Hypercoverings, or Fibrant Homotopies

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Given that a Grothendieck topology is essentially about abstracting a notion of ‘covering’, it is not surprising that modified Čech methods can be applied. Artin and Mazur used Verdier’s idea of a hypercovering to get, for each Grothendieck topos, E, a pro-object in Ho(S) (i.e. an inverse system of simplicial sets), which they call the étale homotopy type of the topos E (which for them is ‘sheaves for the étale topology on a variety’). Applying homotopy group functors gives pro-groups πi(E) such that π1(E) is essentially the same as Grothendieck’s π1(E).

Grothendieck’s nice π1 has thus an interpretation as a limit of a Čech type, or shape theoretic, system of π1s of ‘hypercoverings’. Can shape theory be useful for studying ́etale homotopy type? Not without extra work, since the Artin-Mazur-Verdier approach leads one to look at inverse systems in proHo(S), i.e. inverse systems in a homotopy category not a homotopy category of inverse systems as in Strong Shape Theory.

One of the difficulties with this hypercovering approach is that ‘hypercovering’ is a difficult concept and to the ‘non-expert’ seem non-geometric and lacking in intuition. As the Grothendieck topos E ‘pretends to be’ the category of Sets, but with a strange logic, we can ‘do’ simplicial set theory in Simp(E) as long as we take care of the arguments we use. To see a bit of this in action we can note that the object [0] in Simp(E) will be the constant simplicial sheaf with value the ordinary [0], “constant” here taking on two meanings at the same time, (a) constant sheaf, i.e. not varying ‘over X’ if E is thought of as Sh(X), and (b) constant simplicial object, i.e. each Kn is the same and all face and degeneracy maps are identities. Thus [0] interpreted as an étale space is the identity map X → X as a space over X. Of course not all simplicial objects are constant and so Simp(E) can store a lot of information about the space (or site) X. One can look at the homotopy structure of Simp(E). Ken Brown showed it had a fibration category structure (i.e. more or less dual to the axioms) and if we look at those fibrant objects K in which the natural map

p : K → [0]

is a weak equivalence, we find that these K are exactly the hypercoverings. Global sections of p give a simplicial set, Γ(K) and varying K amongst the hypercoverings gives a pro-simplicial set (still in proHo(S) not in Hopro(S) unfortunately) which determines the Artin-Mazur pro-homotopy type of E.

This makes the link between shape theoretic methods and derived category theory more explicit. In the first, the ‘space’ is resolved using ‘coverings’ and these, in a sheaf theoretic setting, lead to simplicial objects in Sh(X) that are weakly equivalent to [0]; in the second, to evaluate the derived functor of some functor F : C → A, say, on an object C, one takes the ‘average’ of the values of F on objects weakly equivalent to G, i.e. one works with the functor

F′ : W(C) → A

(where W(C) has objects, α : C → C′, α a weak equivalence, and maps, the commuting ‘triangles’, and this has a ‘domain’ functor δ : W(C) → C, δ(α) = C′ and F′ is the composite Fδ). This is in many cases a pro-object in A – unfortunately standard derived functor theory interprets ‘commuting triangles’ in too weak a sense and thus corresponds to shape rather than strong shape theory – one thus, in some sense, arrives in proHo(A) instead of in Ho(proA).

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