nLab
Cech homotopy

Idea and discussion

In Čech cohomology, in its traditional form, you resolve the space by a inverse system or pro-object, $\check{C}(X,-)$, 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), $A$, we throw $H^n(-,A)$ 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 ${\check{H}}^n(X,A)$.

Now dualise … . Replace cohomology by homotopy $[A,-]$, 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 $[A,\check{C}(X,-)]$ is a pro-object and so we should take limits not colimits to get thet thing we want, but $\lim$ is not an exact functor on the usual categories available here, such as groups, sets, modules, etc.

There is another problem, coherence! $\check{C}(X,-)$ 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.

Handling the problem of coherence

• Using homotopy coherence directly As explained in Čech methods, given an open cover $\beta$ of a space $X$ and a refinement $\alpha$ of $U$ for a given set in $V$ there may be many different sets in $U$ containing it, so the seemingly natural way to get a simplicial map

$$\check{C}(X,\alpha )\to \check{C}(X,\beta )$$\check{C}(X,\alpha)\to \check{C}(X,\beta)

involves a choice, We choose a ‘refinement mapping’ $\phi :\alpha \to \beta$ so that for each $V\in \alpha$, $V\subseteq \phi (V)$. There is an obvious extension to $n$-simplices. We send $⟨V_0,\dots ,V_n⟩$ to $⟨\phi (V{)}_0,\dots ,\phi (V{)}_n⟩$. This still does not get us a functor from $\mathrm{Cov}(X)$ to $\mathrm{SSets}$. 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 $(\alpha ,\beta )$, we do get a homotopy coherent diagram?

$$\check{C}(X,-):\mathrm{Cov}(X)\to \mathrm{SSets}.$$\check{C}(X,-) : Cov(X)\to SSets.

with the coherence being spelled out completely.

• Using the Vietoris complex The Vietoris complex is an alternative way of obtaining a simplicial complex (or simplicial set) from a space $X$ and an open cover $\alpha$. 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 $X$. 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, $V(X,-):\mathrm{Cov}(X)\to \mathrm{SSet}$. We therefore get a well defined pro-homotopy type $V(X)$ in the homotopy category $\mathrm{Ho}(\mathrm{pro}-\mathrm{SSet})$. (The category $\mathrm{pro}-\mathrm{SSet}$ has various model category structures and cofibration category structures, with the same basic notion of homotopy equivalence.

How are the two ways around the coherence problem related?

There is a result Dowker's Theorem which shows that the geometric realisations of $V(X,\alpha )$ and $\check{C}(X,\alpha )$ 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 by 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 $\check{C}(X,-)$ is a homotopy coherent diagram, (by a second route).

That is a start on it anyhow!