I am by the way not quite convinced that the envelopping algebra can be constructed in the way I suggested for a general operad $O$. What I use is that $U(Lie(V)) = S(Lie(V))$, i.e., that there is a $\Sigma$-module $S$ such that the enveloping algebra of a free algebra is isomorphic to $S$ applied to the free algebra. I don't know if that is true in general.

The category of projective (or flat) modules over a commutative ring with tensor product as monoidal operation.

But the posed problem assumed that we wouldn't necessarily have cokernels. I get the impression that that was the whole point of the question and I think I have managed to avoid them in my answer.

I don't understand your comment about existence of the symmetric algebra. As far as I can see the sum over all n of the symmetric tensors in $V^{\otimes n}$ has a product given by the product in the tensor algebra followed by symmetrisation. This only requires that idempotents have kernels. On the other hand a formula for the star product can not use bases as it is supposed to make sense in an arbitrary symmetric monoidal category.

That is true but it seems that the general construction of the enveloping algebra requires cokernels and we are not allowed to assume that.

Exactly what kind of category are we talking about? If it is abelian you can construct the enveloping algebra as a quotient as in the usual case. If not how do you construct the symmetric algebra?

It is enough that the source complex is projective (or the target complex injective) to construct things inductively. If you like you can replace $B$ in my example by $\mathbb Q/\mathbb Z$ in degrees $0$ and $1$ with a suitable map $A \to B$ and get the same counterexample.

I would like to add that while the failure of the Whitney trick opens up for the possibility that something new can happen in 4 dimensions (and Freedman et al closed that possibility in the topological case) it seems to be the special properties of differential geometry that gives an actual realisation of that possibility. In the case of Donaldson's work it seems to be the fact that 2-forms are of degree half the dimension of the manifold.

Well, in this case this is the answer (from the horse's mouth so to speak) and you needed to post the question in order to find out that the concept was coined by Mazur so I don't see how it could be bad form.

Personally I must admit that both examples would take longer (though far from impossible of course) to parse than versions following Noah's rule. That I think should be the reason for doing it one way or other. (It is very tricky to apply such a principle however as it is very individual. I realised that after having done some programming I tended to put in more parentheses in formulas which probably makes them more difficult to read for some, perhaps most, mathematicians.)

For me at least it started out that I learnt that it was a bad idea to end a sentence and start the next one by formulas. This makes a lot of sense and should be followed. However, once I started doing that my sense of esthetics changed and I ended up disliking sentences that start with a formula. I don't really see any rational justification for this more strict rule but I can live with it as it seems easier to apply reflexively than the rule that really does make sense.

My real point about the Steinberg extension was left out of my comment (because I forgot to make it...), I wanted to speculate that what Bloch did was relating the different kernels.

..... Note however that actual loops may wander across Bruhat cells so it doesn't seem to give a complete cocycle in the topological case.

$K_2$ is defined in terms of the Steinberg extension of $SL_n$ so perhaps the point is that you get a map from the kernel of the universal central extension to $K_2$ for all groups (if I remember correctly even for $SL_n$ and a field these kernels only stabilise from $n=3$ and onwards). In any case going to the cocycle question I think that from the Steinberg point of view it is clear that we get sections over the Bruhat cells and hence (in principle at least) a cocycle desciption for the algebraic loop group. .....