The motivation comes from rational homotopy theory. The attaching map of (from the cell structure of ) gives a graded lie algebra structure on : If , is (note that in the case we get ). If is simply connected, then as a Hopf algebra.
One of the tool for studying rational homotopy group is the homology of the loop space is the Eilenberg-Moore spectral sequence (fiber product analogue of Kunneth formula). As a special case, we have , converge strongly if is simply connected (more generally, it states , take , then get the above special case.
This goes wildly wrong if and is finite (LHS is just at degree since , but , so simply connectedness is crucial).
The claim is that the best approximation to that can see is the pro-unipotent (malcev) completion. What is it? It is the initial object in the category of pro-unipotent groups (inverse limit of unipotent groups) receiving a group homomorphism from (this depends on field ).
Here is Quillen’s Construction of pro-unipotent completion of a group: Let augmentation ideal of the group ring (generated by for ). The completed group ring is a completed Hopf algebra (using the completed tensor product). Define to be the group-like elements and such that .
If is a finite dimensional vector space over (e.g. is finite generated), then is pro-unipotent completion.
For a pro-unipotent group its Lie algebra is easy to compute: it is just the inverse limit of the individual Lie algebra, and we have mutually inverse map and (because of the unipotent assumption). Note that we can also make sense of for by using the formula and the binomial formula (which makes sense since is nilpotent).
Assume . We give some example:
, , and . Setting , we get . ; is also group-like. The pro-unipotent group is isomorphic to .
, , since , so , so so the pro-unipotent completion is trivial.
Thus the intuition is that pro-unipotent completion can only see non-torsion phenomenon. The lie algebra of the pro-unipotent completion is the primitive elements of the completed group algebra and we have .
- free group on generators. The group ring isomorphic to , , . so completed free lie algebra. Generally if and and .
The difference between profinite completion (which is inverse limit over taken over all finite index subgroup ) and pro-unipotent completion is that we are completing along versus .