above, but nonetheless give different theories in large degrees? Or at least give examples, as in Q2?
Disclaimer I am aware of the correct definition of $\mathrm{HH}$ as a funtor in $\mathsf{Alg}$ in terms of free models, where one needs to correct the $0$-forms $B$ by $1$-forms on $B$ and build a double complex. Similarly, $\mathrm{HC}$ in arbitrary characteristic needs a double complex built from the previous one, making it $2$-periodic, which incorporates the universal derivation from $0$-forms to $1$-forms. This gives me a "technical" answer, I am looking for something a bit less technical but still convincing.
