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).

Grothendieck’s Abstract Homotopy Theory

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Let E be a Grothendieck topos (think of E as the category, Sh(X), of set valued sheaves on a space X). Within E, we can pick out a subcategory, C, of locally finite, locally constant objects in E. (If X is a space with E = Sh(X), C corresponds to those sheaves whose espace étale is a finite covering space of X.) Picking a base point in X generalises to picking a ‘fibre functor’ F : C → Setsfin, a functor satisfying various conditions implying that it is pro-representable. (If x0 ∈ X is a base point {x0} → X induces a ‘fibre functor’ Sh(X) → Sh{x0} ≅ Sets, by pullback.)

If F is pro-representable by P, then π1(E, F) is defined to be Aut(P), which is a profinite group. Grothendieck proves there is an equivalence of categories C ≃ π1(E) − Setsfin, the category of finite π1(E)-sets. If X is a locally nicely behaved space such as a CW-complex and E = Sh(X), then π1(E) is the profinite completion of π1(X). This profinite completion occurs only because Grothendieck considers locally finite objects. Without this restriction, a covering space Y of X would correspond to a π1(X) – set, Y′, but if Y is a finite covering of X then the homomorphism from π1(X) to the finite group of transformations of Y factors through the profinite completion of π1(X). This is defined by : if G is a group, Gˆ = lim(G/H : H ◅ G, H of finite index) is its profinite completion. This idea of using covering spaces or their analogue in E raises several important points:

a) These are homotopy theoretic results, but no paths are used. The argument involving sheaf theory, the theory of (pro)representable functors, etc., is of a purely categorical nature. This means it is applicable to spaces where the use of paths, and other homotopies is impossible because of bad (or unknown) local properties. Such spaces have been studied within Shape Theory and Strong Shape Theory, although not by using Grothendieck’s fundamental group, nor using sheaf theory.

b) As no paths are used, these methods can also be applied to non-spaces, e.g. locales and possibly to their non-commutative analogues, quantales. For instance, classically one could consider a field k and an algebraic closure K of k and then choose C to be a category of étale algebras over k, in such a way that π1(E) ≅ Gal(K/k), the Galois group of k. It, in fact, leads to a classification theorem for Grothendieck toposes. From this viewpoint, low dimensional homotopy theory is ssen as being part of Galois theory, or vice versa.

c) This underlines the fact that π1(X) classifies covering spaces – but for i > 1, πi(X) does not seem to classify anything other than maps from Si into X!

This is abstract homotopy theory par excellence.