Locality and descent
Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
A hypercover is the generalization of a Cech nerve of a cover: it is a simplicial resolution of an object obtained by iteratively applying covering families.
be the geometric embedding defining a sheaf topos into a presheaf topos .
in the category of simplicial objects in , hence the category of simplicial presheaves, is called a hypercover if for all the canonical morphism
in are local epimorphisms (in other words, is a “Reedy local-epimorphism”).
Here this morphism into the fiber product is that induced from the naturality square
of the unit of the coskeleton functor .
A hypercover is called bounded by if for all the morphisms are isomorphisms.
The smallest for which this holds is called the height of the hypercover.
(Dugger-Hollander-Isaksen 02, def. 4.13) see also (Low 14-05-26, 8.2.15)
(Dugger-Hollander-Isaksen 02, prop. 7.2
In this form the notion of hypercover appears for instance in (Brown 73).
In some situations, we may be interested primarily in hypercovers that are built out of data entirely in the site . We obtain such hypercovers by restricting to be a discrete simplicial object which is representable, and each to be a coproduct of representables. This notion can equivalently be formulated in terms of diagrams , where is some simplicial set and is its category of simplices.
Consider the case that is simplicially constant. Then the conditions on a morphism to be a hypercover is as follows.
in degree 0: is a local epimorphism.
in degree 1: The commuting diagram in question is
Its pullback is , Hence the condition is that
is a local epimorphism.
in degree 2: The commuting diagram in question is
So the condition is that the vertical morphism is a local epi.
Similarly, in any degree the condition is that
is a local epimorphism.
Existence and refinements
For a cover, the Cech nerve projection is a hypercover of height 0.
Over general sites
Given any site and given a diagram of simplicial presheaves
where the vertical morphism is a hypercover, then there exists a completion to a commuting diagram
where the left vertical morphism is a split hypercover, def. 2.
Moreover, if is a dense subsite then as above exists such that it is simplicial-degree wise a coproduct of (representables by) objects of .
(e.g. Low 14-05-26, lemma 8.2.20)
(see also Low 14-05-26, lemma 8.2.23)
Over Verdier sites
It is sufficient that all the are monomorphisms.
Examples include the standard open cover-topology on Top.
A basal hypercover over a Verdier site is a hypercover such that for all the components of the maps into the matching object are basal maps, as above.
Over a Verdier site, every hypercover may be refined by a split (def. 2) and basal hypercover (def. 4).
This is (Dugger-Hollander-Isaksen 02, theorem 8.6).
Let be a hypercover. We may regard this as an object in the overcategory . By the discussion here this is equivalently . Write for the category of abelian group objects in the sheaf topos . This is an abelian category.
Forming in the sheaf topos the free abelian group on for each , we obtain a simplicial abelian group object . As such this has a normalized chain complex .
For a hypercover, the chain homology of vanishes in positive degree and is the group of integers in degree 0, as an object in :
Descent and cohomology
The following theorem characterizes the ∞-stack/(∞,1)-sheaf-condition for the presentation of an (∞,1)-topos by a local model structure on simplicial presheaves in terms of descent along hypercovers.
This is the central theorem in (Dugger-Hollander-Isaksen 02).
The following theorem is a corollary of this theorem, using the discussion at abelian sheaf cohomology. But historically it predates the above- theorem.
(Verdier’s hypercovering theorem)
For a topological space and a sheaf of abelian groups on , we have that the abelian sheaf cohomology of with coefficients in is given
by computing for each hypercover the cochain cohomology of the Moore complex of the cosimplicial abelian group obtained by evaluating degreewise on the hypercover, and then taking the colimit of the result over the poset of all hypercovers over .
A proof of this result in terms of the structure of a category of fibrant objects on the category of simplicial presheaves appears in (Brown 73, section 3).
The concept of hypercovers was introduced for abelian sheaf cohomology in
An early standard reference founding étale homotopy theory is
- Michael Artin, Barry Mazur, Étale Homotopy , Lecture Notes in Mathematics 100, Springer- Verlag, Berline-Heidelberg-New York (1972).
The modern reformulation of their notion of hypercover in terms of simplicial presheaves is mentioned for instance at the end of section 2, on p. 6 of
A discussion of hypercovers of topological spaces and relation to étale homotopy type of smooth schemes and A1-homotopy theory is in
A discussion in a topos with enough points in in
A thorough discussion of hypercovers over representables and their role in descent for simplicial presheaves is in
On the Verdier hypercovering theorem see
Split hypercover refinement over general sites is discussed in