The profinite completion of a (discrete) group is the limit (in the category of topological groups) over the diagram with objects the quotient groups where is a normal subgroup of with finite index, and morphisms induced from the lattice of subgroups of .
Note that the profinite completion actually is a profinite group, and there is a canonical homomorphism .
More formally, we note that for any group , the family of its normal finite index subgroups forms a cofiltered category under inclusion. (Denote it by .) The assignment of to gives a functor from to the category of finite groups. It is thus a profinite group in the sense given in that entry, i.e. a pro-object in the category of finite groups. This pro-object is the profinite completion of .
The above topological version of this, with which we started, is obtained by means of the equivalence between the category of pro-(finite groups) and that of the groups internal to profinite spaces that is by taking the limit in the category of topological groups of the diagram of (discrete) finite groups that the above construction gives one.
and that functor does have a left adjoint. If we restrict that ‘pro-adjoint’ to the subcategory of given by the ‘constant’ pro-objects, then the result is the pro-finite completion construction that is given above. Because of this, if we think of the natural functor to be an inclusion, i.e. think of an object as a pro-object indexed by the one arrow category, we can give a universal property for the pro-finite completion of a group . This universal property gives a universal cone from to finite groups, and just encodes the obvious fact that any homomorphism from to a finite group factors through one of its finite quotient groups. If we write for the pro-finite completion, the universal cone is a map in .
The subcategory of consisting of the pro- groups and the continuous homomorphisms between them will be denoted .
This notation now has two definitions, but, as the corresponding categories are equivalent, this causes no problem.
The categories of the form form varieties in . Recall that a variety in any algebraic context means a subcategory of ‘algebras’ closed under products, subobjects and quotients. We note the condition on implies the closure of under finite products, so is what is called a pseudovariety. The category is monadic over the category of spaces. This means that free objects exist in all the . A good reference for this is Gildenhuys and Kennison, (1971), see below.
Consider the profinite completion of the fundamental group of an complex projective variety . Since has an underlying topological space, its fundamental group of loops can be defined in the usual way. But one can also define the algebraic fundamental group . This is a profinite group, which is isomorphic to the profinite completion of .
The profinite completion of the integers is
(Beware there are two possible interpretations of this term. One is handled in the section above, being profinite completion of the homotopy type of a space. The entry linked to here treats another more purely topological concept.)
Profinite completion of groups is a special case of a general process that ‘completes’ a category together with a ‘forgetful functor’ to some ‘base’ category, replacing it by a category which is equational/monadic over the base.
Artin and Mazur in their lecture note on étale homotopy introduced a process of profinite completion, generalising that for groups in as much as the profinite completion of an Eilenberg-Mac Lane space having as fundamental group has the profinite completion og as its fundamental group. (WARNING: This needs a bit more detail to make it true! so this part of the entry needs more work.)
L. Ribes and P. Zalesskii, 2000, Profinite groups , volume 40 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge , Springer-Verlag, Berlin.
J. Dixon, M. du Sautoy, A. Mann and D. Segal, 1999, Analytic pro-p groups, volume 61 of Cambridge Studies in Advanced Mathematics , Cambridge Univ. Press.