In Čech cohomology, in its traditional form, you resolve the space by an inverse system or pro-object, , of nerves of open covers (or if you prefer the nerves of the corresponding sheaf of groupoids). For constant and Abelian coefficients (and we will keep to those for the moment), , we throw at this pro-object. As cohomology switches ‘variance’ we get a direct system or ind-object of Abelian groups, and usually we then take its (direct) limit (i.e. its colimit) to get .
Now dualise … . Replace cohomology by homotopy , but although there will be some very useful long exact (Puppe) sequences available in homotopy, we cannot go this way to get them! In dualising, things can go slightly ‘wrong’. We have is a pro-object and so we might want to take limits not colimits to get the thing we want, but is not an exact functor on the usual categories available here, such as groups, sets, modules, etc.
There is another problem, coherence! is a pro-object in the homotopy category of simplicial sets, not in the category of simplicial sets itself. It is a homotopy coherent system, but the choices of covering refinement maps cause the problem.
We are very near the topics discussed in shape theory, Čech methods and strong shape theory (when that has been written!). One interpretation of Čech homotopy is shape theory, but that does not usually handle the ‘homotopy is dual to cohomology’ idea so this will need more work here.
As explained in Čech methods, given an open cover of a space and a refinement of for a given set in there may be many different sets in containing it, so the seemingly natural way to get a simplicial map
involves a choice. We choose a ‘refinement mapping’ so that for each , . There is an obvious extension to -simplices. We send to . This still does not get us a functor from to . The fact that we had to choose make it unlikely in the extreme that given three (or more) covers, each a refinement of the next, that the resulting diagram of simplicial sets will turn out (by some miracle) to be commutative. However once we choose a refinement maps for each pair , we do get a homotopy coherent diagram
with the coherence being able to be spelled out completely, (see Coverings, Hypercoverings and Homotopy Coherence, thesis (1999) from Bangor, by Alinor Abdul Kadir, available here).
The Vietoris complex is an alternative way of obtaining a simplicial complex (or simplicial set) from a space and an open cover . The disadvantage of it is that whilst the nerve of the open cover would be a finite simplicial set if the open cover is finite (for instance this is useful when handling compact spaces as we can replace the indexing set of all open covers by that of finite open covers without changing the result up to isomorphism), the Vietoris complex has as many vertices as there are points in . It has the advantage that the induced maps between the simplicial complexes are independent of the choice of refinement maps so we do get a pro-simplicial set, . We therefore get a well defined pro-homotopy type? in the homotopy category . (The category has various model category structures and cofibration category structures, with related basic notions of homotopy equivalence. These are discussed in the entry on pro-homotopy theory.)
How are the two ways around the coherence problem related?
There is a result Dowker's Theorem which shows that the geometric realisations of and are homotopy equivalent. (This explains why Čech and Vietoris homology are isomorphic.) Given a commutative diagram of spaces (or simplicial sets), if you change some of the spaces to homotopy equivalent spaces then you can build from those new spaces, a homotopy coherent diagram equivalent to the original one. Using Dowker’s theorem we get that is a homotopy coherent diagram, (by a second route).
One obvious case to investigate of the above is to take a pointed space , apply the homotopy group functor to the Čech or Vietoris complex to get a progroup, . This will lead to long exact sequences for pairs, fibrations etc., but these will be of progroups so are less amenable to study. If, on the other hand, you take the limit of these progroups, as mentioned above, you destroy the exactness of the sequences and thus, to a large extent, the simplicity of their use.
The Čech homotopy group of a pointed space is defined to be .
If is compact metric then it has countable cofinal subcategories of covers, so the progroups can be replaced up to isomorphism by inverse sequences, indexed by the natural numbers. (These are sometimes called towers.)
Here are two specific examples:
The Warsaw circle, : The progroup is isomorphic to the constant tower with value , the bonding maps being the identity maps.
The dyadic solenoid?: This time the progroup is isomorphic in to the inverse sequence
where if is the generator of , .
This second example is certainly not isomorphic to a constant one. Note that it has trivial limit group, so is it possible to get some ‘useful’ limiting information from the progroup by taking some other type of ‘limiting’ approach? Yes. There are two related ways.
The derived functors of lim?. Applied to the above progroup you get a definite measure of the non-triviality of the progroup. (This example is in that entry.)
(A second method will be put here but will need additional work over on proper homotopy theory.)
That is a start on it anyhow!