See file Gluing.tex.
Goal: Develop the gluing formalism begun in the thesis.
Can we obtain Archimedean cohomology by glueing rigid/l-adic at finite primes with Betti/de Rham at infinity?? What would the gluing data be? Maybe the period and p-adic period maps would play a role. Someone, either GS or Deninger remarks somewhere, I think in Motives vol, that CH-hat should inject into arch.
The point is that you should work at the level of the model category in the following sense. Let M be a stable model category (as in Hovey’s book). Let T=Ho(M) be the corresponding homotopy category. As explained by Hovey, T is naturally endowed with a triangulated structure. Then, you have a good way of understanding distinguished triangles.
Let u:A–>B and v:B–>C be two maps in M (which you might think of intuitively as maps in T, but the fact that you get something in M is crucial here). We would like to understand when u and v define canonically a distinguished triangle of shape A–>B–>C–>A[1] in T. A necessary and sufficient condition is: i) the map vu is null in T; ii) the canonical map B/A–>C (obtained by u) is an isomorphism (where B/A denotes the homotopy cofiber of the map u, which is called the cone of u in the context of complexes in an abelian category). Under assumption i), you see that condition ii) may be expressed as N=0, where N denotes the homotopy cofiber of the map B/A–>C.
Now, if you are in the case where M is the model categoy underlying SH(X) or DM(X) for a scheme X, the condition for an object of T to be null is local for the Nisnevich topology (by construction): an object N of SH(X) (resp. of DM(X)) is trivial iff there exists a Nisnevich covering f:U–>X such that f^(N)=0 (in the case of Beilinson motives, this condition is local with respect to the h-topology, in the case of l-adic sheaves, this is local for the étale topology, etc). In fact, N is trivial iff for any point x of X, its pullback along the map Spec(O^h_{X,x})–>X is trivial, or, in other words, iff it is stalkwise trivial (this stalk interpretation comes from the part on continuity in our big paper).
By putting everything above together, you will see how being a cofiber sequence is a local property.
nLab page on Gluing