Classifying topoi and groupoids

(1)

Universit`

a degli Studi di Pisa

Corso di Laurea Magistrale in Matematica

Tesi magistrale

Classifying topoi

and groupoids

Candidato

Guglielmo Nocera

Matr. 501522

Relatore

Prof. Riccardo Benedetti

Controrelatore

Prof. Andrea Maffei

13 July 2018

(2)

- Que voulez-vous savoir au juste?

- Ce que tu penses de l’histoire universelle en général et de l’histoire générale en particulier. J’écoute.

- Je suis bien fatigu´e, dit le chapelain.

- Tu te reposeras plues tard. Dis-moi, ce Concile de Bˆale, est-ce de l’histoire universelle?

- Oui-da. De l’histoire universelle en g´en´eral. - Et mes petits canons?

- De l’histoire g´en´erale en particulier. - Et le mariage des mes filles?

- Á peine de l’histoire événementielle. De la microhistoire, tout au plus.

- De la quoi? hurle le duc d’Auge. Quel diable de langaige est-ce l´a? Serait-ce aujourd-hui ta Pentecˆote?

- Veuillez m’excuser, messire. C’est, voyez-vous, la fatigue. Raymond Queneau, Les fleurs bleues

(3)

Acknowledgements

Ringrazio in primo luogo la mia famiglia, in particolare i miei genitori e mio nonno. Ringrazio il mio relatore magistrale, Prof. Riccardo Benedetti, per avermi “seguito e assecondato” per tutto l’anno; il prof. Roberto Frigerio, mio relatore triennale; il prof. Andrea Maffei, mio controrelatore magistrale. Ringrazio anche quei professori che hanno impresso per argomenti, per vicinanza o per stile, un’impronta marcata al percorso di studi: dal primo all’ultimo anno, oltre ai citati, Luigi Ambrosio, Carlo Mantegazza, Elisabetta Fortuna, Ilaria Del Corso, Fabrizio Broglia, Mauro Di Nasso, Marco Forti, Marco Abate, Bruno Martelli, Luciano Mari, Enrico Arbarello.

Ringrazio i miei compagni d’anno, Alice, Andrea, Claudio, Davide, Emanuele, Francesco Florian, Federico, Giada, Giacomo, Luca, Ludovico, Luigi, Tess, e in particolare Benzo, Fabio e Gioacchino; e oltre a loro l’ormai incamminato pianista Michele Franceschi. Ringrazio anche gli amici o compagni di studio degli altri anni, e in particolare Alessan-dro Candido, AlessanAlessan-dro D’Angelo con il quale ho condiviso un interminabile processo d’indagine dei dipartimenti di Matematica del pianeta, Andrea Marino, Antonio, Dario Ascari, Dario Balboni, Federico Franceschini, Dario Rancati, Davide Falessi, Francesco, Laura, Lorenzo Portinale, Marco Costa, Morena, Rosario, Sebastiano.

Ringrazio Alberto, Leo e Marco.

Ringrazio Ivan Di Liberti per consigli e referenze, e Giulio Bresciani per il grande aiuto. Ringrazio i professori Fabio Bentivoglio, Grazia Caligaris, Massimo Piccolomini e Pasquale Maiano per aver voluto mantenere contatti e influenza sulle mie letture e riflessioni.

E infine ringrazio il prof. Franco Tardi senza il quale, forse, non avrei scelto di studiare Matematica.

(6)

(7)

Introduction

The aim of this dissertation is to enlighten some connections between the theory of Grothendieck topoi and the notion of topological groupoid. The key concept is that of classifying topos, that is defined for an arbitrary topological category. The classifying topos is called this way because it classifies bundles on a topological category, in a sense made clear by a theorem by Diaconescu. On one hand, this is a special case of the definition of classifying topos of a first-order theory, and on the other hand it recalls an analogue property of the classifying space BG of a group: namely, the fact that there exists a correspondence between homotopy classes of maps [X, BG] and principal G-bundles on X for every topological space X.

This leads us to consider, on one hand, the relationship between classifying space and classifying topos, and on the other hand what happens when we extend ourselves from the case of a group to the case of a groupoid, which is a kind of structure that has gained a great importance in Mathematics in the last decades.

More specifically, in the first chapter we introduce the notion of Grothendieck topos and the general definition of the classifying topos BC of a topological category C. Then we make examples of the “classifying property” starting with the classical theorem that connects the classifying space of a group with the principal G-bundles on a topological space, and then studying the classifying topos of a group BG (namely, the category of right G-sets), proving a theorem by Diaconescu that rephrases the classifying property of the classifying space in a categorical context. We also explain the connections be-tween topos cohomology and group cohomology, proving that Hn_{(BG, A) ∼}_{= H}n_{(G, A)}

for every G-module A.

Besides, a brief overview of the more general statement of the “classifying property”, in term of the so-called geometric theories, is given.

In the second chapter we examine in more detail the notion of homotopy for a topos, in order to establish a connection with the homotopy of the classifying space. We present the theory of ´etale homotopy, starting with the example of ´etale coverings on a scheme and then generalising to the case of general Grothendieck topoi. We are

(8)

then able to consider the homotopy progroups of a Grothendieck topos, and to state the so-called toposophic Whitehead theorem, that connects isomorphisms in (´etale) homotopy with isomorphisms in (topos) cohomology.

We then deepen the context of simplicial objects and sheaves on them, in order to define the nerve of a topological category. This allows us to define and study the clas-sifying space of a topological category, in a way that extends the case of groups.

In the third chapter we prove the comparison theorem: for an s-´etale topological category C, there is a weak homotopy equivalence between the topoi Sh(BC) and BC. As an application, we restrict to the case of topological groupoids, and consider the Haefliger groupoid Γq_{. This groupoid “classifies foliations”, in the sense that the}

existence of certain foliations on an open manifold is equivalent to the existence of a lifting in a diagram involving the classifying space of Γq_{. We prove a theorem by Segal,}

following Moerdijk’s alternative proof, according to which BΓq _{can be replaced, up to}

homotoopy, by the classifying space of M (Rq), the monoid of smooth embeddings of Rq into itself.

In the fourth chapter we consider a sort of “inverse question”: given a Grothendieck topos E , is it true that it can be represented as the classifying topos of a groupoid G? The answer is, in general, negative. It is positive when taking topoi with “enough points”. Also, every topos can be represented as the classifying topos of a “localic groupoid”. These results are due, respectively, to Butz and Moerdijk (1998) and to Joyal and Tierney (1977). We examine the proof of the first theorem. Finally, we give account on a recent result (2017) on an infinitary version of the theorem, and say a few words about the theorem by Joyal and Tierney, as prove by Joyal and Moerdijk in 1990.

(9)

Chapter 1 Topoi and groups

Outline of the chapter. • We define the notion of Grothendieck topology and of Grothendieck topos.

• We define a topological category and the topos of equivariant sheaves on it, i.e. its classifying topos.

• We prove a theorem by Radu Diaconescu, connecting the notion of classifying topos and of principal G-bundle, for a discrete group G. We also give the general statement for topological categories.

• We expand the classifying property encoded in the Diaconescu theorem to the general context of geometric theories, thus justifying the notion (and also for later use).

• We investigate, in the special case of discrete groups, the relationship between topos cohomology (which we define) and group cohomology, noting that the co-homology of the classifying topos BG of a group is isomorphic to that of the group (hence to that of the Eilenberg-MacLane classifying space).

• As a possible solution to a problema arising from the basic definitions, we briefly speak about the so-called Grothendieck universes in the appendix.

1.1 The classifying topos of a topological category

1.1.1 Grothendieck topoi

The definition of a sheaf on a topological space is well-known. The idea of the Grothendieck topos is to generalise this from spaces to categories, by introducing a

(10)

notion of “Grothendieck topology” on a category. More precisely, one specifies what the coverings of that category should be. For example:

Example 1.1. Take a topological space X. Consider the category O(X) that has the open sets of X as objects and the inclusions as arrows. A subfamily F ⊆ Ob(O(X)) is a covering if and only if S

U ∈FU = X.

Remark 1.2. Here we need to specify something about the use of the words “cover-ing” and “cover”. Actually, “coverings” in the previous example are just open covers. Taking the disjoint union C =F

U ∈FU provides a continuous surjection C −→ X, i.e.

an epimorphism in Top from C onto X. But this won’t be, in general, a topological covering. However, as we are going to see later, in many important (and historically relevant) cases we will not consider just open covers, but only some special covers whose disjoint union will be an actual topological covering onto X. In any case, the confusion that arises in English doesn’t exist in the original French terminology: one speaks of recouvrements (collection of sets whose union is the whole space), and not of revˆetements, which are surjective local homeomorphisms with trivializing open sets.

To know the coverings is the key information when talking about sheaves: for example, separatedness and glueing conditions that define sheaves on a space depend on the coverings of the space. We will then give a definition that, as we will prove, extends the idea of sheaves on a space. But first, in order to understand the construction of this idea, we present an “approximate” definition:

Definition 1.3. Let C be a small category with pullbacks. A Grothendieck pre-topology on C is defined by specifying, for each object U of C, a set P (U ) of families of morphisms of the form {U αi

−→| i ∈ I}, that are called covering families of the pretopology, such that

(i) For any U , the family whose only member is U −→ U is in P (U ).id

(ii) If V −→ U is a morphism of C and {Ui −→ U | i ∈ I} is in P (U ), then

{V ×U Ui π1 −→ V } is in P (V ) (iii) If {Ui αi −→ U | i ∈ I} ∈ P (U ) and {Vij βij

−→ Ui | j ∈ Ji} ∈ P (Ui) for each i, then

{Vij αiβij

−−−→ U | i ∈ I, j ∈ Ji} ∈ P (U ).

We could now define a sheaf for the pretopology P to be a presheaf F such that the diagram F (U ) −→Y i∈I F (Ui) ⇒ Y i,j F (Ui×U Uj)

(11)

1.1. The classifying topos of a topological category 11 is an equalizer for every covering family {Ui −→ U | i ∈ I}. However, if F satisfies this

condition for the family R = {αi | i ∈ I}, it will clearly also satisfy it for any family

S which contains R, and indeed the converse is true if every morphism in S factors through one of the αi.

Therefore, different pretopologies may give exactly the same sheaves. To remove this ambiguity, we restrict our attention to those families R which are “saturated” in the sense that (V −→ U ) ∈ R implies (Wα −→ U ) ∈ R for any Wαβ −→ V . Such a family isβ called a sieve on the object U . We now write a new definition in terms of sieves, which will simplify the concept of pretopology by adapting it to this saturated families. Remark 1.4. Note that we are dealing with small categories. Our main non-trivial example will be the ´etale site, which, as set, is the collection of all ´etale schemes over a fixed scheme X; and all Grothendieck sites will be set-based. This is not a matter of simplicity: if C is a class, then SetsCop has as objects proper classes, hence it cannot be handled by class theory, so it is not even a category. We will come back to this problem later.

Remark 1.5. Given an object C of a category C, a subobject of C is an equivalence class of monomorphisms towards C. Consider a small category C - i.e. a category such that the class of objects C0 and the class of arrows (morphisms) C1 are sets. Then if

C ∈ C a sieve on C is a subobject of Yon(C) ∈ SetsCop, where Yon is the Yoneda embedding Yon(C) = HomC(−, C). Alternatively, this is a family of morphisms, with

a common codomain C, that satisfies a sort of “ideal property”, i.e. f ◦ g ∈ S for all f ∈ S, g ∈ C1, if they are composable.

Remark 1.6. Given a sieve R on U , every morphism f : U −→ V induces the natural “preimage” sieve f∗(R) = {g | cod(g) = V, f g ∈ R}.

Definition 1.7. A Grothendieck topology on a small category C is a function J which assigns to each object U of C a set J (U ) of sieves on U , in such a way that

(i) the maximal sieve tU = {f | cod(f ) = U } is in J (U ) for all U

(ii) (stability axiom) if R ∈ J (U ), then f∗(R) ∈ J (U ) for any f : V −→ U

(iii) (transitivity axiom) if R ∈ J (U ) and S is any sieve on U such that f∗(S) ∈ J (V ) for all h : V −→ U which belong to R, then S ∈ J (R).

If R ∈ J (U ) we say that R J -covers U .

Remark 1.8. Note that the first and third conditions together imply that if R ∈ J (U ) and S is a sieve on U containing R then S ∈ J (U ). Also, if we are given a pretopology P on C we can replace it by a topology having the same sheaves, by defining a sieve to be a J -covering if and only if a P -covering family is contained in it.

(12)

Remark 1.9. This whole construction is actually a generalisation of the notion of topological covering. If C is the topology O(X) of a topological space as above, with as morphisms the inclusions of open sets, a sieve on U is any family of open sets included in U which is closed by inclusion. Now for every U ∈ τ take J (U ) to be the family of all possible sieves. It is then trivial that the maximal sieve is in J (U ), and if we have a sieve S ∈ J (U ) and h : V −→ U is an inclusion of open sets, then h∗(S) is the family of all open sets in S that are also contained in V ; now this is again a family of open sets which is closed by inclusion, hence a sieve on V .

Finally, if S ∈ J (U ) and R is another sieve on U , then R ∈ J (U ) authomatically, so axiom (iii) is trivial.

Definition 1.10. Let C be a small category and J a Grothendieck topology. A presheaf (of sets) is a contravariant functor from C to Sets, so

P sh(C) := SetsCop with natural transformations as morphisms.

Remark 1.11. The change from pretopology to topology has another advantage in addition to removing the ambiguity mentioned earlier. One can note that we have been able to dispose with the assumption that C has pullbacks, since a sieve can alwas be pulled back along a morphism of C, even if the individual morphisms in it cannot. The reason for this is that each sieve R on U can be identified with a sub-presheaf of the representable functor hU (cfr.Remark 1.5), namely the presheaf

V 7→ {α ∈ R | dom(α) = V }; and the presheaf category SetsCop does, of course, have pullbacks. We make use of this identification of sieves with sub-presheaves if hU in

defining sheaves for a topology.

Definition 1.12. A basis for a Grothendieck topology on a category C with pullbacks is a function K which assigns to each object U a collection K(U ) consisting of families of morphisms with codomain U , such that

(i’) If f : U −→ U is an isomorphism, then {f : U0 −→ U } ∈ K(U ) as a singleton. (ii’) If {fi : Ci −→ C | i ∈ I} ∈ K(C), then for any morphism g : D −→ C the family

of pullbacks {π2 : Ci×C D −→ D | i ∈ I} is in K(D)

(iii’) If {fi : Ci −→ C | i ∈ I} ∈ K(C), and if for each i ∈ I one has a family

{gij : Dij −→ Ci | j ∈ Ii} ∈ K(Ci), then the family of composites {fi ◦ gij :

Dij −→ C | i ∈ I, j ∈ Ii} is in K(C).

Definition 1.13. Let (C, J ) be a site, F a presheaf on C. We say that F is a sheaf for the topology J if, for every object U of C and every R ∈ J (U ), each morphism

(13)

1.1. The classifying topos of a topological category 13 R −→ F in SetsCop has exactly one extension to a morphism hU −→ F ; and F is a

separated presheaf if it satisfies the above condition with “exactly one” replaced by “at most one”. We denote the full subcategory of SetsCop whose objects are J -sheaves by

Sh(C, J ).

Definition 1.14. A Grothendieck topos is a category that is equivalent to the category of the sheaves on a site C with a Grothendieck topology J .

Definition 1.15. A topos morphism (or geometric morphism) is a pair of functors E f∗

f∗

F , f∗ _{called the “inverse image” functor and f}

∗ called the “direct image” functor,

such that f∗ is left adjoint to f∗ and f∗ commutes with finite limits (i.e. is “left exact”).

Remark 1.16. Note immediately that the category Sets is a Grothendieck topos. Namely,

Sets ∼= Sets∗op,

where ∗ is the one-point topological space represented as one object and the identity morphism (and thus coinciding with the category O(∗)). The topos of sets is in many ways the simplest of all topoi, and the latter remark justifies that we think of it as the analogous in topos theory of the one-point topology. We will investigate more deeply this special condition of Sets in Chapter 4, when introducing the notion of point of a Grothendieck topos E (which is, not surprisingly, a Grothendieck topos morphism Sets E).

Remark 1.17. Note that P sh(C) is, in general, a proper class: for example, Sets. There is an approach that avoids this, that can be regarded as a problem in certain contexts, by restricting the class Sets to arbitrarily large Von Neumann classes (which are actually sets) that satisfy certain properties of closure. However, the existence of these latter is not guaranteed by the ZFC axioms, and is actually equivalent to the existence of strongly inaccessible cardinals. We illustrate briefly this approach in the Appendix to the present chapter.

1.1.2 Topological categories

Definition 1.18. A topological category C is given by a set of objects C0 and a

set of morphisms C1, as every small category, but both C0 and C1 are endowed with a

topological structure such that the following maps (the structure maps) are continuous: s : C1 −→ C0

(14)

which associates to a morphism its domain object (“source”), t : C1 −→ C0

which associates to each morphism its codomain, m : C1×C0 C1 −→ C1

for composition (see Remark 1.19), and

u : C0 −→ C1

which associates to each object its identity morphism.

Remark 1.19. m : C1 ×C0 C1 −→ C1 actually represents composition, this means

that the fiber product represents the compatibility of the morphisms which we want to compose, once we make clear that the maps on C0 are source and target, and are

located in this precise position:

C1×C0 C1 C1 C1 C0 π1 π2 s t

Indeed, C1×C0C1 can be seen as the couple of morphisms (α, β) such that s(α) = t(β),

i.e. such that α ◦ β is defined.

Definition 1.20. A topological category C is said to be s-étale if, in addition, the source map is also étale, i.e. a local homeomorphism, and étale if both s and t are local homeomorphisms.

Definition 1.21. A C-sheaf is a sheaf p : S −→ C0 over the topological space C0,

equipped with a continuous right action α : S ×C0 C1 −→ S, denoted α(x, f ) = x · f .

So x · f is defined whenever p(x) = t(f ), and it satisfies the usual identities for actions: (x · f ) · g = x · (f ◦ g), x · idp(x) = x, p(x · f ) = s(f ).

A map between C-sheaves is defined as a map of sheaves (over the topological space C0) which respects the action.

Definition 1.22. For a topological category C, we define its classifying topos BC as the category of sheaves on C with as morphisms the maps of C-sheaves.

(15)

1.2. Diaconescu’s Theorem 15 Proposition 1.23. Let C be a discrete category, i.e. such that C0, C1 are discrete

spaces. This is a topological category, since all the structure maps are continuous, and the category of sheaves BC coincides with the category of presheaves SetsCop.

Remark 1.24. Note that this agrees with the geometrical tradition: if X (instead of C0) is a discrete space, sheaves coincide with presheaves since the glueing conditions

(existence and uniqueness) are automatically satisfied. This is what the following proof says in abstract.

Proof (of Proposition 1.23): Let F ∈ SetsCop, and

S = a

c∈C0

F (c)

with the discrete topology. Then the map p which associates to an x ∈ F (c) the object c ∈ C0 is a sheaf of spaces since all topologies are discrete, and we can define the

action α(x, f ) = F (f )(x), where if x ∈ F (c) then F (f )(x) ∈ F (f (c)) (by f (c) we mean the target t(f )) since F is a functor. This action clearly satisfies the compatibility conditions, and it is continuous, again by discreteness.

Remark 1.25. We must of course prove that the classifying topos of a group is a topos. The proof is not immediate, at least in the general case of a topological category, and the simplest way to do it seems to use an axiomatic characterisation of Grothendieck topoi, provided by the Giraud theorem (cfr. [MM92]). One can verify those axioms and conclude that BC is a Grothendieck topos. There is actually a canonical site that can be constructed for this topos, but this construction, as it is, needs the fact that we already know that we have a Grothendieck topos.

1.2 Diaconescu’s Theorem

1.2.1 The Eilenberg-MacLane space

We recall here the standard construction of the Eilenberg-MacLane space. Cfr. [Hat01], pages 89-91 for proofs and details. Consider a group G, and let En = Gn+1 conceived

as a set, endowed with the diagonal right action (g0_{, . . . , g}n_{)g = (g}0_{g, . . . , g}n_g)1_{. The}

system {En}n∈N forms a simplicial set, whose ordinary geometric realization EG is

contractible. Now quotient every En with respect to the action of G; the resulting

(simplicial) space BG is covered by EG, and therefore is aspherical (πn = 0 ∀n ≥ 2);

moreover, π0(BG) = 0 and π1(BG) = G. A space with these homotopy groups is called

1_{Hatcher’s book and almost everyone use the left action: the result is analogous (i.e. the resulting}

(16)

a K(G, 1) space (meaning that the only nontrivial homotopy group is the first and it equals G), and the following important theorem holds:

Theorem 1.26. For every group G, the Eilenberg-MacLane space K(G, 1) is unique up to homotopy equivalence.

Definition 1.27. BG = K(G, 1) will be called the classifying space of G.

Throughout this dissertation we will encounter different types of “classifying” prop-erties. The classifying property of BG is the first, and will also be the base for under-standing all the other ones. We state it here:

Theorem 1.28. There is a bijective correspondence between isomorphism classes of covering spaces with group G, which we call principal G-bundles or G-torsors (see later) and homotopy classes of maps X −→ BG.

1.2.2 The classifying topos of a group and Diaconescu’s

The-orem

Now we define the classifying topos. One of the simplest possible actions of a group on a set is the action by multiplication on itself, from the right or from the left. We choose multiplication from the right for a special reason that will be clear in a few lines. So, G endowed with the right action is a “right G-set”, and we call it ˜G in order to differentiate it from the acting group. This is actually a special object amongst all right G-sets, as one can imagine, and we will soon see one aspect of this special condition. What we are going to study now is the relationship between right G-sets and principal G-bundles. We define then the main categories of this section:

Definition 1.29. For a discrete group G, we define G (by abuse of notation) as the category having only one object {G} and as morphisms λg, g ∈ G, seen as multiplication

from the left. We then define BG as the category of right G-sets. When the context is topological, we will always endow a G-set with the discrete topology, thus respecting the topology of G.

Remark 1.30. Note immediately that this is a special case of Definition1.22, since G is endowed with the discrete topology.

Proof: The one-object category G is discrete, since it has only one object and the family of morphisms (i.e., left multiplications by elements) inherits the discrete topology from the fact that G is assumed to be discrete. The sheaves on a discrete category coincide with the presheaves (Proposition 1.23), and thus BG = SetsGop, which are precisely functors from the only object {G} to Sets (i.e. the choice of a set) which invert the

(17)

1.2. Diaconescu’s Theorem 17 morphisms of G. Let F be one such functor. S = F ({G}) is a set. Now composition of multiplications is defined from the left in G, and therefore contravariance gives a right action on the set S: morphisms λg in G are isomorphisms2, thus F (λg) must be

an isomorphism in HomSets(S, S), i.e. a permutation.

Note that

Lemma 1.31. The topos of right G-sets is generated under colimits by G.

Proof: In fact, every right G-set can be seen as disjoint union of orbits. So it suffices to show that each orbit is a colimit of G.

Let O = _Stab(x)G be an orbit. Consider a system of generators for G (e.g., all elements) and call it {gi}i∈I. Now consider copies Gi, i ∈ I of ˜G and maps ρ_g−1

i gjGi −→ Gj,

where ρ_g−1

i gj ∈ Hom( ˜G, ˜G) is the multiplication from the right, that is a morphism in

BG by construction of the classifying topos. It is clear that the colimit of such objects and maps is the quotient _Stab(x)G :

G G

G/Stab(x)

·g_i−1gj

π

Definition 1.32. Let X be a topological space. Recall that a sheaf on X is a local homeomorphism E −→ X. A principal G-bundle on X is a surjective sheaf equipped with a continuous fiberwise action α : G×E −→ E ×XE −→ E (denoted α(g, e) = g ·e)

which is free and transitive on each fiber. The map

(α, π2) : G × E −→ E ×X E

is a homeomorphism of sheaves over X, since G × E satisfies the universal property of E ×X E (this is easy to verify, using that the action is free and transitive on the

fibers). So the map p : E −→ X is a covering projection, since it is the quotient map relative to a free and properly discontinuous group action.

The principal G-bundles over X form a category: morphism are given by maps ϕ : E −→ E0 which preserve the actions. Note that such maps must be isomorphisms of sheaves (this comes from the fact that G acts freely and transitively on the fibers).

2_{Note that they are isomorphisms in Hom}

G(G, G) since they are invertible, but they are not

(18)

Definition 1.33. Call P rin(X, G) this category.

Theorem 1.34 (Diaconescu for groups). There is a natural equivalence of categories Hom(Sh(X), BG) −→ P rin(X, G), where Hom(Sh(X), BG) denotes the category of topos morphisms between Sh(X) and BG.

Proof: Let f : X −→ BG be a morphism of topoi. Consider the right G-set ˜G defined above; this is then an element of BG, which in virtue of its “canonical” nature will be the key of the construction of the correspondence.

In fact, the principal bundle corresponding to f is (E, p) = f∗G. Note that the map p˜ is determined explicitly by f∗, in the sense that f∗, commuting with limits, sends the terminal object of BG, which is 1, in the terminal object of Sh(X), which is clearly X; therefore it is easy to see that the map p is precisely f∗(c), where c : ˜G −→ 1.

Now E is a sheaf; let us see that it is a principal G-bundle. In order to check surjectivity observe first of all that ˜G × ˜_{G ⇒ ˜}G −→ 1 is a coequalizer in BG: in fact, if a map h : ˜G −→ D equalizes π1 and π2, then applying it to a couple (g1, g2) one has that

h is constant. Now applying f∗ to the coequalizer, since it preserves products and coequalizers one obtains that E ×X E ⇒ E −→ X is a coequalizer in Sh(X), and

therefore in Sets; so it is surjective.3

Now we turn to the action. For each g ∈ G, the left multiplication λg(x) = g · x defines

a map λg : ˜G −→ ˜G in the category BG. Thus one obtains a map f∗(λg) : E −→ E of

sheaves, and this defines an action α of G on E. To see that it is free and transitive, note that the map

¯ λ :X

g∈G

˜

G −→ ˜G × ˜G λ¯g(x) = (g · x, x)

is an isomorphism in BG, having as its inverse the right G-sets morphism that sends (y, x) in the copy of x situated in the yx−1-component of the sum. Since f∗ preserves sums, products and isomorphisms, it sends this map ¯λ into an isomorphism of bundles

¯ α :X

g∈G

E −→ E ×X E α¯g(y) = (f∗(λg)(y), y).

This means precisely that the action by G on E is principal.

Now let us give the map from Prin(X, G) to Hom(Sh(X), BG) and then show that these two maps give a natural equivalence. Suppose that p : E −→ X is a principal G-bundle over X. If S is any object from BG consider the “tensor product” S ⊗GE obtained

from S × E by the identifications (s · g, e) ∼ (s, g · e). We denote the equivalence class by s ⊗ g. The natural map pS : S ⊗GE −→ X, pS(s ⊗ e) = p(e), is a well-defined local

3_{To see this, recall from [}_Mac98_{] that a coequalizer of sets is actually a quotient projection from}

(19)

1.2. Diaconescu’s Theorem 19 homeomorphism (since S has the discrete topology). Thus S ⊗GE is a sheaf on X.

The construction is functorial in S, as one can easily verify, and this defines a functor − ⊗GE : BG −→ Sh(X)

which turns out to be the inverse image component of a Grothendieck topos morphism from Sh(X) to BG. To see this, it suffices to check that the “tensor” functor preserves colimits and finite limits. Moreover, this verification can be done on stalks, i.e. on the fibers of the sheaf. Let us then see how these stalks behave:

(S ⊗GE)x ∼= S ⊗GEx ∼= S

since the sheaf map “ignores” S, so we see the stalk only on E, and the latter iso-morphism is defined in the following way: chosen any y ∈ Ex (which makes the map

not natural) send s 7→ s ⊗ y; this is an isomorphism since the action of G is free and transitive on the fibers. Thus, since (S ⊗ E)x ∼= S for any x, then the “tensor” functor

preserves finite limits and colimits.

Finally, this correspondence is a natural isomorphism. First of all, we can consider tensors of G-sets with exactly the same definition as above, and we note that

S ⊗GG ∼˜ = S s ⊗ g 7→ s · g.

Now we want to prove that, given a Grothendieck topos morphism f : X −→ BG, then − ⊗Gf∗( ˜G) ∼= f∗.

Actually both of these functors are cocontinuous in their argument, and the topos of right G-sets is (freely) generated under colimits by G (cfr. Lemma 1.31) So to show that they agree it suffices to show that they agree at G. But G ⊗Gf∗(G) ∼= f∗(G).

Conversely, for any principal bundle E there is a canonical isomorphism of principal bundles ˜G ⊗GE ∼= E.

Remark 1.35. Note that in the statement of Diaconescu’s theorem the notion of ho-motopy has apparently disappeared. One speaks of topos morphisms and not anymore of continuous maps. Actually, homotopy is just hidden in the statement, because of the following fundamental fact:

Proposition 1.36. If X and Y are sober4 _{topological spaces, then there is a bijection}

between the set of continuous maps from X to Y and the set Hom(Sh(X), Sh(Y )) of topos morphisms between the two categories of sheaves.

4_{A topological space is sober if every irreducible closed set has a unique generic point (i.e. is of the}

form {x} for a unique point x ∈ X). Every Hausdorff space is sober, since only points are irreducible (and they are closed).

(20)

Proof: The fact that a continuous map f induces a map between the categories of sheaves is well-known: f∗ is given by pullback of sheaves, while f∗ is more easily given

in term of sheaves as presheaves (i.e. contravariant functors): f∗(F ) = F ◦ f−1 : O(Y )op−→ O(X)op −→ Sets

for any f : O(X)op _{−→ Sets.}

Conversely, let ϕ : Sh(X) −→ Sh(Y ) be a topos morphism. Then we can consider ϕ∗ and restrict it to subobjects of the terminal object. A subobject of the terminal object is an equivalence class of sheaves that go monomorphically into Y (the terminal sheaf, or the identity sheaf). But to have a monomorphism means to have an open set, and two open sets are isomorphic as sheaves if and only if they are in O(Y ), i.e. they coincide. So we restrict ϕ∗ to ϕ∗ : O(Y ) −→ O(X) that preserves finite intersections and unions (so sends the empty set in the empty set). For x ∈ X, define Fx = Y \S{U ∈ O(Y ) : x /∈ ϕ∗(U )}: this is closed and irreducible, since if there are

two disjoint open sets U and V in Fy then x ∈ ϕ∗(U ) ∩ ϕ∗(V ) = ϕ∗(U ∩ V ) = ∅,

impossible. Now if Fx is irreducible there exists a unique y ∈ Y s.t. Fx = {y}. Take

ϕ(x) = y. This gives a map from X to Y s.t. for any U open in Y and x ∈ X we have ϕ(x) ∈ U if and only if x ∈ ϕ∗(U ), hence one can check that ϕ is continuous.

1.2.3 Diaconescu’s Theorem for topological categories

Remark 1.37. Having have defined the classifying topos of a general topological cat-egory, we are now able to generalise Diaconescu’s theorem to that context. The gener-alisation involves involves principal bundles on a topological category. Such a bundle is defined as follows:

Definition 1.38. Let C be a topological category. A C-bundle on a space X is a sheaf E −→ X, equipped with a continuous fiberwise left C-action, given by maps

π : E −→ C0, a : C1×C0 E −→ E.

The map a is defined for all pairs (g, e) where g ∈ C1, e ∈ E and s(g) = π(e), and is

denoted a(g, e) = g · e. That a is an action is expressed by the usual identities 1e· e = e

and g · (h · e) = (g ◦ h) · e; that is fiberwise means that p(g · e) = p(e). Such a bundle is said to be principal if the following three conditions hold.

(i) The stalk Ex is never empty

(ii) For any two points y ∈ Ex ans z ∈ Ex there are a w ∈ Ex and arrows α :

(21)

1.2. Diaconescu’s Theorem 21 (iii) For any point y ∈ Ex, and any pair of arrows α, β in C with s(α) = π(y) = s(β)

and α · y = β · y, there exists a point w ∈ Ex and an arrow γ : π(w) −→ π(y) in

C such that γ · w = y in Ex and αγ = βγ in C.

With the obvious notion of action preserving map, these principal C-bundles over X form a category denoted Prin(X, C).

Theorem 1.39 (Diaconescu theorem for s-´etale categories). For any topological space X and any s-´etale category C, there is a natural equivalence

Hom(X, BC) ∼= P rin(X, C).

Proof: Cfr. [Moe95]. This is quite similar to the case of groups. The case of a discrete category (i.e., a category with the discrete topology on C0 and C1, which is clearly a

topological category; this includes the case of discrete groups as represented before) is used as a key step for the proof, and contains most of the new arguments necessary to abstract from the case of groups.

We will use this result in its generality while presenting Moerdijk’s alternative proof of Segal’s theorem on the Haefliger groupoid Γq in Chapter 3.

Remark 1.40. Note that a discrete group is s-´etale, and a principal G-bundle is a principal G-bundle also in the categorical sense, so this is actually an extension of Diaconescu’s Theorem for groups.

Remark 1.41 (A degeneracy case: the equivariant sheaves on a point). Note that the theorem cannot possibly hold for the classifying topos of an arbitrary topological category, since such a Grothendieck topos may be degenerate. As an example, take G to be a nontrivial topological group acting on a topological space X from the right. Let XG be the associated translation category: it has X as space of objects, and X × G as

space of arrows, where (x, g) is an arrow x·g −→ x. An XG-sheaf is a sheaf p : S −→ X

on X, with an action by G on S so that p is G-equivariant. Thus B(XG) is the category

of G-equivariant sheaves on X.

Now, when X is a point and G is connected, G = XG is a one-object topological

category (which is not s-´etale!). Since the action of a connected group on a discrete set must be trivial, B(XG) = BG collapses to the category of sets.

(22)

1.3 The classifying topos of a theory

1.3.1 Internal G-torsors

Remark 1.42. Let E be a Grothendieck topos over Sets, i.e. a Grothendieck topos with a Grothendieck topos morphism γ : E −→ Sets. Note that the object γ∗(G) is an internal group object in E , i.e. the datum of one object and three arrows (in this case, the inverse images of the correspondings maps in Sets) 1, m (multiplication) and i (inverse) such that the group axioms hold, as expressed via commuting diagrams. Note that this is equivalent to the request that Hom(X, γ∗(G)) is a group (and not just a set) for every X ∈ E .

Let Prin(E , G) denote the subcategory of E consisting in principal γ∗(G)-bundle objects (or G-torsors in E ). By this, we mean objects E ∈ E such that there exists a right action of γ∗(G) on E, i.e. a morphism µ = µE : E × γ∗(G) −→ E (any Grothendieck

topos has finite limits) such that both diagrams

E × 1 E × γ∗(G) E × γ∗(G) × γ∗(G) E × γ∗(G) E E × γ∗(G) E 1×e ∼ = _µ µ×1 1×m µ µ (1.1) commute.

Prin(E , G) is referred to as the category of principal G-bundles (or G-torsors) internal to E . This is equivalent to give for every X ∈ E an action of the group HomE(X, G)

on the set HomE(X, E), natural in X.

With these definitions, the following generalisation of the Diaconescu theorem holds: Proposition 1.43. For every Grothendieck topos E ,

Hom(E , BG) ∼= Prin(E , G).

Hence the Diaconescu theorem can be seen in the context of the general theory of classifying topoi, which arises from the definition that will be given now.

1.3.2 Internal models of a theory

Definition 1.44 ([MM92, page 238]). A formula ϕ of the first-order language L is said to be geometric if it can be obtained from atomic formulas by conjunction ∧, disjunction ∨, and existential quantification ∃x ∈ X. More precisely, the collection of geometric formulas is the smallest collection of formulas such that

(23)

1.3. The classifying topos of a theory 23 (a) the atomic formulas R(t1, ..., tn), t = t0, ⊥, > are all geometric formulas;

(b) if ϕ and ψ; are geometric formulas, then so are ϕ ∨ ψ; and ϕ ∧ ψ;

(c) if ϕ(y1, ..., yn) is a geometric formula, then so is the formula ∃x ∈ Xϕ(xl, ..., xn),

where X is any sort and x is a variable of that sort.

Definition 1.45. Let T be a geometric theory in a first-order (possibly infinitary) language L, and let E be a Grothendieck topos. We want to define a model of T in E : there is a straightforward way to interpret variables, constants, relation and function symbols of L inside E . For example, a variable X is interpreted as an object X(M ) (where M is the interpretation) of E , an n-ary relation symbol is interpreted as a subobject

R(M ) ⊆ X₁(M )× · · · × X(M ) n ,

and so on. A complete treatment can be found in [MM92], pages 532 to 534. Moreover, one can define what it means for a sentence in the language L to be true or false: again, this is completely analogue to the case of set-based model theory, with the suitable generalisations. So one can establish whether a theory is satisfied by an interpretation M inside a topos E , and define

Mod(T, E )

as the category of T -models in E : the arrows are the homomorphisms of models, which are given by

HX : X(M ) −→ X(M

0₎

in E , one for each variable X, respecting the interpretation of relation and function symbols as well as that of constants (of course this is again expressed as commutative diagrams in E ).

The following theorem illustrates the connection between geometric formulas and topos theory:

Theorem 1.46 ([MM92, page 239]). Let f : F −→ E be any topos morphism, let M be an interpretation of the language L in E , and let f∗M be the induced interpretation in F5_{. Then for any geometric formula ϕ(x}

1, ..., xn),

f∗({(x1, . . . , xn) | ϕ}M) = {(x1, . . . , xn) | ϕ}f

∗_M

where equality is that as subobjects of X₁(f∗M )×· · ·×Xn(f∗M ) ∼= f∗({X₁(M )×· · ·×Xn(M )}).

Note that topos morphisms arre currently called geometric morphism

(24)

Definition 1.47. Let T be a first-order theory in a (possibly infinitary) language L. Then its classifying topos B(T ) is defined as a Grothendieck topos such that for every cocomplete6 topos E

Hom(B(T )) ∼= Mod(E , T ).

It has been proven (cfr. [MM92, pages 561 ff.]) that this topos exists for every first-order geometric theory.

Actually, Stephen Awodey has extended this to higher-order theories.

Example 1.48. In the case of Diaconescu theorem, BG is the classifying topos for the theory of principal G-bundles. For example, if E = Sh(X), it is easy to see that Prin(E , G) = Prin(X, G) = Mod(T, Sh(X)), where T is the following L|G|,ω-theory in

the language which has one variable X and one unary operation symbol g : X −→ X for each element of G (including the identity 1):

T ` ∀x 1(x) = x T ` ∃x ∈ X

For each g except 1, T ` g(x) = x →⊥ T ` ∀x, y _

g∈G

g(x) = y.

(Note that T has as many sentences as the cardinality of G, if G is infinite, because we have to state the freeness axiom by listing all the sentences g(x) 6= 1. Note also that the third axiom could not be stated as

For each g except 1, T ` ∀x g(x) 6= x because this is not a geometric formula.)

1.4 Cohomology comparison

Let us turn to the classifying space and topos of a group. We will see that there is a notion of cohomology for topoi that coincides in the case of BG with the cohomology of the group (which is also the cohomology of the classifying space BG). We will then see that this extends rather easily to discrete categories.

(25)

1.4. Cohomology comparison 25

1.4.1 Groups

It is known that, by definition, abelian group homology and cohomology coincide with (simplicial) homology and cohomology of the corresponding classifying space.

We recall the purely algebraic definition of the cohomology of an abelian group, because it will be more useful later.

Definition 1.49. Let G be a group. Consider the ring Z[G], and the following Z[G]-modules: En = Gn+1 with the diagonal right action of G. Consider the projective

resolution of Z in the category of Z[G]-modules (the action on Z is the identity for integers and the trivial one for elements of G) given by

Z ←− E0 ←− E1 ←− . . . .

The En are free over Z[G] (because G is, with the multiplication action, having as

a basis any of its own elements), hence projective. If A is any G-module and we apply the functor Hom(·, A) the resulting sequence

Hom(Z, A) −→ Hom(E0, A) −→ Hom(E1, A) −→ . . . .

gives by definition the cohomology of the group. Note that this may be synthetized as

Hn(G, A) ∼= Extn_Z[G]_{(Z, A)}

since we have taken a projective resolution of Z. Note that an alternative way to compute the same cohomology would be to have an injective resolution of A in Z[G] − M od and consider the right derived functor of Hom(Z, ·) applied to that resolution. Definition 1.50 (Topos cohomology). Let E be a Grothendieck topos. Consider the category Ab(E ) of all abelian group objects in E . This is an abelian category with enough injectives7_. _{Therefore the following construction is well-defined.} _Consider

an abelian group object A ∈ Ab(E ). Let 1 be the terminal object in E8_{. Then let}

Γ(·) = Hom(1, ·), and define Hn_{(E , A) = R}n_{Γ(A) (recall that the category has enough}

injectives).

Lemma 1.51. Ab(BG) = G-Mod.

7_{See [}_Joh77_{]: the proof is analogous to that of the classical theorem in homological algebra}

accord-ing to which the category of Z-modules has enough injectives.

8_{An important property of topoi is that the terminal object always exist. This is actually the}

constant presheaf (the one sending every object to one fixed one-point-set, and each morphism to the identity; this definition clearly doesn’t depend on the chosen one-point-set, up to equivalence of topoi), that is in fact a sheaf.

(26)

Proof: One can follow an approach similar to [GM03], pages 101-103.

Theorem 1.52. If G is an abelian group and A is a G-module then Hn_{(G, A) ∼}₌

Hn_{(BG, u(A)), where}

u : Ab(BG) = G-Mod −→ BG is the forgetful functor.

Proof: We have defined the cohomology as Rn_(Hom(1

BG, ·))(uA). We know that this

co-incides, in abstract, with Ln_{(Hom(·, uA))(1}

BG). Now, considering the forgetful-faithful

adjunction, there exists

F : Sets −→ G-Mod

such that F 1 = Z (this is just the “free Z-module functor”, which is left adjoint to u) and

Hom(1, uA) ∼= Hom(F 1, A) ∼= Hom(Z, A) hence the desired expression for the cohomology.

Remark 1.53. Led by this, we would suspect that BG and BG have in common some notion of homotopy. Now, BG is not a topological space, so such an homotopy notion should be defined.

This is not just a matter of symmetry. Homotopy for topoi is an immediate general-isation and collection of the notions that have been necessary to construct the étale homotopy of schemes, i.e. a notion of homotopy for schemes that could take into ac-count also the algebraic structure. In that case the topos is the étale topos consisting of the sheaves on the site of all étale coverings on the scheme X.

1.4.2 Discrete categories

In the case of a discrete category, we can easily build a generalisation of the notion of group cohomology. This will turn out to be useful in Chapter 3, so we state it here, also because it is a nice review of the tools used in the case of groups, but a little more in abstract.

Definition 1.54. Let C be a discrete category, and let A be an abelian group in BC = SetsCop_{. Define}

C•(C, A) = Y

c0←−···←−cn

(27)

1.5. Appendix: Grothendieck Universes 27 and a coboundary d : Cn−1_{(C, A) −→ C}n_{(C, A) given by}

(da) c0 f1 ←−...←−fncn = n−1 X i=0 (−1)iadi(c0←−···←−cn)+ (−1) n A(fn)adn(c0←−···←−cn),

where di(c0 ←− · · · ←− cn) is the usual simplicial boundary

di(c0 f1 ←− . . . fn ←− cn) =        c1 ←− . . . ←− cn (i = 0) c0 ←− . . . ←− ci−1 fi◦fi+1 ←−−−− ci+1 ←− . . . ←− cn (0 < i < n) c0 ←− . . . ←− cn−1 (i = n)

The resulting complex has a cohomology which we call cohomology of the category C with coefficients in A.

Not surprisingly, a result similar to what we proved in Theorem1.52 holds:

Theorem 1.55. For any discrete category C, and any abelian presheaf A in C (i.e. an element of Ab(SetsCop

)) there is a canonical isomorphism H•(C, A) ∼= H•(BC, A).

We postpone the proof to Chapter 2, where we will talk about the nerve of a category.

1.5 Appendix: Grothendieck Universes

Definition 1.56. A Grothendieck universe (or a universe for short) is a transitive set U cointaining ω which is closed under powersets, pairing, and “functional unions”: if f : x −→ U with x ∈ U , thenS

y∈xf (y) ∈ U .

Theorem 1.57. U is a universe if and only if U = Vκ for some strongly inaccessible

cardinal κ.

Proof: If κ is strongly inaccessible, verifications of the universe axioms are trivial. If U is a universe, let α be the least ordinal not in U , and call it height of U . If we show that α is an uncountable regular strong limit cardinal, then by induction one shows that Vβ ⊆ U for all β < α, and then U = Vα.

So let us observe that α is first of all a limit ordinal: if α = β + 1 observe that β ∈ U, {β} = {β, β} ∈ U, {0, 1} ∈ U , then the function 0 7→ β, 1 7→ {β} realises α = β ∪ {β} as functional union. So α ∈ U .

(28)

a contradiction just as above. So cof (α) = α, and therefore α is a cardinal (being a limit and having cofinality equal to itself) and it is regular.

Now for every β ∈ U the function U 3 2β −→ |2β_{| realises |2}β_{| as functional union,}

hence α is a strong limit cardinal. Being regular, it is strongly inaccessible.

Corollary 1.58. If arbitrarily large strongly inaccessible cardinals exist, every set X belongs to a Grothendieck universe.

Definition 1.59. Let U be a universe. A set is called a U -set if it belongs to U , and U -small if it is isomorphic (i.e., in bijection) to a set belonging to U . A category is called an U -category if all Hom-sets are U -sets, and a U -small category if in addition its class of objects is an U -set.

Remark 1.60. Let A ⊆ U , B ⊆ U , U a Grothendieck universe. Then AB _{is not}

necessarily a subset of U . (For example, if U = Vκ, UU ∈ Vκ+1.) Compare this “size

problem” with the one stated in Remark1.4.

Definition 1.61. Let C and A be categories. Then P sh(C, A) := ACop, and

P shU(C) := (U -Sets)C

op

.

When the universe is fixed, we call this just P sh(C). Actually, if U is fixed we always assume C and A to be U -small.

Proposition 1.62. If C and A are U -small categories, then P sh(C, A) is U -small. In particular, it is a set.

In the following chapters, one can suppose as well that a universe is fixed. Bibliographical note. For basics on topos theory, see [Joh77] and [MM92].

For topological categories and the Diaconescu theorem see [Moe95]. For the classfying topos of a theory, see [MM92].

For Grothendieck universes, see [KS06], especially section 17.1, [Zhe14] and [Wil69] for the proof of the equivalence theorem.

(29)

Chapter 2 ´

Etale homotopy

Outline of the chapter. • Both as the fundamental motivating example and for later reference, we describe the basic properties of the étale topos of a scheme. • We outline the construction of the étale homotopy type of the étale site of a

scheme, following [SS10].

• We speak about the general construction for arbitrary locally connected pointed topos, following [AM69] and [Moe95]. This provides homotopy progroups that generalize the homotopy groups of a topological space in a precise sense.

• We state the so-called toposophic Whitehead theorem, that allows to verify iso-morphisms in homotopy progroups by passing to cohomology (except that in degree 0 and 1).

• As a preparation for the next chapter, we treat more in detail the notion of simplicial object, defining the nerve of a topological category and two kinds of geometric realization: one for simplicial sets and one, more general, for simplicial spaces, that will allow us to deal with topological categories. We follow [Moe95].

2.1 The ´

etale topos

The étale topos serves as a motivating and helpful example to approach étale homotopy. So we will use it as a guide to expose the theory, but we will also state here some details that will not be mentioned again throughout the chapter but that will be used in Chapter 4. We mainly follow [AGV72, Tome 2, Exposés VII-VIII].

Definition 2.1. Let X be a scheme. The étale schemes over X, f : Y −→ X, form a category Xet´, taking as morphisms the étale morphisms over X, i.e. étale morphisms

Y −→ Y0 such that the following diagram commutes: 29

(30)

Y Y0 X

We want to make this category into a site. We take as coverings all ´etale coverings, J (Y ) = {all families {ϕi : Ui −→ Y } of ´etale schemes over Y such thatS ϕi(Ui) = Y }.

These form a pretopology, which in turn generates a Grothendieck topology by taking sieves. So we can define the small ´etale site as (Xet´, J ).

Definition 2.2. The topos of sheaves on the étale site of X is called the small étale topos over X and denoted by Sh(Xét) or fXét.

Proposition 2.3. A morphism of schemes f : X −→ X0 induces a topos morphism fet´ : fX´et −→ fX´et0 .

Proof: Observe first that by pullback via f one can obtain from an étale scheme over X0 an étale scheme over X. So we have a functor f_et_´∗ : X_´_et0 −→ Xét. This actually

preserves finite limits and covering families. So one has also a functor between the topoi of sheaves, which we call: ˜f´et

∗ : fXet´ −→ fX´et0 , ˜f∗et´(F ) = F ◦ f´et∗. It can be proved

that ˜fet´

∗ has a left adjoint ˜fet´∗, that in fact extends f ∗ ´ et (when we see X 0 ´ et inside fX 0 ´ et via

the Yoneda embedding).

Remark 2.4. Now consider a scheme X, and a geometric point ξ −→ X, i.e. a morphism of schemes of the form u : ξ = Spec k(s)sep −→ X, where s ∈ X and sep

denotes separable closure.1

Now consider the “fiber functor” f Xet´ ˜ u∗ ´ et −→ fξ´et Γξ −→ Sets where Γξ(F ) = F ({ξ}) (global sections on ξ).

Proposition 2.5 ([AGV72, Exposé VIII, Théorème 3.5]). For every geometric point ξ of X the fiber functor commutes with colimits and finite limits.

Moreover, the family of all geometric points of X has the property that if f : F −→ V is a homomorphism of sheaves over Xet´, f is an isomorphism if and only if its pullbacks

via all fiber functors, for every ξ geometric point, are.

This implies that the fiber functor is the inverse image of a topos morphism Sets fXet´.

1_{One could also define a geometric point as a morphism Spec L −→ X where L is separably closed,}

(31)

2.1. The ´etale topos 31 We will turn back to this in Chapter 4: the second part of this theorem will imply that “the ´etale topos has enough points”.

We will also need the following theorem by Monique Hakim. To be precise, Hakim did not use the language of models of first-order theories, so this is actually a rephrasing of her result.

Definition 2.6. Let (X, OX) be a scheme. A sheaf of OX-algebras (or just an OX

-algebra) is a sheaf F on X (not necessarily quasicoherent!) such that F (U ) is an OX(U )-algebra for every open set U of X, and such that the restriction maps respect

the ring structure and the multiplication for elements of the structure sheaf, in the sense that V ⊂ U =⇒ (rm)|V = r|Vm|V for every r ∈ OX(U ), m ∈ F (U ).

Definition 2.7. A ring R is called strictly henselian if it is a local domain, its residue field k is separably closed, and for every f ∈ R[t], α ∈ k such that s is a simple root of the reduced polynomial ¯f there exists a ∈ R such that f (a) = 0 and ¯a = α.

Definition 2.8. An OX-algebra F is called strictly henselian if and only if all F (U )

are strictly henselian as rings.

Theorem 2.9 ([Hak72, III,2-4]). The ´etale topos of a scheme X is the classifying topos for the theory of strict henselian OX-algebras. Alternativerly, for every cocomplete

topos E , we have a natural equivalence

Hom(E , fX´et) ∼= Mod(Sh(X), T )

where T is the theory of strictly henselian rings (just as algebraic objects).

Example 2.10. If X = Spec k for a field k, then the ´etale topos is the classifying topos for the theory of strictly henselian k-algebras.

Remark 2.11. There are not many other simple examples. For example, if X has two points things are already not so intuitive: if R is a discrete valuation ring with maximal ideal m, a sheaf F on X just consists of two sets F (X) and F ({0}) together with a map F (X) → F ({0}). So to give an R-algebra structure on F means to give an R-algebra structure on the ring F (X) and an Rm-algebra structure on F ({0}) such

that the map F (X) → F ({0}) is a map of R-algebras. Now F ({0}) does not have to be the localization F (X)m (it is just some Rm-algebra with a map from F (X)m). So

many cases can occur other than the quasicoherent one. It is important to remark that

Theorem 2.12. For any locally constant abelian group object A in fXet´, H∗( fXet´, A) ∼=

(32)

Proof: Straightforward from the definition of topos cohomology: one takes an injective resolution and computes the cohomology. Recall that the ´etale cohomology is defined as the right derived functor of the global section functor

Γ(X, −) : Sh(Xet´) −→ Ab

F 7→ F (X)

that associates to a sheaf in the ´etale topos the abelian group given by evaluating that sheaf in X (trivially seen as ´etale scheme on itself). Cfr. [Mil80].

2.2 Etale homotopy of schemes

´

Let X be a scheme. The standard theory of sheaf cohomology provides cohomology groups with sheaf coefficients, that capture a great amount of the structure of the scheme, whereas the homotopy groups associated to the underlying topological space fail to describe the algebraic structure. The idea of ´etale homotopy is to attach to a scheme X a CW-complex (or, more in abstract, a “simplicial object”, cfr. later) whose homotopy properties reflect something more of the algebraic structure of the scheme itself.

The key idea to build this object is to consider the small étale topos, and take into account all étale coverings “together”: namely, to take some limit of groups associated to each of them and build a homotopy invariant for the scheme. This strategy needs some refinements, because the coverings do not satisfy the “filtering” properties that would allow us to take a limit, or are not coherent with other already existing notions like étale cohomology. For example, attempting to compute étale cohomology via coverings method leads to a failure:

Example 2.13. Consider a covering U ∈ Xet´. One can define the ˇCech cohomology

groups with coefficients in a locally constant sheaf A, and then take the limit: ˇ

H_´_etn(X, A) = lim_−→Hˇn(U , A).

The problem is that this does not coincide with the ´etale cohomology group H_et_´n(X, A) defined as in [Mil80] (cfr. before).

In [AGV72], Expos´e V, Sec. 7, Jean-Louis Verdier showed that the correct idea in order to solve this discrepancy was to introduce the so-called hypercoverings, which generalise coverings. For example, to a covering Y −→ X we can associate a countable

(33)

2.2. Etale homotopy of schemes´ 33 list of schemes defined by

Yn= Y ×X · · · ×X Y (n times).

This is a kind of hypercovering, and one can see that in some sense it allows more freedom in each dimension with respect to a covering.

The notion of hypercovering provided the correct limit for the cohomology, but had another problem: just like the category of coverings, the category of hypercoverings on a scheme is not cofiltering:

Definition 2.14. A cofiltering category I is a category satisfying (i) for every objects a and b there exists c with arrows to a and b

(ii) for every couple of arrows a ⇒ b there exists c −→ a such that the compositions are equal.

A cofiltered system of objects in a category C is thus a functor i 7→ Ci from a

cofiltering category I to C. A limit for such a system is an object l with morphisms l −ϕ→ Ci i (compatible with the cofiltering morphisms) such that for every d

ψi

−→ Ci

(again with compatibility) there is a unique factorizing morphism d −→ l such that the resulting diagram commutes.

The fact that the category of hypercoverings is not cofiltering is in fact a problem. So the last step of the construction is to substitute the category Hyp of hypercoverings on X with its homotopy category HC, i.e. the category with the same objects but with as arrows the homotopy classes of maps. This is cofiltering, but comes with a loss of precision while passing, roughly speaking, from Top to H (the homotopy category of CW-complex and homotopy classes of maps between them). In some cases (see references of [SS10]) this precision can actually be recovered, obtaining a proper “´etale topological type”.

It is important to remark where the “homotopy type” obtained by means of HC lives: it will be an object in the category pro − H:

Definition 2.15. Let C be a category. The objects of the pro-category pro − C

are functors F : I −→ C, where the category I is cofiltering. The morphisms are defined as:

Hompro−C({Ci}, {Dj}) = lim_←− j∈J

lim −→

i∈I

Hom(Ci, Dj).

(34)

Thus we will not obtain an actual CW-complex (up to homotopy) but a “cofiltered system” of them, much in the same way as hypercoverings are systems of coverings. We will see this more in detail.

2.3 The general construction

Our aim here is to present the basic steps of the theory summarised above, but re-ferring to general topoi and not just to the context of ´etale coverings of a scheme. This amounts to substitute the ´etale topos with an arbitrary Grothendieck topos. The result will be a “homotopy type” that will live, as above, in pro − H.

Definition 2.16. Let ∆ be the simplicial model category, which has as objects [m] = {0, . . . , m} for m integer ≥ 0, and as arrows order-preserving functions from m (seen as {1, . . . , m}) to [n]. A simplicial object with values in a category C is a contravariant functor ∆ −→ C. Simplicial objects in C form a category SC, where the morphisms are the natural transformations of such functors.

Definition 2.17. Let C be a category closed under finite limits and colimits (this property is satisfied by both Sets and X´et). For n ≥ 0 one defines functors skn and

coskn from SC to itself, as follows: let ∆/n be the category ∆ truncated at the level

n, i.e. the full subcategory of ∆ whose objects are [m] for m ≤ n. Let SnC be the

category of functors ∆/n −→ C. Since ∆/n is a finite category and C is closed under finite colimits, the obvious truncation functor τn : SC −→ SnC has a left adjoint,

called the left Kan extension ([Mac98, X.4]). We compose it with the truncation to obtain the skeleton functor

skn: SC −→ SnC −→ SC.

If C = Sets then skn(X) is the simplicial subset of X that agrees with X up to the

level n and which has no nondegenerate simplices in dimensions greater than n. Dually, since ∆/n is finite and C is closed under finite limits the “right Kan extension” provides a right adjoint for the truncation. We then consider the composition

coskn : SC −→ SnC −→ SC

and call it the coskeleton functor. We thus have

(35)

2.3. The general construction 35 hence taking A = B we get canonical morphisms

skn−→ A, A −→ coskn(A). (2.1)

Remark 2.18. If C is the category of sets or pointed sets and A ∈ SC, then one can observe that the coskeleton can be built as

coskn(A)m = HomSSets(∆[m], coskn(A)) = HomSSets(skn(∆[m]), A)

where ∆[m] is the simplicial set given by [n] 7→ Hom([n], [m]) (its geometric realisation is just the standard simplex ∆n). In particular

cosk0(A)m = Am+10

and

coskn(∆[r]) = ∆[r]

if n ≥ r.

If X ∈ H0, where H0 is the homotopy category of pointed simplicial complexes, then

one can see that the coskeleton is a Postnikov tower for X, with maps . . . −→ coskn+1(X) −→ coskn(X) −→ . . . −→ cosk0(X).

This means that πm(coskn(X)) = 0 for m ≥ n, and the canonical map X −→ coskn(X)

is universal in the homotopy category among the maps to objects with vanishing πm

for m ≥ n; and (2.1) induces isomorphisms πm(X) ∼= πm(coskn(X)).

Definition 2.19. For an object X• ∈ pro − H0 we define the n-th homotopy progroup

as

πn(X•) = {i 7→ πn(Xi)}

where Xi are pointed CW-complexes, and therefore their homotopy groups are defined.

This is an object of pro − Grp, since πn is a covariant functor. An isomorphism of

homotopy progroups is an isomorphism between such progroups in pro − Grp, and two objects in pro − H0 are said to be weakly equivalent if for every n the homotopy

groups are isomorphic.

Definition 2.20. If X• ∈ pro − H0, we can define

(36)

indexed by couples (n, i) with maps induced by compositions coskm(Xi) . . . coskn(Xi)

coskm(Xj) . . . coskn(Xj)

(2.2)

when n ≤ m in N, i −→ j in I. The vertical maps exist by covariant functoriality of the coskeletron.

Remark 2.21. There is a canonical map X −→ X\_{, induced by (}_2.1_{). This is a weak}

equivalence, i.e. induces isomorphism on all homotopy groups, since the coskeleton is a Postnikov tower. Defining [X\, X] as the set of pointed homotopy classes of maps be-tween the corresponding CW-complexes, one can see from the definition of morphisms in pro − H0 and the costruction of X\ that

[X\, X] = lim_−→[coskn(X), X].

It turns out then that the canonical map X −→ X\_{has an inverse in [X}\_{, X] if and only}

if πn(X) = 0 eventually, so that X is homotopy equivalent to some coskn(X). Thus,

unless this is the case, we have two weakly equivalent objects that are not homotopy equivalent (i.e. not isomorphic in pro − H0).

Observe finally that any map f : X −→ Y in pro − H0 induces a map X\ −→ Y\,

which is an isomorphism if and only if f induces isomorphisms of all coskeletons, or equivalently, by the universal property of the coskeleton, of all homotopy pro-groups. Definition 2.22. A map of simplicial sets f : Y• −→ X• is a trivial fibration if for

every n ≥ 1 the diagram

∂∆n Y•

∆n _X

•

(2.3)

has a dashed filler.

This amounts to require that the map

Xn= Hom(∆[n], X•) −→ HomSSets(∂∆[n], X•) ×Hom(∂∆[n],Y•)Hom(∆[n], Y•) (2.4)

is surjective. If Y• = 1 (the constant simplicial set) we call X a contractible Kan

complex if the condition is satisfied. This generalises to the following:

(37)

2.3. The general construction 37 Definition 2.23. Call a map f : Y• −→ X• between simplicial objects in a topos E

local trivial fibration if the map (2.4) is an epimorphism in E .

Definition 2.24. Let E be a topos. A hypercovering of E is a simplicial object X•

in E s.t. the map X• −→ 1 is a local trivial fibration.

Example 2.25. Consider a scheme X. A simplicial ´etale X-scheme U• is called a

hypercovering if

(i) U0 −→ X is a covering;

(ii) for every n the canonical morphism Un+1 −→ coskn(U•)n+1 is a covering.

Remark 2.26. We compare the latest two notions of hypercovering in the case of the ´etale topos. Observe that the Yoneda embedding of Xet´ into fXet´ sends schemes to

representable presheaves and preserves limits, so from the definition on a scheme we recover the latest one.

This definition gives the right limit in cohomology: denote by HC the homotopy category of these hypercovers, where homotopy here is the standard topological notion associated to simplicial objects and their realisation.

Proposition 2.27. ˇH_{V erdier}n (E , A) := lim_−→

X∈HC = H

n_{(E , A).}

We can now construct the homotopy progroups. Before that, we need the notion of connected components of an object in a topos:

Definition 2.28. Let 0 be the initial object of a topos (e.g., the empty sheaf in Sh(X)). A nonzero object E in E is connected if it cannot be decomposed as a sum of two nonzero objects. The topos is said to be locally connected if an only if every object E in E can be decomposed as a sum (not necessarily finite) of connected objects P

IEi. This decomposition is essentially unique, so one has a well-defined functor

π0 : E −→ Sets that sends E to I.

This functor is left adjoint to the constant sheaf functor ∆ : SSets −→ E . We also call π0(E ) := π0(1).

Suppose now to have a locally connected pointed topos (i.e. with a choice of a topos morphism p : Sets −→ E ). Now for any hypercover X• of E the connected components

form a simplicial set π0(X•). A base-point of such a hypercover over the point p of E is by

definition a vertex x0of the simplicial set p∗(X•). This vertex x0 yields a corresponding

Classifying topoi and groupoids

Universit`

a degli Studi di Pisa

Classifying topoi

and groupoids

Candidato

Guglielmo Nocera

Matr. 501522

Relatore

Prof. Riccardo Benedetti

Controrelatore

Prof. Andrea Maffei

Contents

Acknowledgements

Introduction

Chapter 1

Topoi and groups

1.1

The classifying topos of a topological category

1.1.1

Grothendieck topoi

1.1.2

Topological categories

1.2

Diaconescu’s Theorem

1.2.1

The Eilenberg-MacLane space

1.2.2

The classifying topos of a group and Diaconescu’s

The-orem

1.2.3

Diaconescu’s Theorem for topological categories

1.3

The classifying topos of a theory

1.3.1

Internal G-torsors

1.3.2

Internal models of a theory

1.4

Cohomology comparison

1.4.1

Groups

1.4.2

Discrete categories

1.5

Appendix: Grothendieck Universes

Chapter 2

´

Etale homotopy

2.1

The ´

etale topos

2.2

Etale homotopy of schemes

´

2.3

The general construction