Motivic Homotopy Theory - Math

Ecole Polytechnique Fédérale de Lausanne, Suisse Faculty of Mathematics

Spring 2012

Motivic Homotopy Theory

Bogdan Gheorghe Advisor: Kathryn Hess Bellwald

Abstract The goal of this project is to introduce motivic homotopy theory, which is a homotopy theory for schemes. Given a small category of k-schemes Sch/k, the Yoneda embedding embeds it fully faithfully in the category of simplicial presheaves [Sch/k op , sSet], which admits (several) model structures inherited from sSet. Unfortunately, these model structures do not preserve the colimits of Sch/k. The game is to refine these model structures until they reflect the ’geometry of schemes’ and resemble standard homotopy theories.

Contents Contents

1

Introduction

3

1 Prerequisites 1.1 Prerequisites from Category Theory . . . . . . . . . 1.2 Enriched Category Theory . . . . . . . . . . . . . . . 1.2.1 Monoidal categories . . . . . . . . . . . . . . 1.2.2 Enriched categories . . . . . . . . . . . . . . . 1.3 Presheaves and Sheaves on Grothendieck Topologies 1.3.1 Grothendieck topologies . . . . . . . . . . . . 1.3.2 Sheaves on Grothendieck sites . . . . . . . . . 1.4 Simplicial and Cosimplicial Objects . . . . . . . . . . 1.4.1 (Co)simplicial objects and (co)skeletons . . . 1.4.2 Augmented simplicial objects . . . . . . . . .

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2 Additional Structures on Model Categories 2.1 Model Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 A few categorical prerequisites . . . . . . . . . . . . . . . . . . . . . . 2.1.2 The definition of a model category and examples . . . . . . . . . . . . 2.1.3 The construction of the homotopy category . . . . . . . . . . . . . . . 2.1.4 Functors between model categories . . . . . . . . . . . . . . . . . . . . 2.1.5 Cofibrantely generated model categories and the small object argument 2.1.6 Cellular and combinatorial model categories . . . . . . . . . . . . . . . 2.1.7 Proper model categories . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Simplicial Model Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Simplicial categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Simplicial model categories . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Localization of Model Categories . . . . . . . . . . . . . . . . . . . . . . . . .

34 34 34 36 41 46 49 59 63 65 65 67 71

3 Motivic Homotopy Theory 3.1 Global Model Structures on Simplicial Presheaves . . . . . . . . . . . . . . . . 3.1.1 The global projective model structure . . . . . . . . . . . . . . . . . . 3.1.2 Comparison between injective and projective global models . . . . . .

80 81 81 88

1

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Universal Model Categories and Small Presentations . . 3.2.1 Universal cocompletion . . . . . . . . . . . . . . 3.2.2 Universal homotopy cocompletion . . . . . . . . 3.2.3 Small presentations . . . . . . . . . . . . . . . . . Local Model Structures on Simplicial Presheaves . . . . 3.3.1 Hypercompletion . . . . . . . . . . . . . . . . . . 3.3.2 Characterization of cofibrant and fibrant objects The Category of (Nisnevich) Motivic Spaces . . . . . . . 3.4.1 The category of motivic spaces . . . . . . . . . . 3.4.2 Homotopy theory on motivic spaces . . . . . . . 3.4.3 The category of Nisnevich motivic spaces . . . . Unstable Motivic Homotopy Theory . . . . . . . . . . .

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90 90 92 99 100 104 118 119 120 121 123 126

Index

132

Bibliography

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2

Introduction The idea of the subject came at the end of a lecture in algebraic K-theory given by my advisor for this project, Professor Kathryn Hess Bellwald. I would like to take this opportunity to thank her for all her help and support for this work. Without any doubt I learned far more mathematics during this semester under her supervision than during any other. I am also thankful and in debt to Marc Hoyois, for answering so nicely to my emails and sharing his algebro-geometric intuition. The initial motivation for this project was to define motivic cohomology, in order to understand how it helps in the computation of the K-theory of the integers. Unfortunately, for lack of time, neither of these two topics was covered. Therefore, I would like in this introduction to explain the road to what was my leitmotiv for almost all this work, the definition of motivic cohomology. Motivic cohomology is a cohomology theory for schemes, conjectured in the 60’s by Alexander Grothendieck. I highly recommend the very readable exposition text [Kah07] related to the subject. This cohomology theory was supposed to satisfy axioms making it into a universal cohomology for schemes, in the categorical sense that any other such cohomology should factor through it. To better understand what this cohomology theory is and how it arises, we need to (re)define more precisely what a cohomology theory is. For topological spaces, a cohomology theory is a defined as a (contravariant) functor from Top to, say graded abelian groups, satisfying the Eilenberg–Steenrod axioms. There are many attempts to formulate such axioms for schemes, but none of them really broke through, as did the Eilenberg-Steenrod axioms in topology. As a general idea, the important properties of a cohomology theory are the following three. (1) Being invariant with respect to a reasonable notion of homotopy : this is the role of such a theory, to give (computable) invariants up to homotopy, which is a weaker version of an isomorphism; (2) Commuting with coproducts (or filtered colimits) : the cohomology of a disjoint union is computable in terms of the cohomology of the components; (3) A local-to-global principle (excision) : the cohomology of a compatible local data should compute the entire cohomology. For topological spaces, the last axiom is defined in the sense that, for example, the coho` mology of the pushout Y = X X0 X 0

3

X0

X

X0

pY

is determined by the cohomology of its pieces X, X 0 and X0 . If we start with some category of schemes, there are a few problems. First, what will be the notion of a weak equivalence of schemes and a homotopy between morphisms of schemes ? The notion of a weak equivalence between the underlying topological spaces is certainly not enough since the structural sheaf is also part of the structure of a scheme. Moreover, given a homotopy theory of schemes, how to construct cohomology theories ? An answer to the first question is given by the machinery invented by Quillen, the model categories, introduced in [Qui67]. A model structure on a category M is the data of three classes of maps, the cofibrations C , the fibrations F and the weak equivalences W satisfying some axioms. The weak equivalences give raise to a homotopy theory on M, and to a homotopy category Ho(M) which is the localization M[W −1 ] of the initial category forcing all the weak equivalences to be isomorphisms. The additional data of cofibrations and fibrations have no influence on the homotopy category, but they ensure its existence and allow an explicit construction. It is usually hard to understand and construct (arbitrary) localizations of categories; one of the greatest strength of model categories is to explicitly construct the localization M[W −1 ], as a sort of quotient of M. Model categories have proved to be very useful in algebraic topology and homological algebra, giving a common framework in which the study of objects up to (chain) homotopy is possible. One of their first applications outside these fields, is to the definition of A1 -homotopy theory (also called motivic homotopy theory) in algebraic geometry, which is exactly the model structure on schemes that we look for. The resulting model category is called the (unstable) category of motivic spaces, and we will denote it by MS . In fact, it is the development of the A1 -homotopy theory (and MS ) and the further definition of motivic cohomology that led Vladimir Voevodksy to the award of a Fields Medal in 2002. Before explaining in more details how this homotopy theory for schemes is defined, let’s see how this leads to cohomology theories. The key fact is to use a variant of Brown’s representability theorem, which roughly says that, in the case of topological spaces, every cohomology theory is represented by an object in the homotopy category of spectra1 . Mimicking this property, one could hope that similarly, by inverting some suspension functor in the category of motivic spaces MS , this would give a category where the cohomology theories for schemes live. It turns out that this construction actually leads to the stable motivic category, giving access to cohomology theories for schemes. So how is this model structure on schemes constructed, leading to a homotopy theory of schemes ? I will here briefly describe this construction, and refer to the (non-published) article [Dug] for a further discussion. 1

The category of spectra of topological spaces is the category in which the suspension functor has been inverted.

4

More concretely, let’s endow the category Sch/k of schemes of finite type over an algebraically closed field k with a homotopy theory. The first observation is that from a categorical point of view, the category Sch/k is intractable since it does not contain all colimits. A good (universal) way to solve this problem is by fully faithfully embedding it by the Yoneda embedding, into its category of presheaves. That is, embedding it in the category of functors Sch/k op GGA Set. However, the category of sets is not meant for homotopy theory. In a certain sense, there is not enough room for homotopic deformations in this category, so we should instead consider functors from the category of schemes into sSet, the category of simplicial sets. The category of functors Sch/k op GGA sSet is called the category of simplicial presheaves, and is the starting point for defining a model structure. As any category of functors (or diagrams), it inherits object-wise most of the structure of the target category sSet. In particular, the natural homotopy theory of sSet, which is equivalent to the homotopy theory of Top, is inherited by the category of simplicial presheaves. As we will see, this homotopy theory does not reflect the geometry of schemes, in the sense that some colimits are not computed as geometric intuition would expect them to be. Here, the geometric intuition is interpreted by the underlying topological space of the scheme. So the problem is that the underlying topological space of a colimit of schemes is not the same as the colimit of the underlying topological spaces. A way to deal with this problem is by adding an additional structure to the category Sch/k of schemes, and quotienting in some way by it, in the category of simplicial presheaves. This is encoded in a Grothendieck topology on Sch/k, which gives the notion of a covering. By specifying which family of morphisms will be an ’open cover’ in the category of schemes, there is a way, called (Bousfield) localization, to force these colimits to be preserved in the category of simplicial presheaves. Another problem is that the category of simplicial presheaves lacks the crucial notion of an interval I. However, the affine line A1 (k) may play this role, and again by localization, we may force it to homotopically act as an interval, whatever that role may be. The question now is, did we extract enough geometric properties from Top in order to have an interesting homotopy theory of schemes ? The article [Dug01c] from Dugger explains that all this construction may be enough. Indeed, by starting with a more geometric category than Sch/k, we can apply the same construction and see what comes out of it. Roughly speaking, the construction is the following, starting with any category C. (1) (2) (3) (4)

Cocomplete it by taking presheaves on it; Extend this to simplicial presheaves to add a homotopy theory; Endow C with a Grothendieck topology and localize for having the geometric colimits; Choose a reasonable interval in C, and force it to homotopically behave like one.

It turns out that, by starting with the simplicial category ∆ with interval ∆[1], we get back the homotopy theory of simplicial sets, and by starting with the category of real manifolds ManR with interval R, we get back the homotopy theory of real manifolds ! Moreover, by skipping the last step, in the case of ∆, the homotopy theory is almost the homotopy theory of simplicial sets, except that it does not know that ∆[n] must be contractible, and similarly 5

for ManR we get the homotopy theory of manifolds, without R being contractible. That is, the last step is necessary and there are reasons to believe that this machinery gives a reasonable homotopy theory for schemes. Starting with some category of schemes, the output of this machinery is called the unstable motivic category, and this is as far as this project goes. In order to define motivic cohomology for schemes, the unstable motivic category may be stabilized by inverting two suspension functors, and then picking the right object in the homotopy category of the stable motivic category that represents motivic cohomology. Even though this is explained in a few lines here, complications arise since there are two suspension functors : one associated to the sphere in simplicial sets, and one associated to the sphere in the category of schemes (the multiplicative group), see for example [DLØ+ 07] for more explanations. Let’s now provide an overview of the mathematical content of this project. For each section, we list the important definitions and results, and give the reference it is taken from. When no reference is given, it usually means that this is taken from the internet, mostly ncatlab.org. The first chapter contains various prerequisites. The reader is assumed familiar to be with the basic notions of category theory, simplicial sets, and homotopy groups of topological spaces. The first section contains a brief review of Kan extensions [Mac71]. The second section defines monoidal categories and then categories enriched over monoidal categories [Bor94a]. Our prototype of a monoidal category is the category of simplicial sets with the categorical product, and later the categories of functors in simplicial sets. In the third section we define sites to be categories endowed with a Grothendieck topology, as well as the notion of (pre)sheaves on sites [Art62]. We explore the relation between presheaves and sheaves by means of the sheafification adjoint. In the last section we define (augmented) simplicial objects in general categories, which is a convenient language that will be used later. The second chapter is dedicated to the study of model categories. The usual references, from which all the chapter is taken, are [Hov99] for a general approach and an emphasis on the homotopy category, [Hir03] for an emphasis on localizations but also a huge amount of the general theory, [GJ09] for an emphasis on simplicial examples, and [DS95] for a general introduction. In the first section we first give the definition and first properties of a model category, as well as many examples. We then give the usual construction of its homotopy category, and define the notion of functors between model categories, Quillen adjunctions. We then define cofibrantly generated model categories, which are given with a much smaller amount of data than a usual model category. We explaine the small object argument, which is a generic argument that ’constructs’ model structures from two well-chosen generating sets. We then define cellular and combinatorial model categories, which are model categories that are cofibrantly generated, in a stronger sense. Finally, we define properness in model categories, which is an extra useful property that model categories may enjoy. The second section treats simplicial model categories, which are model categories that are enriched over sSet, where the enrichment is required to be compatible with the model structure. Such categories are very useful as they carry a natural notion of a simplicial mapping space between any two objects. Finally the third section is devoted to the heavy machinery of 6

localization, from [Hir03]. This section is important as it will be used many times later on. The third and last chapter finally treats motivic homotopy theory. In the first section, we explain how to endow a category of C-diagrams in a model category M, with a model structure primarily coming object-wise from M. Most of the proof for the model structure is taken from [Hov99]. We then specialize it to C-diagrams in sSet and characterize cofibrant and fibrant objects. In the second section we give the first step towards a homotopy theory of schemes, by showing first how to universally cocomplete a category (by formally adding colimits), and then how to universally homotopy cocomplete it (by formally adding homotopy colimits) [Dug01c]. Starting with a category of schemes, these categories may now be endowed with a model structure from the first section. Finally, we define the notion of small presentation of a model category. In the third section, we take care of the third step mentioned above, and localize with respect to the Grothendieck topology. This is done by a more general procedure called hypercompletion [DHI04]. In the fourth section, we specialize the construction to categories of schemes. We first start by studying the categorical properties of such categories, then endow it with the first model structure. Then, we define the Nisnevich topology, which is the Grothendieck topology that will be used for A1 -homotopy theory. All the work done previously, allows us to formally localize the model structure with respect to this topology. Finally in the last section, we do the last step and localize with respect to the interval A1 . This gives the unstable motivic category that defines a homotopy theory for schemes.

7

1. Prerequisites 1.1

Prerequisites from Category Theory

We will assume familiarities with the basic notions of category theory such as (locally small) categories, functors, natural transformations, all kinds of (small) limits and colimits, completeness and cocompletness, adjunctions and categories of functors. There are many good introductions to the subject, for example the book of Borceux [Bor94a] or the standard [Mac71]. The notion of simplicial sets as well as homotopy groups of topological spaces is also recommended. For a category C, we will denote its class of objects by Ob(C) or simply by C. All the concrete categories are assumed to be locally small, i.e., the hom-sets between any two objects are actual sets (elements of the category Set of sets). Most of the usual categories are indeed locally small, even tough in some constructions, we may leave the world of locally small categories. We recall the Yoneda lemma and Kan extensions. Lemma 1.1.1 (Yoneda lemma). Let C be a (locally small) category, and let C ∈ C be an object. For any functor F : C GGA Set, there is a (natural) bijection Nat(C(C, −), F ) ∼ = F (C)

∈ Set,

between the natural transformations C(C, −) =⇒ F and the elements of F (C). Similarly, for any functor F : C op GGA Set, there is a (natural) bijection Nat(C(−, C), F ) ∼ = F (C)

∈ Set,

between the natural transformations C(−, C) =⇒ F and the elements of F (C). Given two functors i

C

A

F M, it is sometimes useful to be able to extend the functor F to a functor G : A GGA M. It may not be possible to extend it strictly, such that G ◦ i = F , but only up to a natural transformation, either G ◦ i =⇒ F or in the other direction F GGGA G ◦ i. When such 8

CHAPTER 1. PREREQUISITES

extensions exist, there are two (extremal) universal ones that are called the left and right Kan extension of F along i. In particular, such extensions exist when i is a fully faithful embedding (if C is a subcategory of A for example) and if M is complete and cocomplete. In this case, the natural transformations G ◦ i =⇒ F and F =⇒ G ◦ i are in fact natural isomorphisms. Definition (Left and right Kan extensions). A left Kan extension of F along i is a functor ε L : A GGA M with a natural transformation F =⇒ L ◦ i, that is universal, in the sense described below. Dually, a right Kan extension of F along i is a functor R : A GGA M with a natural ε transformation R ◦ i =⇒ F , that is universal, in the sense described below. For a left Kan extension, the universality means that for any other functor L0 : A GGA M ε0

µ

with a natural transformation F =⇒ L0 ◦ i, there is a unique natural transformation L =⇒ L0 such that the composite µ∗id

ε

F =⇒ L ◦ i =⇒ L0 ◦ i is equal to ε0 . Dually, for a right extension, the universality means that for any other ε0

functor R0 : A GGA M with a natural transformation R0 ◦ i =⇒ F , there is a unique natural µ transformation R0 =⇒ R such that the composite µ∗id

ε

R0 ◦ i =⇒ R ◦ i =⇒ F is equal to ε0 . The functor i : C GGA A induces a functor by precomposition i∗ : MA GGA MC . Observe that the functor categories MA and MC may not be locally small categories if either C or A are not small, but in our application the categories C and A will be small. The best possible scenario is if i∗ has a left adjoint or a right adjoint. Indeed, if there exists a left adjoint Li A ∗ GGA Li : MC DG G ⊥G M : i , then the left Kan extension of F along i is the functor Li (F ) ∈ MA with the unit of the adjunction F =⇒ Li (F ) ◦ i as natural transformation. Dually, if there is a right adjoint C GGA i∗ : MA DG G ⊥ G M : Ri ,

then the right Kan extension of F along i is the functor Ri (F ) with the counit of the adjunction Ri (F ) ◦ i =⇒ F as natural transformation. It is important to emphasize the fact that it is not necessary that i∗ admits a left or right adjoint in order for a functor F to admit a left or right Kan extension. However, we have the following theorem that gives the existence of such adjoints. Theorem 1.1.2. If C is small and M is complete, then i∗ admits a right adjoint Ri C GGA i∗ : MA DG G ⊥ G M : Ri ,

9


and therefore any functor F : C GGA M admits a right Kan extension along i. Dually, if M is cocomplete, then i∗ admits a left adjoint Li A ∗ GGA Li : MC DG G⊥G M : i ,

and therefore any functor F : C GGA M admits a left Kan extension along i. Proof. This is Corollary 2 in Section X.3 in [Mac71]. Corollary 1.1.3. In addition, if the functor i : C ,GGA A is full and faithful, then the unit F =⇒ Li (F ) ◦ i and the counit Ri (F ) ◦ i =⇒ F are natural isomorphisms. Proof. This is Corollary 3 in Section X.3 in [Mac71]. For example, if we set C = ∆ the simplex category, consider the diagram ∆

i

sSet

ρ Top, where i : ∆ ,GGA sSet : [n] G [ GA ∆(−, n) is the Yoneda embedding, and where we define ρ : ∆ ,GGA Top : [n] G [ GA ∆n . Since ∆ is small and Top is cocomplete, ρ admits a left Kan extension along i ∆

i

sSet Re

ρ

Top, called the geometric realization. Moreover, this functor admits a right adjoint GGA Re : sSet DG G ⊥ G Top : Sing, called the singular functor. See Proposition 3.2.1 for a generalization of this construction.

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1.2

Enriched Category Theory

Recall that we assumed that all our categories are locally small. In this section, we will define the notion of an enriched category, which may be seen as a generalization of an ordinary category. It often happens that in a category C, the hom-sets C(A, B) have more structure than just being sets. In fact, most of the standard categories admit extra structure on their hom-sets, for example (1) the hom-set Top(X, Y ) between two topological spaces can be endowed with the compactopen topology; (2) the hom-set Ab(G, H) between two abelian groups is an abelian group under the addition of homomorphisms; (3) the hom-set R Mod(M, N ) between two (left) R-modules is also an abelian group under the addition of R-linear maps; (4) the hom-set VecF (V, W ) between two F-vector spaces is again an F-vector space; and the list can continue for many other categories. A category C is said to be enriched over another category V 1 if the hom-sets are objects of V, denoted by C(A, B) ∈ V, and if there are composition maps C(A, B) × C(B, C) GGA C(A, C)

∈ V,

satisfying an associativity and unit axiom. Implicitly, here we made use of the categorical product in V, by computing C(A, B) × C(B, C). However, the categorical product does not always have good properties (it is not always in adjunction with Hom for example). We will now define the categories with good products, on which we will be able to enrich other categories.

1.2.1

Monoidal categories

We will enrich categories over what is called a monoidal category, that is, a category equipped with a bifunctor −⊗− : V ×V GGA V that will play the role of the above product of the homobjects. This introduction to monoidal categories and enriched categories is essentially from Section 6 in the book [Bor94b] of Borceux. Another good reference with many examples is given in Chapter 4 of [Hov99]. Definition (Monoidal category). A monoidal category V is a category V together with • a bifunctor − ⊗ − : V × V GGA V called the tensor product; • a distinguished object I ∈ V called the unit; • for every triple of objects A, B, C ∈ V a natural isomorphism ∼ =

αABC : (A ⊗ B) ⊗ C GGA A ⊗ (B ⊗ C), called the associator of A, B and C; 1

A more rigorous definition is given later in 1.2.2.

11


• for every object A ∈ V a natural isomorphism ∼ =

lA : I ⊗ A GGA A, called the left unit of A; • for every object A ∈ V a natural isomorphism ∼ =

rA : A ⊗ I GGA A, called the right unit of A; such that for any objects A, B, C, D ∈ V, the following two diagrams of associativity coherence and unit coherence commute. αA⊗B,C,D ((A ⊗ B) ⊗ C) ⊗ D (A ⊗ B) ⊗ (C ⊗ D)

(A ⊗ I) ⊗ B

αAIB

A ⊗ (I ⊗ B)

αABC ⊗ id αA,B,C⊗D

(A ⊗ (B ⊗ C)) ⊗ D

rA ⊗ id

id ⊗lB

αA,B⊗C,D ⊗ id A ⊗ ((B ⊗ C) ⊗ D) A ⊗ (B ⊗ (C ⊗ D)) id ⊗αBCD

A ⊗ B.

Observe that all the arrows in both diagrams are isomorphisms in V. We usually refer to such a monoidal category by (V, ⊗, I) or even simply by V if the context is clear, letting understood the product, the unit (unique up to isomorphism), and the associators and unit isomorphisms. Definition (Symmetric). A monoidal category (V, ⊗, I) is called symmetric if in addition there are natural isomorphisms ∼ =

sAB : A ⊗ B GGA B ⊗ A, such that the following diagrams of associativity coherence, unit coherence and symmetry coherence commute. (A ⊗ B) ⊗ C

sAB ⊗ id

αABC

αBAC

A ⊗ (B ⊗ C)

B ⊗ (A ⊗ C)

sA⊗B,C (B ⊗ C) ⊗ A

(B ⊗ A) ⊗ C

id ⊗sAC αBCA 12

B ⊗ (C ⊗ A)


sAI

A⊗I ra

A

I ⊗A la

sAB

B⊗A

A⊗B

sBA A ⊗ B.

Again, all arrows in those three diagrams are isomorphisms in V. All the monoidal categories that we will consider will indeed be symmetric. Before giving some examples, let’s give a last useful property that we would like monoidal categories to satisfy. Definition (Closed symmetric monoidal category). A symmetric monoidal category V is said to be closed, if for all objects A ∈ V, the endofunctor − ⊗ A has a right adjoint. This right adjoint is usually denoted by (−)A . Observe that it follows that A ⊗ − also has a right adjoint, since these two functors are naturally isomorphic by ∼ = s−,A : − ⊗A =⇒ A ⊗ −. The right adjoint is denoted by (−)A because it really corresponds to an exponentiation, as is showed in the following examples. Example 1.1 (Monoidal categories). (1) The most basic example of a monoidal category is the category Set of sets with the cartesian product × as tensor product and the unit being a singleton {∗}. Moreover, it is clearly symmetric since A × B ∼ = B × A naturally, and it is moreover closed. The adjoint functor of − × B is the covariant hom-functor (−)B = Set(B, −), and the adjunction is sometimes called the exponential law ∼ Set(A, Set(B, C)). Set(A × B, C) = (2) Let V be a category that admits all finite products. By choosing a terminal object ∗ and a fixed product A × B for all objects A, B ∈ V, the category V is a monoidal category with the categorical product × as tensor product and ∗ as unit. This product is also symmetric since A × B ∼ = B × A (by a unique isomorphism) by definition of the categorical product. If the monoidal structure is closed, the right adjoint (−)B gives the notion of an internal-hom, also denoted by AB = V(B, A) ∈ V. (3) In particular, the category Top of topological spaces is symmetric monoidal with the cartesian product ×, the unit ∗ and the natural isomorphisms X ×Y ∼ = Y ×X. Moreover, a candidate for a right adjoint to −×Y is the exponential functor that gives an internalhom (−)Y = Top(Y, −) : Top GGA Top : Z G [ GA Top(Y, Z), where Top(Y, Z) is endowed with the compact-open topology. However, the isomorphism Top(X × Y, Z) ∼ = Top(X, Top(Y, Z))

13


only holds when Y is locally compact and Hausdorff, and in fact it turns out that Top is not closed. This is a reason why the category of topological spaces is sometimes restricted to the full subcategory of compactly generated spaces, which is a closed symmetric monoidal category under the cartesian product. (4) Similarly, the category sSet of simplicial sets is symmetric monoidal with the cartesian product and the unit {∗}∗ . Moreover, it is a closed symmetric monoidal category, where the right adjoint of − × Y• is given by the internal-hom (−)Y• = sSet(Y• , −) : sSet GGA sSet : Z• G [ GA sSet(Y• , Z• ) which is given by sSet(Y• , Z• )n := sSet(Y• × ∆[n], Z• ). (5) The category Ab of abelian groups is a monoidal category with tensor product ⊗ and unit Z. Moreover, it is symmetric since A ⊗ B ∼ = B ⊗ A naturally, and it is closed by the tensor-hom adjunction ∼ Ab(A, Ab(B, C)), Ab(A ⊗ B, C) = where Ab(B, C) is given the structure of an abelian group under pointwise addition. (6) More generally, if R is a commutative ring the category R Mod of R-modules is a monoidal category with tensor product ⊗R and unit R. It is symmetric since M ⊗R N ∼ = N ⊗R M naturally, and it is closed by the same tensor-hom adjunction R Mod(M

∼ R Mod(M, R Mod(N, L)), ⊗R N, L) =

where R Mod(N, L) is given the structure of an R-module under pointwise addition and pointwise multiplication by R. (7) An example that we will often use is the induced monoidal structure on a category of diagrams. If C is a small category and (V, ⊗, I) is a monoidal category, the category of functors [C, V] admits a monoidal structure with pointwise tensor product F ⊗ G(C) := F (C) ⊗ G(C), and where the unit is given by the constant functor I˜: C GGA V : C G [ GA I(C) = I. Moreover, it is symmetric if V is, and in addition closed if V is. This example is treated in more details in Example 1.2 at page 19, or when V = sSet at the beginning of Section 3.4. (8) The pointed versions Top∗ and sSet∗ are also symmetric monoidal categories under the smash product ∧, and the unit given by ∗+ , where we consider the adjunctions GGA (−)+ : Top DG G ⊥ G Top∗ : U

and

GGA (−)+ : sSet DG G ⊥ G sSet∗ : U,

where (−)+ adds a disjoint base point, i.e., X+ := X ∗, and U is the forgetful functor. They are both symmetric, and sSet∗ is again closed with a similar adjoint (−)Y• . `

14


Most of these categories admit an internal-hom bifunctor V op × V GGA V, that will be denoted by V(−, −), as already did in the previous examples. As we will see, it turns out that any closed symmetric monoidal category admits such an internal-hom functor. The covariant functor V(I, −) : V GGA Set represented by the unit I is called the underlying set functor, as it gives back the underlying set of the objects in many situations. For example in Set since Set(∗, X) ∼ = X, in Top since Top(∗, X) ∼ = X, in R Mod since R Mod(R, M ) ∼ = M, and so on. Let’s now consider a closed symmetric monoidal category (V, ⊗, I), where the adjunctions are denoted by GGA − ⊗ A : V DG G ⊥ G V : V(A, −). The unit of this adjunction gives morphisms B GGA V(A, B ⊗ A), which specializes to I GGA V(A, I ⊗ A) ∼ = V(A, A), by letting B = I. It turns out that in the previous examples when I = ∗, the image of this morphism gives back the identity morphism on A. For example in Set, this morphism is {∗} GGA Set(A, A) : ∗ G [ GA idA . The counit of the adjunction gives evaluation morphisms evAB : V(A, B) ⊗ A GGA B, which for example in Set are Set(A, B) × A GGA B : (f, a) G [ GA f (a). These evaluation morphisms induce a composition cABC : V(A, B) ⊗ V(B, C) GGA V(A, C), which is given as the adjoint morphism of the composite evAB ⊗id evBC ∼ V(B, C)⊗B GGA V(A, B)⊗V(B, C)⊗A ∼ C. = V(A, B)⊗A⊗V(B, C) GGA B ⊗V(B, C) =

Proposition 1.2.1. On a symmetric monoidal closed category (V, ⊗, I), there is a bifunctor V(−, −) : V op × V GGA V : (A, B) G [ GA V(A, B), whose postcomposite with the underlying set functor V(I, −) gives the hom-functor V(−, −) : V op × V GGA Set : (A, B) G [ GA V(A, B). Proof. We first have that V(A, −) is a functor, which is the right adjoint to − ⊗ A. For the other variable, for any morphism f : A GGA A0 , define V(f, B) : V(A0 , B) GGA V(A, B) 15


to be the corresponding morphism by adjunction, to the composite evA0 B

id ⊗f

V(A0 , B) ⊗ A GGA V(A0 , B) ⊗ A0 GGA B, and thehe functoriality of the bifunctor V(−, −) follows. For the second part, the composite gives (A, B) G [ GA V(A, B) G [ GA V(I, V(A, B)), which is isomorphic in Set to V(A, B) since V(I, V(A, B)) ∼ = V(I ⊗ A, B) ∼ = V(A, B).

1.2.2

Enriched categories

In this projet, the main purpose for defining monoidal categories is to further define categories enriched over them2 . Definition (Enriched category). Let (V, ⊗, I) be a monoidal category. A V-category or a category enriched over V is • a collection of objects Ob(C); • for every pair of objects A, B ∈ Ob(C), an object C(A, B) ∈ V; • for every triple of objects A, B, C ∈ Ob(C), a composition morphism C(A, B) ⊗ C(B, C) GGA C(A, C)

∈ V;

• for every object A ∈ Ob(C), a unit morphism I GGA C(A, A)

∈ V;

such that the associativity axiom and the unit axiom hold, i.e., the following two diagrams commute (C(A, B) ⊗ C(B, C)) ⊗ C(C, D)

cABC ⊗ id

C(A, C) ⊗ C(C, D)

αC(A,B),C(B,C),C(C,D) cACD

C(A, B) ⊗ (C(B, C) ⊗ C(C, D)) id ⊗cBCD C(A, B) ⊗ C(B, D)

cABD

2

C(A, D)

Another important aspect of monoidal categories is to define algebras over them. Hovey is more focused on this purpose in [Hov99].

16


I ⊗ C(A, B)

lC(A,B)

uA ⊗ id C(A, A) ⊗ C(A, B)

lC(A,B)

C(A, B)

id ⊗uB

id cAAB

C(A, B) ⊗ I

C(A, B)

cABB

C(A, B) ⊗ C(B, B).

Since the axioms are always similar, a shorter way to state and remember them would be to say that • the two ways of getting from (C(A, B) ⊗ C(B, C)) ⊗ C(C, D) to C(A, D) should be the same (associativity); • the two ways of getting from I ⊗ C(A, B) to C(A, B) are the same (left unit); • the two ways of getting from C(A, B) ⊗ I to C(A, B) are the same (right unit). The ordinary theory of locally small categories can be seen as the theory of categories enriched over Set. In a similar way, a category with all finite limits and colimits is a preabelian category if and only if it is enriched over the category Ab of abelian groups. We will mostly be interested in two types of enriched categories • categories enriched over spaces, usually over sSet; • monoidal categories enriched over themselves. Proposition 1.2.2. Let (V, ⊗, I) be a closed symmetric monoidal category. Then V can naturally be enriched over itself. Proof. Since it is closed, the tensor product − ⊗ A has a right adjoint V(A, −) : V GGA V, and Proposition 1.2.1 shows the existence of a bifunctor V(−, −) : V op × V GGA V. The unit of the adjunction, specialized at I gives the unit morphism I GGA V(A, A), and the counit (evaluation) of the adjunction, applied two times gives a composition morphism V(A, B) ⊗ V(B, C) GGA V(A, C). It remains to check the commutativity of the diagrams, which is a fastidious but a straightforward checking.

17


Observe that in the definition of a V-category C, there is no mention of C being a category. Applying Proposition 1.2.1 and Proposition 1.2.2 we get that starting with a closed symmetric monoidal category V, if we enrich it over iself with internal-hom V(A, B), applying the functor V(I, −) : V GGA Set gives back the hom V(A, B) of the underlying category. There is a more general statement which says that given a V-category C, there is an underlying category, also denoted by C. The general construction, that appears in Chapter 6 of [Bor94b] is as follows. We first define V-functors between V-categories and V-natural transformations between them, and the small V-categories with these notions form a 2-category, denoted by V-Cat. All these definitions are natural and are what we expect them to be. We can then define the V-representables and we get an enriched Yoneda lemma. Moreover, for every monoidal category V, there is a forget 2-functor3 U : V-Cat GGA Cat that admits a left adjoint and really behaves like a forget functor. It satisfies U (C)(A, B) = V(I, C(A, B)) for any V-category C, and in particular applying it to V gives U (V)(A, B) = V(A, B), as expected. This functor is in fact a functor V-Cat GGA Set-Cat, which is the identity on objects. In view of these remarks, we may often abuse notation and identify an enriched category C with the underlying category, also denoted by C. The last notion that we need from enriched category theory is the notion of a tensor and cotensor. Given a category C enriched over a symmetric closed monoidal category V, this provides a way to compute tensor products C ⊗ V ∈ C for an object C ∈ C and V ∈ V, as well as the adjoint operation, the power V C ∈ C. Definition (Tensor and cotensor). Let V be a symmetric monoidal closed category, C be a V-category and pick two objects C ∈ C and V ∈ V. • The tensor of V and C, if it exists, is an object denoted by V ⊗ C ∈ C together with isomorphisms in V C(V ⊗ C, C 0 ) ∼ = V(V, C(C, C 0 )) natural in C 0 ∈ C. We say that C is tensored (over V) when all tensors V ⊗ C ∈ C exist. • The cotensor of V and C, if it exists, is an object denoted by C V ∈ C together with isomorphisms in V C(C 0 , C V ) ∼ = V(V, C(C 0 , C)) natural in C 0 ∈ C. We say that C is cotensored (over V) when all cotensors C V ∈ C exist. If C is tensored and cotensored over V, it follows that there are natural isomorphisms C(V ⊗ C 0 , C) ∼ = C(C 0 , C V )

∈ V,

that can be interpreted as an enriched adjunction. 3

Roughly speaking, this forget 2-functor is a usual functor that has the additional structure of sending V-natural transformations to natural transformations.

18


Example 1.2 (Enriched categories). (1) Any locally small category C is enriched over (Set, ×, {∗}). The enriched hom-sets are the usual hom-sets C(A, B) ∈ Set, the composition is the usual one and the unit morphism is {∗} GGA C(A, A) : ∗ G [ GA idA . Moreover, if C has coproducts, the tensor of S ∈ Set and A ∈ C must satisfy C(S ⊗ A, B) ∼ = Set(S, C(A, B)) ∼ =

Y

C(A, B) ∼ =C

S

! a

A, B ,

S

and if C has products, the cotensor of S ∈ Set and A ∈ C must satisfy C(B, A ) ∼ = Set(S, C(B, A)) ∼ = S

Y

C(B, A) ∼ = C B,

S

! Y

A .

S

Therefore, if C has products and coproducts, then C is tensored and cotensored where ` the tensor can be given by the copower S A and the cotensor can be given by the Q power S A. For this reason, a tensor is sometimes called a copower and a cotensor is called a power. (2) Let V be a symmetric monoidal closed category. Then the V-category V is both tensored and cotensored, with tensor V ⊗ A and cotensor AV . In particular, all the previous symmetric monoidal closed categories Set, sSet, sSet∗ , compactly generated spaces, Ab, R Mod (for R a commutative ring), . . . are tensored and cotensored over themselves. (3) Consider the closed symmetric monoidal category Ab under the tensor product ⊗ and unit Z. For a non-necessary commutative ring R, the category ModR of right Rmodules can be enriched over Ab under pointwise addition of R-linear maps. However, under no additional conditions on R (such as commutativity for example), the abelian groups ModR (M, N ) are not R-modules, in general. This category is tensored and cotensored over Ab where, for two objects M ∈ ModR and A ∈ Ab, the tensor is the right R-module A ⊗Z M ∈ ModR satisfying ModR (A ⊗Z M, N ) ∼ = Ab(A, ModR (M, N ))

∈ Ab,

and the cotensor of A and M satisfies ModR (N, M A ) ∼ = Ab(A, ModR (N, M ))

∈ Ab.

It is given by the right R-module which is Ab(A, M ) endowed with the right action Ab(A, M ) × R GGA Ab(A, M ) : (f, r) G [ GA f ∗ r, where (f ∗ r)(a) := f (a) · r ∈ M . (4) An example of an enriched category that we will often use is a category of C-diagrams in a monoidal category V, that can be enriched either over itself or over V. More precisely, let C be a small category and consider the category of functors [C op , sSet] that we will denote by M := [C op , sSet]. The only reason of taking C op instead of C is because the category M of functors is called the category of simplicial presheaves on C and is of 19


important use in algebraic geometry. Consider the symmetric closed monoidal structure on sSet given by the categorical (cartesian) product ×, the constant simplicial set {∗}∗ as unit and the internal-homs are given by sSet(X• , Y• ) ∈ sSet where sSet(X• , Y• )n := sSet(X• × ∆[n], Y• )

∈ Set.

This monoidal structure induces two different structures on M. First, it induces a monoidal structure on M and therefore induces a structure of M-enriched category on itself. Second, we can also see M as enriched over sSet, and it is tensored and cotensored. The induce monoidal structure is the pointwise one. For F, G ∈ M, their product is (F × G)(C) := F (C) × G(C) ∈ sSet, the unit is the constant simplicial presheaf ∗ : C G [ GA ∗ that sends any object C ∈ C to the unit of ∗ of sSet. Moreover, it is symmetric and closed since sSet is (as already pointed out in Examples 1.1. ˜• : C G For any simplicial set K• consider the simplicial presheaf K [ GA K• which is constant on objects. We can now define the tensor and cotensor of F ∈ M with K• ∈ sSet by ˜ ˜• × F K• ⊗ F := K and F K• := F K• , i.e., f (−)

taking the operation after the embedding sSet ,GGA M. The category M := [C op , sSet] of simplicial presheaves is enriched over sSet by defining MsSet (F, G) ∈ sSet to be ˜ M(F × ∆[−], G), i.e., ˜ MsSet (F, G)n := M(F × ∆[n], G)

∈ Set.

It follows that the enrichement is tensored and cotensored by the above formulas. A follow-up of this section appears in section 2.2 where we first explain in more details what means to be enriched over sSet, and then explore the compatibility between a model structure and an enriched structure over simplicial sets.

1.3

Presheaves and Sheaves on Grothendieck Topologies

In this section we will add another type of structure on a category, a Grothendieck topology. The role of a Grothendieck topology on a category is to give accessible the notion of what an open cover is. The goal is to mimic the natural notion of open cover that appears in the category Top of topological spaces, or in the category ManR of real manifolds. This can be seen as adding a geometric flavour in the category. The notion of an open cover {Uα ,GGA X} allows a transition from local informations on Uα to global informations on X, and the presheaves that respect these transitions will be called sheaves.

20


1.3.1

Grothendieck topologies

Definition (Grothendieck (pre)topology, site). Let C be a category that admits all pullbacks. A Grothendieck pretopology on C is an assignment to each object X ∈ C of a collection of families of morphisms {Uα GGA X}α ⊆ Mor(C) called covering families (of X), satisfying the axioms ∼ =

∼ =

(1) any isomorphism U GGA X gives a covering family of X with one morphism {U GGA X}; (2) for any covering family {Uα GGA X}α of X and any morphism Y GGA X, the projections Uα ×X Y GGA Y from the pullback squares Uα ×X Y

Y

y Uα

X

form a covering family {Uα ×X Y GGA Y }α of Y ; (3) for any covering family {Uα GGA X}α of X and every covering families {Vα,β GGA Uα }β for each Uα , the composite {Vα,β GGA Uα GGA X}α,β is again a (finer) covering family of X. A category C with the additional structure of a Grothendieck topology is called a site. If in addition the category C is small, then it is called a small site. The data of all these covering families defines a pretopology on C, and generates a unique topology on C. Even though there can be different pretopologies that generate the same topology, it is not really restrictive to only work with pretopologies. The important point is that sheaves only depends on the topology, so it is enough to consider a pretopology that generates the wanted topology. It is therefore usually convenient to work with pretopologies instead of topologies. A Grothendieck topology is a similar definition as a pretopology, that is given in term of covering sieves instead of covering families. A covering sieve can be seen as the closure under precomposition of morphisms in the category, of a covering family, and is therefore a much ’bigger’ data than a covering family. As the definition of a topology is more technical and gives no additional insight in what these topologies are, we will only illustrate what a sieve intuitively is. If a covering family of X can be seen as a 1-iterated covering containing only morphisms Uα GGA X (as on the left picture), a sieve can be seen as the closure by precomposition of a covering family (as on the right picture)

21


Uα Uβ .. .

Vα,β X

Uα

···

Wα,β,γ

Uβ

···

.. .

.. .

···

Wβ,γ,δ

X

Vβ,γ

··· and contains the closure by precomposition of a given covering family. Of course, the diagram of a covering sieve (of X here) is much more complicated than this drawing, but this already gives an idea that it is more convenient to work with covering families. Working with pretopologies can be compared with working with a basis of a vector space, the choice is not unqiue but they all generate the same space by some closure operations. We will usually identify a pretopology with the topology it generates and therefore work with a topology that is defined in terms of covering families. The only axiom of a (pre)topology that may sound mysterious is the second axiom with the pullback property, which is explained in the following examples. Example 1.3 (Grothendieck topologies, sites). (1) The prototype example of a small site is the category top(X) for a topological space X. The category top(X) is the poset under inclusion ⊆ of the open subsets of X, i.e., it is defined by • Objects : inclusions U ,GGA X; • Morphisms : inclusions U ,GGA V between the underlying open subsets of X, such that the triangle commutes X U

V.

The notion of a covering family in top(X) is the usual one, that is, a family of morphisms S {Uα ,GGA W }α in the category top(X) is a covering family if and only if Uα covers W . In top(X), colimits are unions and limits are intersections. More precisely, the ’pullback over X’

22


V

U

W

X ∼ =

is the intersection U ∩V ,GGA X ∈ top(X). Since the only isomorphisms are U ,GGA U , the first axiom of a pretopology is satisfied. The second axiom can be restated as if {Uα ,GGA W }α is a covering family of W and V ,GGA W is another morphism in top(X), then the morphisms in the pullbacks Uα ×W V = Uα ∩ V

V

Uα

W

give a covering family {Uα ∩ V ,GGA V }α of V . In words, this says that the intersection with V of a covering of W gives of covering of V . The last axiom says that if {Uα ,GGA W }α is a covering of W and if {Vα,β ,GGA Uα }β are coverings of each Uα , then the composite {Vα,β ,GGA Uα ,GGA W }α,β is a covering of W . This can be seen as a finer covering that refines the covering by Uα ’s. (2) The large version of the small sites top(X) is the big site Top of all topological spaces. A collection {Uα GGA X}α is defined to be a covering family in Top if and only if it is so in top(X). In particular the morphisms Uα ,GGA X are required to be injections. (3) We will be mostly interested in topologies on categories of schemes. Let k be an algebraically closed field, and let Sch/k be the category of schemes over k, i.e., morphisms of schemes (X, OX ) GGA (Spec k, Ok ). We will usually denote a scheme (X, OX ) by the underlying topological space X. As we did with topological spaces, let’s fix a k-scheme S ∈ Sch/k, and denote by Sch/S the category of k-schemes over S. There is a long list of Grothendieck topologies that can endow Sch/S, see for example [Sta, Chapter : Topologies on Schemes]. For example the Zariski topology on Sch/S has as covering families the collections fα

{Uα GGA X}α where each morphism fα is an open immersion and

S

fα (Uα ) = X.

The étale topology also defines a Grothendieck topology on Sch/S, where a collection fα

of morphisms {Uα GGA X}α is defined to be a covering family if and only if each S morphism fα is étale and again fα (Uα ) = X. 23


Motivic homotopy theory is defined in terms of a topology strictly in between the Zariski and the étale, the Nisnevich topology, which is defined and motivated in Section 3.4. There are other conditions that can be imposed on the morphisms such as smooth, syntomic, . . . , where the result generates a Grothendieck topology thanks to the fact that all these properties are preserved by pullbacks and by composition, see for example Proposition 6.8.3 in [Gro65]. (4) As in the case of topological spaces, all the topologies on the small site Sch/S extend to topologies on the big site Sch/k of all k-schemes. Moreover, there are many other useful subcategories of Sch/k that admit Grothendieck topologies, for example the subcategories Sm/k of smooth k-schemes, the subcategory of schemes of finite type, . . . . Let C be a small category. Recall that the category of functors [C op , Set], sometimes op denoted by SetC or by Pre(C), is called the category of presheaves (of sets) on C, and a functor F : C op GGA Set is called a presheaf (of sets). The smallness condition on C ensures op that the category of functors SetC is a (locally small) category.

1.3.2

Sheaves on Grothendieck sites

On a small Grothendieck site, the notion of coverings gives a sense to a local to global iα

principle, as it is the case in the prototype category top(X). Let {Uα ,GGA U }α be an open cover of U in top(X) and let F : top(X)op GGA Set be a presheaf (on X). A collection {fα }α of elements fα ∈ F (Uα ) is called compatible if, with the notations of the two commutative squares Uα ∩ Uβ

iαβ

Uα

iβα Uβ

iα

iβ

F (iα )

F (U )

=⇒

F (iβ )

U

F (Uβ )

F (iαβ )

F (iβα )

for any α 6= β, the equality is satisfied F (iαβ )(fα ) = F (iβα )(fβ )

∈ F (Uα ∩ Uβ ).

A presheaf F is called a sheaf if • for any open cover {Uα ,GGA U }α , and • for any compatible collection {fα }, there exists a unique element f ∈ F (U ) such that F (iα )(f ) = fα 24

F (Uα )

for all α.

F (Uα ∩ Uβ ),


Example 1.4 (Sheaf on top(X)). Let X ⊆ Rn be a real manifold and consider F : top(X)op GGA Set : U G [ GA F (U ) = C 0 (U, R) i

the presheaf of R-valued continuous functions. The functor F sends an embedding U ,GGA V to the precomposition i∗ : F (V ) GGA F (U ), which is in fact the restriction to U i∗ : C 0 (V, R) GGA C 0 (U, R) : f G [ GA f

U.

iα

Let now {Uα ,GGA U }α be an open cover of U . A collection of elements {fα } where fα ∈ F (Uα ) for all α, i.e., fα : Uα GGA R is compatible if fα

Uα ∩Uβ

= fβ

Uα ∩Uβ

for any α, β such that Uα ∩ Uβ 6= ∅. Therefore, if {fα } is a compatible collection, then there is an element f ∈ F (U ) such that each restriction to Uα is fα . Moreover, this function is necessarily given by f : U GGA R : u G [ GA fα (u)

for any α such that u ∈ Uα ,

which is a well-defined function by the compatibility of the collection of functions {fα }. Since this condition is verified for any collection of compatible elements and for any open cover, it follows that F = C 0 (−, R) is a sheaf. With the notation of the the above two commutative squares, the morphisms F (iβα ) and F (iαβ ) give, for a fixed α, two morphisms of F (Uα ) into the term F (Uα ∩ Uβ ) = F (Uβ ∩ Uα ) and this induces the diagram Q Y

F (iαβ ) Y

F (Uγ ) Q

γ

F (iβα )

F (Uα ∩ Uβ ).

α,β

A collection of elements {fγ }γ is an element in the product F (Uγ ), and the fact that it is Q compatible is the same as saying that it gets equalized by the two morphisms F (iαβ ) and Q F (iβα ). Moreover, the fact that there exists an element f ∈ F (U ) such that F (iα )(f ) = fα for all α means that the two composites Q

Q

F (U )

F (iγ )

Q Y

F (iαβ ) Y

F (Uγ ) Q

γ

F (iβα )

F (Uα ∩ Uβ ).

α,β

are the same, and the fact that such an f is unique means that F (U ) is in fact the equalizer iα

of this diagram. Therefore, a presheaf F is a sheaf if and only if, for any open cover {Uα ,GGA U }α in top(X), the induced diagram is an equalizer. Moreover, since the intersection Uα ∩Uβ can be interpreted as the pullback Uα ×U Uβ , this categorifies the notion of a sheaf. Definition (Sheaves on a Grothendieck site). Let C be a small Grothendieck site. A presheaf F : C op GGA Set is called a sheaf if for any open covering {Uα GGA X}α in the site C, the induced diagram 25


Q

F (X)

Q

F (iγ )

Y

F (iαβ ) Y

F (Uα ) Q

α

F (iβα )

F (Uα ×X Uβ )

α,β

is an equalizer in Set. In the category of presheaves, the full subcategory of sheaves will be denoted by Sh(C) ⊆ Pre(C). If we define the indiscrete topology to have as covering families only the isomorphisms ∼ =

{X GGA Y } ⊆ Mor(C), the sheaf condition is satisfied by any presheaf, and thus the category of sheaves for this topology is the category of all the presheaves Pre(C). In particular, we also study the category of presheaves by only studying the sheaves on a Grothendieck site. In the functor category of presheaves [C op , Set], the limits and colimits are computed object-wise, and therefore Pre(C) is a complete and cocomplete category, since Set is. An important question about the subcategory of sheaves Sh(C) ⊆ Pre(C) is to see how limits and colimits are computed, and to see if it is either complete or cocomplete. Since Sh(C) is also a functor category (as Pre(C)), the right notion of limit and colimit is the object-wise one, i.e., the limit and colimit computed in the larger category of presheaves. Consider X : J GGA Pre(C), a J -diagram such that X(j) is in fact a sheaf for each j ∈ J , that is, they satisfy the condition Q

X(j)(U )

X(j)(iγ )

Q Y

X(j)(iαβ ) Y

X(j)(Uα ) Q

α

X(j)(iβα )

X(j)(Uα ×X Uβ )

α,β

for each covering in the site {Uα GGA U }α . The limit presheaf limj∈J X(j) of this diagram is in fact again a sheaf, since the fact that limits commutes with limits (see for example Section IX.8 in [Mac71]) gives that the diagram Q

(limj∈J X(j))(U )

X(j)(iγ ) Y α

Q

(limj∈J X(j))(Uα ) Q

X(j)(iαβ ) Y

(limj∈J X(j))(Uα ×U Uβ )

X(j)(iβα ) α,β

is isomorphic to the diagram Q

limj∈J

X(j)(U )

X(j)(iγ )

Q Y

X(j)(iαβ )

X(j)(Uα ) Q

α

! Y

X(j)(iβα )

X(j)(Uα ×U Uβ ) ,

α,β

which is an equalizer since object-wise it is. The category Sh(C) of sheaves is therefore complete and limits are computed as in the category of presheaves, i.e., object-wise. In particular, the inclusion functor i : Sh(C) ,GGA Pre(C) preserves limits, and may therefore admit a left adjoint. In fact, if the inclusion functor admits a left adjoint, this left adjoint will help compute the colimits in Sh(C), by first computing it in Pre(C) and the pushing it back to Sh(C) since left adjoints preserve colimits.

26


Theorem 1.3.1 (Sheafification). Let C be a small Grothendieck site. The inclusion of sheaves into presheaves i : Sh(C) ,GGA Pre(C) admits a left adjoint GGA a : Pre(C) DG G ⊥ G Sh(C) : i, called the sheafification functor. Idea of the proof. A proof with more details can be found in Chapter 2 of [Art62]. For any object U ∈ C, define a category JU with • Objects : coverings {Uα GGA U }α of U ; f

• Morphisms : an arrow {Uα GGA U }α∈A GGA {Vβ GGA U }β∈B is a function A GGA B fα

with morphisms Uα GGA Vf (α) ∈ C such that all triangles commute Uα

fα

Vf (α)

U. For any U ∈ C, a presheaf F induces a functor   Y Y GGA F (Uα ×U Uβ ) , [ GA eq  F (Uα ) GGA FU : JUop GGA Set : {Uα GGA U }α G α

α,β

φ

where the equalizer is F (U ) if F is a sheaf. Moreover, any morphism V GGA U ∈ C in the site also induces a functor J(φ) : JU GGA JV : {Uα GGA U } G [ GA {Uα ×U V GGA V }. The following square Uα ×U V

Uα ×U Uβ ×U V

Uα

Uα ×U Uβ

commutes, and the upper right corner can be replaced by Uα ×U V ×V Uβ ×U V . There are therefore induced natural transformations FU =⇒ FV ◦ J(φ) and thus a morphism colim FU GGA colim FV . This defines a functor (−)+ : Pre(C) GGA Pre(C) : F G [ GA F + which is defined by F + (U ) := colimJU FU . This is where the magic happens, because in turns out that F ++ is in fact a sheaf for any presheaf F . This follows from the following result. 27


Fact (Lemma 2.1.2 in [Art62]). Let F ∈ Pre(C) be a presheaf. • If for all coverings {Uα GGA U }α the natural map F (U ) ,GGA

Y

F (Uα )

α

is an injection, then F + is a sheaf. • The presheaf F + satisfies the above condition. To show that the sheafification is left adjoint to the inclusion functor, observe first that for every covering {Uα GGA U }α there is a canonical map   Y Y GGA F (U ) GGA FU ({Uα GGA U }α ) = eq  F (Uα ) GGA F (Uα ×U Uβ ) α

α,β

that is induced by F (U ) GGA F (Uα ). Moreover, since JU is connected, the colimit of the constant diagram JUop GGA Set : {Uα GGA U }α GGA F (U ) Q

is canonically isomorphic to F (U ). This induces a map between the colimits, and therefore a map of presheaves F GGA F + , which is an isomorphism if F is already a sheaf. Therefore, any morphism of presheaves F GGA G, where G is a sheaf gives F+

F

G∼ = G+ , that commutes by naturality, and thus any morphism F GGA G where G is a sheaf factorizes trough F + . In other words, if we denote by Pre(C)+ the full subcategory of Pre(C) we restrict the objects to the ones of the form F + for any presheaf F , the functor (−)+ is left adjoint to the inclusion GGA (−)+ : Pre(C) DG G ⊥ G Pre(C)+ : i. Moreover, by applying twice the functor (−)+ , since F ++ is a sheaf for any presheaf F , i.e., Pre(C)++ ∼ = Sh(C), we get the desired adjunction GGA (−)++ : Pre(C) DG G ⊥ G Sh(C) : i. The sheafification functor will usually be denoted either by (−)++ , or by a, or by L2 . Given X : J GGA Pre(C) a J -diagram of presheaves such that X(j) is in fact a sheaf for each j ∈ J , its colimit colim X(j) ∈ Pre(C) does not necessarily give a sheaf. This is due to the fact that colimits do not need to commute with limits (in particular, the equalizer that defines sheaves). As pointed out before, the right notion of a colimit in the functor category Sh(C) is the object-wise one, i.e., the colimit computed in the category of

28


presheaves. Moreover, since the sheafification is left adjoint to the inclusion and in particular preserves colimits, we can safely push the limit back in the category of sheaves ++

colim X(j)

∼ colim X(j)++ ∼ = = colim X(j) ∈ Sh(C),

where the first two colimits are computed in the category of presheaves. As a summary, the limit of a diagram of sheaves computed in the category of presheaves is again a sheaf, since the inclusion i : Sh(C) ,GGA Pre(C) preserves limits. Therefore, limits of sheaves seen as presheaves are already sheaves and are simply computed object-wise. This property does not hold for colimits, and colimits of sheaves seen as presheaves are not necessarily sheaves. Using the adjunction GGA (−)++ : Pre(C) DG G ⊥ G Sh(C) : i, we compute object-wise the colimit in the larger category [C op , Set] of presheaves, and push the limit back in Sh(C) by using the right adjointness of the sheafification functor. Given a complete and cocomplete category M, there is a more general notion of presheaves and sheaves on C with values in M, which are functors in the category [C op , M]. If M is complete and cocomplete, the same results as for presheaves of sets hold. In particular : • For any small category C, the category of presheaves [C op , M] is complete and cocomplete and limits and colimits are computed object-wise. • If in addition C is a Grothendieck site, there is a similar notion of a sheaf. A presheaf F : C op GGA M is a sheaf if and only if, for every covering {Uα GGA U }α in the topology of C, the induced diagram Q

F (X)

F (iγ )

Q Y

F (iαβ ) Y

F (Uα ) Q

α

F (iβα )

F (Uα ×X Uβ )

α,β

is an equalizer in M. Denote the subcategory of sheaves by Sh(M) ⊆ Pre(M). • The inclusion functor i : Sh(M) ,GGA Pre(M) is part of an adjunction GGA L2 : Pre(M) DG G ⊥ G Sh(M) : i. • Limits of sheaves are computed object-wise and colimits of sheaves are given by the sheafification of the colimit of the underlying presheaves (sheafification of the objectwise colimit). As a general idea, all the structure of the category of values M is (object-wise) inherited by the category of presheaves [C op , M], and some of it by the subcategory of sheaves Sh(M) by using the adjunction GGA L2 : Pre(M) DG G ⊥ G Sh(M) : i. 29


For example, the abelian structure of an abelian category M is (object-wise) inherited by the category presheaves Pre(M), and after sheafification by the subcategory of sheaves Sh(M). Moreover, if M has enough injectives, then so does Pre(M), but not necessarily Sh(M). In this project, we will be interested in presheaves with values in simplicial sets, that is, categories of functors [C op , sSet]. The homotopy theory of simplicial sets will give a homotopy theory on [C op , sSet] and most of the work in this project is to find such a suitable homotopy theory.

1.4

Simplicial and Cosimplicial Objects

Given a small category C, the category [C op , sSet] of presheaves on C with values in simplicial sets sSet is called the category of simplicial presheaves (on C). A functor in this category F : C op GGA sSet = [∆op , Set] can also be seen as a bifunctor on C op and ∆op F : C op × ∆op GGA Set, i.e., a presheaf on the product C × ∆, or as a functor F : ∆op GGA [C op , Set], which is a simplicial object in the category of presheaves [C op , Set]. This section treats with the very basic definition of simplicial and cosimplicial objects in arbitrary categories, which are very useful in homotopy theory. As we will see, the category sSet of simplicial sets plays a very important role in homotopy theory, in some sense, it plays a similar role than does the category Set of sets in the theory of (locally small) categories. In fact, by adding a simplicial dimension to the objects in M, a homotopy theory of these objects naturally arises.

1.4.1

(Co)simplicial objects and (co)skeletons

Definition (Simplicial and cosimplicial objects). A simplicial object in a category M is a functor ∆op GGA M, and a cosimplicial object in a category M is a functor ∆ GGA M. For example, a simplicial object in the category Set of sets is a simplicial set. Alternatively, as in the category of simplicial sets, a simplicial object in a category M can be given as a sequence of objects M0 , M1 , M2 , . . . with face maps di : Mn GGA Mn−1 and degeneracy maps sj : Mn GGA Mn+1 , satisfying the simplicial identities. In a general category M, say complete, an example of a simplicial object is the Cech complex of an object X 30


···

X ×X ×X

X ×X

X,

where we did not draw the degeneracies for visual reasons. More generally, any morphism X GGA Y ∈ M defines a Cech complex ···

X ×Y X ×Y X

X ×Y X

X,

which is a simplicial object in M. Denote the n-truncated simplicial category 0

1

.. .

···

2

n,

by ∆n . This is the full subcategory of the simplicial category ∆, with objects 0, 1, . . . , n. op induces by precomposition a functor The inclusion functor ∆op n ,GGA ∆ i∗ = trn : M∆

op

op

GGGGA M∆n ,

called the n-truncation. As its name indicates, this functor sends a simplicial object M• to its n-truncated part M0 , M1 , . . . , Mn with the same faces and degeneracies. Let’s now assume that M is complete and cocomplete. Since ∆op is small, left and right Kan extensions give left and right adjoints to the precomposition by i functor i∗ = trn : M∆

op

op

GGGGA M∆n .

The left adjoint given by left Kan extension is called the n-skeleton ∆op fn : M∆n GGA sk : trn , DG G ⊥G M op

and the right adjoint, defined by right Kan extension is called the n-coskeleton trn : M∆

op

∆op gn n GGA : cosk DG G ⊥G M

op is full subcategory, the unit In addition, since ∆op n ⊆∆ ∼ =

op

M• GGA trn ◦ skn M• and the counit

∼ =

∈ M∆n

trn ◦ coskn M• GGA M•

op

∈ M∆n

are isomorphims (see Corollary 4 in Section X.3 in [Mac71]). By abuse of language, the two composites fn ◦ trn : M∆ skn = sk

op

GGA M∆

op

and

g n ◦ trn : M∆ coskn = cosk

op

op

GGA M∆ ,

are also called the n-skeleton and the n-coskeleton, and we will be careful to specify which functor we consider, when needed. Therefore, for an n-truncated simplicial object X• its skeleton and coskeleton are defined by fn X• sk

and 31

g n X• , cosk


while for a simplicial object X• , its skeleton and coskeleton are defined as the skeleton and coskeleton of its n-truncated part. In particular, the n-skeleton and n-coskeleton do not see the information contained outside of the n-truncation. Moreover, for a simplicial object op op X• ∈ M∆ and an n-truncated simplicial object Y• ∈ M∆n , the adjunctions skn a trn a coskn give two natural bjections op op op op M∆ (skn (Y• ), X• ) ∼ = M∆n (Y• , trn (X• )) and M∆n (trn (X• ), Y• ) ∼ = M∆ (X• , coskn (Y• )). op

In particular, given an n-truncated simplicial object Y• ∈ M∆n there are two universal ways in how to extend it to a simplicial object : • the skeleton skn Y• is the way such that any map from it to a simplicial object is determined by what it does on the n-truncation; • the coskeleton coskn Y• is the way such that any map into it from a simplicial object is determined by what it does on the n-truncation.

1.4.2

Augmented simplicial objects

An augmented simplicial object is defined as a simplicial object with an extra object M−1 in the −1th dimension with a unique morphism M0 GGA M−1 . As a concrete example, an augmented simplicial set is a simplicial set K• with an extra set K−1 of −1-simplices such that each 0-simplex is associated to a unique −1-simplex. This extra set K−1 could for example represent a set of colours, and the face map d : K0 GGA K−1 associates a colour to each vertex. Consider the augmented simplex category ∆+ that is ∆op with an initial object −1, i.e., it is defined as −1

0

1

2

.. .

··· ,

where all possible compositions −1 GGA m give the same morphism. Definition (Augmented simplicial object). An augmented simplicial object in M is a functor M• : ∆op + GGA M. As for simplicial objects, an augmented simplicial object in M can be seen as a sequence of objects M−1 , M0 , M1 , . . . ∈ M with faces and degeneracies that satisfy some simplicial identities. A third and more useful way to interpret an augmented simplicial object is the following. Let N be an object in M, and write rN for the constant simplicial object with rNn = N for every n ≥ 0 and with only identity maps as faces and degeneracies. For any simplicial object M• , a morphism of simplicial objects M• GGA rN (that is, a natural transformation) is the same as a morphism Mm GGA N ∈ M for any fixed m, since rN has only identities as structure maps. In particular, such a morphism M• GGA rN defines an augmentation map M0 GGA N and this is equivalent to an augmented simplicial object M−1 = N, M0 , M1 , . . .. 32


In many cases, augmented simplicial objects are augmented by a morphism arising from a colimit. Indeed, a simplicial object M• : ∆op GGA M can be interpreted as a ∆op -diagram in M and the morphisms Mn GGA colim M• define an augmentation, i.e., M• GGA r colim M• is an augmented simplicial object. Similarly, the natural morphism M• GGA hocolim M• is also an augmented simplicial object. The language of augmented simplicial objects gives a formalism in which a morphism M• GGA rN is naturally seen as an object, and more op precisely, an object of the category of functors M∆+ . The notion of a morphism between augmented simplicial objects is given by natural transformations of functors, and thus augmented simplicial objects form a category. All constructions with simplicial objects can be extended to augmented simplicial objects. In parop ticular, the embeddings ∆op n ,GGA ∆+ also give n-truncations, n-skeletons and n-coskeletons by Kan extensions, i.e., there are adjunctions ∆ fn : M∆n GGA sk DG G ⊥ G M + : trn op

op

and

op ∆op g n. n GGA trn : M∆+ DG : cosk G ⊥G M

We will later use this language to see hypercovers as objects themselves, rather than a morphism from a simplicial object to a constant simplicial object.

33

2. Additional Structures on Model Categories The language of model categories, introduced by Quillen, is an efficient machinery for doing homotopy theory. In fact, a model category is equipped with more than enough structure for homotopy theory, that is, it contains more data than only a class of weak equivalences. The extra data turns out to be very useful, because it gives a presentation of its homotopy category, which, at first sight, only looks like a localization. However, the axioms for a model category are not too restrictive, since many categories admits (at least) such a model. For example, the category Top of topological spaces, the category sSet of simplicial sets, the category ChainR of chain complexes of R-modules, categories of diagrams with target a model category, . . . , all admit model structures. There are many written introductions to model categories, the handbook of Dwyer and Spalinski [DS95], the monograph of Hovey [Hov99] or the one of Hirschhorn [Hir03] are all good references. In this first section we present the definition of a model category that will be used throughout this report. We will give all the necessary definitions as well as their first properties in order to understand and be able to use model categories. The crucial notions of a cofibrantly generated model category, as well as a cellular and a combinatorial model category are explored. The second section studies simplicial model categories, which are model categories enriched over sSet, in which the simplicial mapping space between any two objects has a homotopy theoretic content. In the third section we develop the notion of localization of model categories, as explained by Hirschhorn in [Hir03]. This notion is crucial for the next chapter, where it will be used more than once.

2.1 2.1.1

Model Categories A few categorical prerequisites

Let C be a category. We need first introduce a few definitions that will be used in the category of arrows Arr(C) = C 2 . Definition (Retract). An object A ∈ Ob(C) is said to be a retract of B ∈ Ob(C) if there id

exists a factorization of A GGA A through B 34

CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES

id A

B

A. p

i

A morphism A GGA B is itself said to be a retract, if there is a retraction B GGA A id A

i

B

p

A.

Moreover, a class of objects W ⊆ Ob(C) is said to be closed by retracts or closed under retracts, if given any object B ∈ W and any retract A ∈ C of B, then A is also in W . i

If A GGA B is a retract, it immediately follows that A is a retract of B. These terms come from what happens in topology, where an inclusion A ,GGA X is a retract if there f

exists a retraction X GGA A A. Similarly, a morphism A GGA B is said to be a retract of g X GGA Y if it is so in the category of arrows Arr(C), i.e., if there exists a commutative diagram in C id A

X

A

g

f B

f Y

B.

id If the objects of C have an underlying set, and if a morphism A GGA B ∈ Arr(C) has an underlying set-theoretic function, if A is a retract of B id A

i

B

p

A,

then i is necessarily injective and p is necessarily surjective. For example, in the category Set of sets, the retraction are exactly the surjective maps1 . In the category R Mod of R-modules, M ,GGA N is a retract if and only if there is another R-module L such that M∼ = N ⊕ L. There are several useful immediate results2 such as the fact that the retract of a monomorphism, epimorphism, isomorphism, is again respectively a monomorphism, epimorphism or 1 2

This statement is equivalent to the axiom of choice. See for example Section 7 in [AHS06].

35


isomorphism; retractions are closed under composition; any retraction is an epimorphism; any functor preserves retract(ion)s; any fully faithful functor reflects retract(ion)s, . . . . Another property that arrows can have in the category Arr(C) is a left/right lifting property with respect to another arrow. f

Definition (Left/Right lifting property). We say that an arrow A GGA B has the left lifting g

property with respect to X GGA Y , or equivalently, that g has the right lifting property with respect to f , if every commutative square in C A

X g

f

B

Y

admits a diagonal filler that makes the two triangles commute. Note that isomorphisms have both the left and right lifting property with respect to f

any other morphism. Moreover, if A GGA B has the left (resp. right) lifting property with g

respect to X GGA Y , then any retract of f has the left (resp. right) lifting property with respect to g. The last property that we need in order to define model categories, is the 2-out-of-3 property of a class of arrows. Definition (2-out-of-3 property). A class of morphisms W ⊆ Arr(C) is said to have the 2-out-of-3 property in C, if for any two composable morphisms f and g, if two out of the three morphisms f, g and f ◦ g are in W , then so is the third. In particular, such a class W is closed under composition. The standard example of such a class of morphisms is the class of isomorphisms in any category, because they have an inverse. In fact, the 2-out-of-3 property is meant to be a weakening of the property of being an isomorphism. The class of morphisms that will have this property will behave similarly to isomorphisms, without necessarily having an inverse morphism. We have now all the ingredients to define a model category.

2.1.2

The definition of a model category and examples

Definition (Model category). A model category is a category M with three classes of morphisms C , F , W ⊆ Mor(M) called the cofibrations, the fibrations and the weak equivalences satisfying the following axioms (CM1) The underlying category M is complete and cocomplete; (CM2) The class of weak equivalences W has the 2-out-of-3-property; (CM3) The three classes C , F and W are closed under retracts; (CM4) Any commutative square 36


A

X H

i

p

B

Y

where i ∈ C is a cofibration, p ∈ F is a fibration, and either i or p is a weak equivalence, admits a lifting H; f

(CM5) Any morphism A GGA B admits two functorial factorizations f

A

B p

i

f

A

q

j

C

B,

C

where i ∈ C ∩ W is a cofibration and a weak equivalence and p ∈ F is a fibration, and where j ∈ C is a cofibrations and q ∈ F ∩ W is a fibration and a weak equivalence. In addition, a morphism in C ∩ W is called an acyclic cofibration and a morphism in F ∩W is called an acyclic fibration. Moreover, as already done in the definition, a cofibration will often be decorated as X GA Y , a fibration as X GGA A Y and a weak equivalence as ∼ X GGA Y . Since there is usually no confusion, we will use the symbol M both for the underlying category and for the model category, i.e., the category M with the extra data of cofibrations C , fibrations F and weak equivalences W . When we need to be precise, we will refer to the model category as the quadruple (M, C , F , W ). Note that since the category is complete and cocomplete, there is a initial object, denoted by ∅, and a terminal object, denoted by ∗. An object X is called cofibrant if the unique morphism ∅ GA X is a cofibration, fibrant if the unique morphism X GGA A ∗ is a fibration, and bifibrant if it is both cofibrant and fibrant. The theory of model categories would be much easier if all objects would be bifibrant. In fact, these objects allow extensions and liftings of morphisms, in a similar manner than projective modules allow liftings of epimorphisms and injective modules allow extensions of monomorphisms. For example, if ∼ A is cofibrant and X GGA A Y is an acyclic fibration, axiom (M4) gives a diagonal filler !

∅ !

H

X ∼

A

Y,

37


that is exactly the same as a lifting of the morphism A GGA Y X H

∼

A

Y.

Dually, if X is fibrant, any morphism A GGA X admits an extension along any acyclic ∼ cofibration A GA B A

X.

∼

H

B These axioms of a model category are slightly stronger than the original definition of Quillen in [Qui67]. First, Quillen only asks for finite limits and finite colimits to exist in the underlying category, instead of requiring all (small) limits and colimits. However, most categories that admit interesting model structures are complete and cocomplete3 . Second, Quillen did not ask for the functoriality of the two factorizations. However, it is rather hard to come up with a model category that does not admit functorial factorizations, see for example the Remark 4.10 in [Isa04]. To explain what the functoriality is, the best approach is to see the category of morphisms of M as the functor category M2 . There is a composition functor f

g

g ◦f

M3 GGA M2 : A → B → C G [ GA A → C. Observe that choosing a factorization for each map f ∈ Mor(M) = M2 by, say an acyclic cofibration if followed by a fibration pf , such as f

Xf if

Yf pf

∼ Af ,

is the same as giving an association if

f

pf

M2 GGA M3 : Xf → Yf G [ GA Xf → Af → Yf which is a section of the composition functor and where if ∈ C ∩ W and pf ∈ F . Asking for a functorial factorization is asking that this is in fact a functor (and similarly for the 3

The category of bounded chain complexes is an example of a category that is only finitely complete and cocomplete.

38


other factorization as a cofibration followed by an acyclic fibration). This means that given a morphism (g1 , g2 ) : f GGA f 0 in M2 f

Xf

Yf

g1

g2

Xf0

f0

Yf0 ,

the choice of the terms Af and A0f is functorial, and there are commutative squares Xf

if

Af

g1 Xf0

pf

Yf

g3

g2

A0f

i0f

Yf0 .

p0f

This interpretation of a functorial factorization is taken from the excellent article [Gar09], where this is explained and motivated in much greater depth. Moreover, there is an equivalent, more compact, definition of a model category defined with the help of weak factorization systems. Definition (Weak factorization system). A weak factorization system4 on a category C is a pair of classes of maps (L, R) that are closed under retracts, and satisfying the two axioms that • each morphism f ∈ Mor(C) factorizes as a map from L followed by a map from R; • every pair of morphisms (l, r) ∈ L × R has the lifting property, i.e., any commutative square in C admits a filler as in the diagram A l

X ∃

B

r Y.

Equivalently to our definition, a model structure on a bicomplete category M is a class W of weak equivalences that has the 2-out-of-3 property, together with two weak factorization systems (C ∩ W , F ) and (C , F ∩ W ). Weak factorizations systems, as well as other types of factorization systems are treated in this same article [Gar09]. An important further observation is that in a model category, two of the three classes C , F and W completely determine the third. Indeed, it follows from the definition that all 4

The word weak is here to remind us that this factorization is not necessarily unique.

39


fibrations have the right lifting property with respect to acyclic cofibrations. In fact, the fibrations turn out to be exactly the morphisms that have the right lifting property with f

respect to all acyclic cofibrations. Suppose that a morphism X GGA Y has the right lifting property with respect to all acyclic cofibrations, and choose a factorization f

X i

Y p

∼ Z.

Since i is an acyclic cofibration, the following square has a diagonal filler id

X

X

∃g

f

i Z

p

Y

and then it follows that f is a retract of the fibration p, and so is itself a fibration. A similar argument shows that the cofibrations are exactly the morphisms that have the left lifting property with respect to all acyclic fibrations. For the last case, if we know all the cofibrations and all the fibrations, this argument shows that we know all the acyclic cofibrations and all the acyclic fibrations. By the 2-out-of-3 property, a morphism X GGA Y is a weak equivalence if and only if it can be written as a composite of an acylic cofibration and an acyclic fibration. The fact that two classes determines the third relies on the fact that we are given two weak factorization systems. In a weak factorization system (L, R), one of the two classes L or R determines the other. Moreover, the fact that the cofibrations are exactly the morphisms that have the (left) lifting property with respect to all acyclic fibrations, implies that the class of cofibrations are closed under composition. Similarly, the class of acyclic cofibrations, fibrations and acyclic fibrations are also closed under composition. More precisely, C , F and W are subcategories of Mor(M), containing all the objects as a domain or codomain of an arrow, and containing all isomorphisms. Given a model category M with cofibrations C , fibrations F and weak equivalences W , there is a canonical model category on the opposite underlying category Mop where the cofibrations are F op , the fibrations are C op and the weak equivalences are W op . Example 2.1 (Example of model categories). (1) Any complete and cocomplete category M, admits two (not really interesting) model structures. These are obtained by setting the weak equivalences and one of C or F to be all the morphisms in Mor(M), while the last one contains exactly all the isomorphisms. The axioms are easily checked. 40


(2) Given a commutative ring R with unit, the category ChainR,≥ of positive chain complexes of modules over R (positively graded differential R-modules) admits two model structures. Both model have as weak equivalence the quasi-isomorphisms, i.e., the maps inducing isomorphisms in homology at every level. In the projective version, the fibrations are the degree-wise epimorphisms while the cofibrations are the degree-wise monomorphisms with projective cokernel, and in the injective version, the cofibrations are the degree-wise monomorphisms while the fibrations are the degree-wise epimorphisms with injective kernel. The extra condition for the cofibrations in the projective version is required so that they have the left lifting property with respect to the acyclic fibrations (degree-wise epimorphisms that are quasi-isomorphisms), and similarly for the extra condition for the fibrations in the injective model. Since the chain with only the 0 R-module is both initial and terminal, every object is cofibrant in the injective model, and every object is fibrant in the projective model. (3) There are two well-known model structures on the category Top of all topological spaces, which, in a sense to be defined below, give the same homotopy theory. The first is due to Quillen, and has as weak equivalences the weak homotopy equivalences, which are the maps inducing isomorphisms on all homotopy groups, for any choice of basepoint. The (Serre) fibrations are the maps satisfying the right lifting property with respect to the inclusions Dn ∼ = Dn × {0} ,GGA Dn × I, and the cofibrations are, roughly speaking the maps that can be built out of the natural inclusions Sn ,GGA Dn+1 . See the Section 2.4 in [Hov99] for a complete proof that these definitions give a model structure on Top. In this model, every object is fibrant. (4) The second model structure on Top is the Hurewicz model (or the Strom model), first proved by Strom in [Str72]. The weak equivalences are in this model only the homotopy equivalences, i.e., the maps having an inverse up to homotopy. There a fewer fibrations since the fibrations are the Hurewicz fibrations, i.e., the maps having the lifting property with respect to all inclusions X ∼ = X × {0} ,GGA X × I for every topological space X. The cofibrations are the closed Hurewicz cofibrations. This model has the nice property that every space is both cofibrant and fibrant. (5) The standard model structure on the category sSet of simplicial sets, gives again the same homotopy theory as topological spaces. As a consequence, it is sometimes more convenient to work with simplicial sets instead of topological spaces, while interested in properties up to homotopy. The weak equivalences are the maps X· GGA Y· that are weak homotopy equivalences in Top after the geometric realization. The cofibrations are exactly the monomorphisms, i.e., the maps that are injective in each degree. The fibrations are the Kan fibrations, which are the maps satisfying the right lifting property with respect to all horn inclusions Λkn ,GGA ∆[n]. Every object is trivially cofibrant, and the fibrant objects are called the Kan complexes.

2.1.3

The construction of the homotopy category

A very convenient aspect of model categories is that they give a presentation of their homotopy categories. More precisely, the associated homotopy category is constructed as a certain localization of the model category. 41


Definition (Homotopy category). Given a model category M, its homotopy category, if it exists, is the localization M[W −1 ] at the class of weak equivalences, usually denoted by Ho(M). L

The localization is more precisely a functor M GGA Ho(M), where every weak equivalence in M becomes an isomorphism in the homotopy category Ho(M), and which satisfies F

the universal property that for any other such functor M GGA A, there is a filler, unique up to unique isomorphism in the diagram M L

F

A.

∃! θ

Ho(M) Observe that the homotopy category does only depend on the weak equivalences. Therefore, any two model structures with same weak equivalences on the same category M, will have isomorphic homotopy categories. The extra structure of a model category, given by the cofibrations C and the fibrations F , allows an explicit construction of the homotopy category of any model category. We will briefly sketch this construction, and refer to Section 5 in [DS95] or Section 1.2 in [Hov99] for more details. The idea is to introduce a homotopy equivalence relation, and to quotient the class of morphisms so that the maps that have inverses up to homotopy will become isomorphisms. The next step is to show that if we restrict our attention to bifibrant objects, the weak equivalences of M are exactly the maps that have an inverse up to homotopy, and so the work is done for bifibrant objects. Finally, we extend this construction to all objects by using a ’homotopy-invariant’ functor from the model category to its full subcategory of bifibrant objects. Mimicking the construction of homotopies from Top, the first ingredient is to define some sort of cylinder object that plays the role of X × I for a fixed object X. Note that in Top, there are two injections i0 X∼ = X × {0} ,GGA X × I

and

i1 X∼ = X × {1} ,GGA X × I,

that play an important role in the definition of a homotopy. Moreover, the projection ∼ X × I GGA X is a weak equivalence, since I is contractible. Definition (Cylinder object). In a model category, a cylinder object for an object X is a factorization of the codiagonal map X

`

X

codiag

X, ∼

Cyl X

42

CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES ∼

where we only ask for Cyl X GGA X to be a weak equivalence. Such a cylinder is called ` good if X X GA Cyl X is a cofibration, and very good if in addition Cyl X GGA A X is a i0

i1

fibration. There are two natural composite denoted by X GGA Cyl X and by X GGA Cyl X. Very good cylinders always exist by factoring the codiagonal map X X GGA X into a cofibration followed by an acyclic fibration. In the Quillen model structure on Top, the object X × I is in general only a cylinder object, while it is at least a good cylinder object in the Hurewicz model. In the model structure on simplicial sets, the object X × ∆[1] is also at least a good cylinder object. Such an object allows the notion of a left homotopy between two morphisms to be defined. `

GGA Y are left Definition (Left homotopy). In a model category, two morphisms f, g : X GGA homotopic, denoted by f ∼l g if there exists a filler in the diagram i0

X

Cyl X

i1

X,

∃ g

f Y for some cylinder object of the source X.

For any pair of objects X, Y ∈ M, this is a reflexive and symmetric relation on M(X, Y ). It need not be transitive, but it is if in addition X is cofibrant. There is a dual notion to this notion of left homotopy. Definition (Cocylinder object). In a model category, a cocylinder object or a path object for an object X is a factorization of the diagonal map diag

X

X × X,

∼ Cocyl X ∼

where we only ask for X GGA Cocyl X to be a weak equivalence. Such a cocylinder is called good if Cocyl X GGA A X × X is a fibration, and very good if in addition X GA Cocyl X ev0 is a cofibration. There are two natural composites denoted by Cocyl X GGA X and by ev1 Cocyl X GGA X. Similarly very good cocylinders always exist by factoring the map X GGA X X into an acyclic cofibration followed by a fibration. In the category of topological spaces, the standard path object of X is the space Top(I, X) with the compact-open topology, and ev0 and ev1 are the evaluation at 0 and 1. The notion of a right homotopy is the dual definition of a left homotopy. Q

43

CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES GGA Y are Definition (Right homotopy). In a model category, two morphisms f, g : X GGA right homotopic, denoted by f ∼r g if there exists a filler in the diagram Y

ev0

Cocyl Y

ev1

Y

∃ g

f X, for some cocylinder object of the target Y .

Similarly, this is a reflexive and symmetric relation, which is also transitive if Y is fibrant. Therefore, if X is cofibrant and Y is fibrant, both ∼l and ∼r are equivalence relations on M(X, Y ). In the category of topological spaces, the exponential adjunction5 Top(X × I, Y ) ∼ = Top(X, Top(I, Y )), and the fact that X × I is a cylinder and Top(I, Y ) is a cocylinder, implies that the notions of left and right homotopy agree on Top. Moreover if X is cofibrant and Y is fibrant, the two equivalence relations agree ∼l =∼r on Top(X, Y ). This is the homotopy equivalence relations, denoted by ∼. Given two objects bifibrant X and Y , denote by [M](X, Y ) the quotient set of morphisms M(X, Y ) /∼ . Moreover, for any three objects X,Y and Z that are cofibrant and fibrant, the composition on the quotient set of maps is a well-defined function [M](X, Y ) × [M](Y, Z) GGA [M](X, Z) : ([f ], [g]) G [ GA [g ◦ f ]. Therefore, this gives a method to construct the localization of the full subcategory of bifibrant objects, with respect to the class of homotopy equivalences between bifibrant objects. However, the homotopy category is the localization (of the whole category) at the class of weak equivalences. In fact, between bifibrant objects, weak equivalences and homotopy equivalences are the same notion. Lemma 2.1.1. Let X and Y be bifibrant objects in a model category M. A morphism X GGA Y is a weak equivalence if and only if it is a homotopy equivalence. Proof. This is Lemma 4.24 in [DS95]. Therefore, between bifibrant objects, the hom-set in the homotopy category may look like [M](X, Y ). The last idea is based on the fact that any object in the model category may be ’replaced’ by a weakly equivalent bifibrant object. This is done by applying both a cofibrant replacement and a fibrant replacement. Definition ((Co)fibrant replacement). Let X ∈ M be an object in a model category. A cofibrant replacement of X is a factorization of the morphism ∅ GGA X as 5

Which applies since I is a locally-compact and Hausdorff topological space.

44


∅

X. ∼ QX

Similarly, a fibrant replacement of X is a factorization of X GGA ∗ as ∗.

X ∼ RX

The point is to replace X up to a weak equivalence, by a cofibrant object QX or a fibrant object RX. Let’s fix for any object X a cofibrant replacement QX and fibrant replacement RX. For convenience, if X is cofibrant let QX = X, and if X is fibrant let RX = X. Then Q extends to a functor from M to the full . subcategory of cofibrant objects where the set of 0 morphisms is the quotient set M(C, C ) ∼l . In fact, the lifting ∅

QY ∴∃

QX

∼ Y

X is uniquely defined up to left and right homotopy equivalence. Similarly, R extends to a functor from M to the appropriate target category. The important result is the fact that RQ and QR are functors from M to the full subcategory of bifibrant objects, where the set of morphisms between two bifibrant objects X and Y is the quotient set [M](X, Y ). Theorem 2.1.2. Let M be a model category. Its homotopy category Ho(M) always exists, and is given by • Objects : the same as M; • Morphisms : the quotient sets Ho(M)(X, Y ) := M(QRX, QRY ) /∼ ; γ

where the localization functor M GGA Ho(M) is the identity on objects, and defined as in the paragraph before the theorem on morphisms. Moreover, a morphism f is a weak equivalence in M if and only if its image γ(f ) is an isomorphism in the homotopy category Ho(M). 45


Since the homotopy category is a localization of the model category at the class of weak equivalences, Ho(M) only depends on the class of weak equivalences W . As a consequence, different model structures on a category M may give isomorphic homotopy categories, as a long as they share the same weak equivalences. Having different model structures giving the same homotopy category is often useful, as we will see in Chapter 3. A first reason when changing the model structure is useful, is for having different classes of cofibrant and fibrant objects, since these objects are good in that they allow extending and lifting morphisms. It is rare that there exists a model in which every object is cofibrant and fibrant, as it happens in Top.

2.1.4

Functors between model categories

Having introduced the homotopy category associated to a model category M, a first question that arises is to ask when a functor F : M GGA N induces a functor from the homotopy category Ho(M) GGA N . In some sense, this is equivalent to asking when a functor F is compatible with the internal homotopy relation of M. Note that this does not depend on whether or not N is a model category. Definition (Left and right derived functor). Let M be a model category, N any category and F : M GGA N a functor. A left derived functor of F is a functor LF : Ho(M) GGA N with a natural transformation σ : LF ◦ γ =⇒ F , that is universal among such pairs. That is, the triangle F

γ

N

=⇒

M

LF Ho(M) commute up to the natural transformation σ, and for any other such pair (LF 0 , σ 0 ), there is α a unique natural transformation LF 0 =⇒ LF such that σ ◦ (α ? idγ ) = σ 0 . Dually, a right derived functor of F is a functor RF : Ho(M) GGA N with a natural transformation σ : F =⇒ RF ◦ γ, that is universal among such pairs. That is, the triangle F

M

=⇒

γ

N

RF Ho(M) commute up to the natural transformation σ, and for any other such pair (RF 0 , σ 0 ), there α is a unique natural transformation RF =⇒ RF 0 such that (α ? idγ ) ◦ σ = σ 0 .

46


If a functor F from a model category M sends all weak equivalences to isomorphisms, the universal property of the localized category Ho(M) says that there is a unique filler M

F

N,

γ LF = RF Ho(M) and the triangle strictly commutes (not only up to a natural transformation). A weaker, and very often used, condition for which a functor F : M GGA N admits a left derived functor is when F sends acyclic cofibrations between cofibrant objects to isomorphisms. Indeed, if F satisfies this property a short argument shows that F sends all weak equivalences between cofibrant objects to isomorphisms (this is Ken Brown’s Lemma). In this case, a possible derived functor is the composite LF = F ◦ Q for some cofibrant replacement Q. Indeed, any change of cofibrant replacement Q is only seen up to isomorphism in N , thanks to the fact that F sends all weak equivalences between cofibrant objects to isomorphisms in N . A dual condition holds for a right derived functor of F . If in addition N is also endowed with a model structure, we can now be interested in when F : M GGA N induces a functor between the homotopy categories. A similar condition as in the previous paragraph may now be weakened, because we do not need a ’complete’ lift Ho(M) GGA N , but only a lift in the homotopy category Ho(M) GGA Ho(N ). Definition (Total left/right derived functor). A total left derived functor of F : M GGA N between model categories, is a functor between the homotopy categories LF : Ho(M) GGA Ho(N ) that is left derived functor of the composite M GGA N GGA Ho(N ). Similarly, a total right derived functor of F is RF : Ho(M) GGA Ho(N ) that is a right derived functor of the same composite. However, we are often interested in functor with more structure than just inducing a functor on (from) the homotopy category. In order to be able to compare two homotopy categories M and N , we would like an adjunction between them. GGA Definition (Quillen adjunction). An adjunction F : M DG G ⊥ G N : G between two model categories is called a Quillen adjunction or a Quillen pair if in addition • either the left adjoint F sends cofibrations to cofibrations and acyclic cofibrations to acyclic cofibrations; • or the right adjoint G sends fibrations to fibrations and acyclic fibrations to acyclic fibrations. The functor F is called a left Quillen functor and G a right Quillen functor. 47


Using the characterization of cofibrations and fibrations, the two conditions are clearly equivalent. More precisely, the fact that F preserves cofibrations is the same as that G preserves acyclic fibrations, and the fact that F preserves acyclic cofibrations is the same as that G preserves fibrations. Therefore, it is also equivalent to asking one of the two mixed conditions • F sends cofibrations to cofibrations and G sends fibrations to fibrations; • F sends acyclic cofibrations to acyclic cofibrations and G sends acyclic fibrations to acyclic fibrations. By the preceding paragraph, one of these conditions is enough to ensure that F admits a total left derived functor LF : Ho(M) GGA Ho(N ) and that G admits a total right derived functor RG : Ho(N ) GGA Ho(M). More importantly, the two induced functors form an adjunction of the homotopy categories. GGA Proposition 2.1.3. Let F : M DG G ⊥ G N : G be a Quillen adjunction. Then, the total left derived functor of F and the total right derived functor of G form an adjunction on the homotopy categories GGA LF : Ho(M) DG G ⊥ G Ho(N ) : RG. Proof. See Lemma 1.3.10 in [Hov99]. Some Quillen adjunctions not only induce an adjunction between the homotopy categories, but an equivalence of the homotopy categories. In fact, the characterization of these special Quillen adjunctions is easy to describe. GGA Definition (Quillen equivalence). A Quillen adjunction F : M DG G ⊥ G N : G is called a Quillen equivalence, if for all cofibrant objects M ∈ M and fibrant objects N ∈ N , the natural bijection N (F M, N ) ∼ = M(M, GN ) restricts to a natural bijection between the sets of weak equivalences ∼ w. e. (M(M, GN )) . w. e. (N (F M, N )) = ∼

That is to say that a morphism F M GGA N ∈ N is a weak equivalence if and only if its ∼ adjoint morphism M GGA GN ∈ M is a weak equivalence. This turns out to be a necessary and sufficient condition for having an induced equivalence of the homotopy categories. GGA Theorem 2.1.4. Let F : M DG G ⊥ G N : G be a Quillen adjunction. This is a Quillen equivalence if and only if the induced adjunction on the homotopy categories GGA LF : Ho(M) DG G ⊥ G Ho(N ) : RG is an equivalence of categories. Proof. The proof can be found as Proposition 1.3.13 in [Hov99].

48


Consider the realization-singular adjunction between topological spaces and simplicial sets GGA Re : sSet DG G ⊥ G Top : Sing. We will see later that this is a Quillen pair6 . Moreover, both the unit and the counit ∼

X• GGA Sing ◦ ReX•

and

∼

Re ◦ SingX GGA X

are weak equivalences. Since they become isomorphisms at the level of the homotopy categories, the realization-singular pair is a Quillen equivalence.

2.1.5

Cofibrantely generated model categories and the small object argument

Given 3 classes of maps in a bicomplete category M, that are closed by retracts and by composition and one of them having the 2-out-of-3-property, it is never easy to check that this corresponds to a model structure on M. In this situation, the axioms to be checked are the lifting property and the existence of the two (functorial) factorizations, that is, showing the existence of two weak factorizations systems. Maybe we should emphasize the fact that the class of weak equivalences is the most important of the three classes. Indeed, it is clear that the homotopy category, which is the localization M[W −1 ], only depends on the class of the weak equivalences. Therefore, the class of weak equivalences should be the first class to be determined, in order to endow a category with a model structure. Afterwards, there is a balance to be found between the cofibrations and the fibrations. By the lifting properties, more cofibrations implies fewer fibrations, and more fibrations implies fewer cofibrations. Furthermore, there should always remain enough of both cofibrations and fibrations, in order to find functorial factorizations. The small object argument is a generic tool that provides weak factorization systems, given as input only a set of morphisms where the domain of each morphism is ’not too big’. This requirement of objects being small enough is very important, since it is one of the only problem that may occur. This machinery outputs two classes of maps I−cell and I−inj such that any morphism of M can be factored by a map from I−cell followed by a map from I−inj. The only missing property of (I−cell, I−inj) for being a weak factorization system is that I−cell is not necessarily closed by retracts. Therefore, if we call I−cof the closure by retracts of I−cell, the couple (I−cof, I−inj) is a weak factorization system, that is, there is the functorial factorization required and every couple (i, p) ∈ (I−cof, I−inj) satisfies the lifting property. This will be seen as one of the two factorization systems of a model category. However, this is only half of the model structure, since a model category is defined with the two weak factorization systems (C , F ∩ W ) and (C ∩ W , F ). Since these two weak factorization systems are certainly not independent, we could not just give as input two sets of maps I and J and hope that the output gives a model structure on M. For example, a relation between them may simply be the fact that every acyclic cofibration (i.e., an element 6

We will see that the realization functor preserves generating acyclic cofibrations and generating cofibrations, and therefore, preserves acyclic cofibrations and cofibrations.

49


of J −cof) is in particular a cofibration (i.e., an element of I−cof). The recognition theorem will give a sufficient condition on the sets I and J in order to have an induced model structure. We will now give some more details and make precise this discussion. A complete development can be found in Chapter 10 and 11 in [Hir03]. The functorial factorization of a morphism X GGA Y will eventually be given by successively factoring it trough bigger and bigger objects Zα for some (infinite) indexing α, until such an object Zβ is big enough so that Zβ GGA Y has the right lifting property with respect to all desired maps. We need first define such infinite compositions, and what is such a notion of ’size’ for objects in a category. Definition (λ-sequence, transfinite composition). Let λ be an ordinal, seen as a poset category. A λ-sequence in a category M is a functor X : λ GGA M, i.e., a λ-diagram X0 GGA X1 GGA · · · GGA Xβ GGA · · ·

∈M ∼ =

satisfying the property that the natural maps colimβ0 X∈Sm/S

.

From Proposition 3.2.2, this model admits the universal property that any functor Sm/S GGA M into a model category M factorizes (uniquely) through MS,proj Sm/S

r = yoneda

MS,proj

=⇒ Re a Sing

γ

M, ∼

where, the natural transformation Re ◦ r =⇒ γ is a natural weak equivalence and the GGA adjunction Re : MS,proj DG G ⊥ G M : Sing is a Quillen adjunction. In particular, in both model, cofibrations are object-wise monomorphisms and fibrations are object-wise Kan fibrations. By definition of these models, the identity adjunction GGA id : MS,proj DG G ⊥ G MS,inj : id 122

CHAPTER 3. MOTIVIC HOMOTOPY THEORY

is a Quillen adjunction. Unfortunately, these two models are not very interesting model structures for at least two reasons. First, they do not take account of the colimits already present in Sm/S, since the embedding Sm/S ,GGA MS of homotopy cocompletion does not preserve homotopy colimits. Second, some colimits are ’geometrically wrong’ in the initial category of schemes and should be corrected in the motivic category. Such an example, coming from Example 2.1.1 in [Dug], is the following. Consider the diagram of affine schemes over a field k A1 − {0}

A1

A1 where the embeddings are z G [ GA z and z G [ GA 1/z. By the usual contravariant equivalence between algebra and geometry, the pushout of schemes corresponds to the pullback of kalgebras k[z]

k[z −1 ]

k[z, z −1 ],

which is just the ground field k. The above pushout of affine lines is therefore the point scheme Spec k. However, by considering the affine line to be something like a real line, geometric intuition would prefer this pushout to be a projective line. The problem may be formulated as that the colimit of the underlying topological spaces is not the underlying topological space of the colimit. This issue is fixed by localizing the global model structure on MS to recover exactly the ’geometric colimits’ we want to have in our category. The resulting model structure is a local model structure and there are different ways to achieve it, as explained in the previous chapter in section 3.3. More precisely, we will adopt here the method by hypercompletion, because it’s the method that explains the best the main idea of recovering our colimits. The colimits to be recovered will be the ones encoded in the Grothendieck topology that Sm/S will be endowed with.

3.4.3

The category of Nisnevich motivic spaces

There are many choices for topologies on schemes, see for example [Sta, Chapter : Topologies on Schemes] for a bunch of them. The two most well-known are the Zariski topology and the étale topology. Roughly speaking, the Zariski topology is the one in which the open covers are given by the scheme-theoretic open immersions (which are also jointly surjective). This is the analogue of the small site top(X) of open subsets of X, for a topological space X. 123


Unfortunately, this topology is not well-suited for A1 -homotopy theory and gives unexpected and unwanted results, in particular because it does not contain enough covers (asking for an open immersion is too much, there are not enough open subsets). A coarser topology with more open covers is the étale topology, where the open covers are defined by the étale morphisms (that are jointly surjective. The étale topology has already proved its utility throughout the étale cohomology of schemes. However, this topology has too many open covers and again can give some unexpected results. For example, a field which is non separably algebraically closed may have a non-trivial étale cohomology, see for example Remark 17.9 in [Mil80]. We will now define the completely decomposed version of the étale topology, the Nisnevich topology, which is in some sense what we get by forcing fields to be acyclic. The Nisnevich topology, which is strictly finer that the Zariski topology and strictly coarser that the étale topology, seem to enjoy many good properties of both topologies, while avoiding some of their defects. We refer to the introduction of chapter 3 in [MV99] for more profound reasons about this choice of topology in A1 -homotopy theory. In what follows, we will denote a scheme (X, OX ) by X, the stalk at a point x ∈ X by OX,x and its residue field by k(x). f

Definition (Completely Decomposed Morphism). A morphism of schemes U GGA X is said to be completely decomposed at x ∈ X if there exists u ∈ U such that f (u) = x and such that the residual field extension k(x) GGA k(u) is an isomorphism. Note that such a point u ∈ U corresponds to a unique filler in the diagram U f Spec k(x)

∗ 7→ x

X.

Moreover, in the situation of a pullback square V f˜ Y

y g

U f X, f˜

f

if U GGA X is completely decomposed at some x = g(y), then its pullback V GGA Y is also completely decomposed at y. Indeed, a filler Spec k(x) GGA U in

124


V

U

y

∴∃ Spec k(y)

∃ Spec k(x)

f˜

f

Y

X

g

induces the required filler Spec k(y) GGA V by the universal property of the pullback. A covering in the Nisnevich topology will be a (surjective) family of completely decomposed étale morphisms. fi

Definition (Nisnevich Coverings). A finite set of morphisms {Ui GGA S} in Sm/S is called a Nisnevich covering if each fi is étale (of finite type) and if for each point x ∈ S there fi0

exists an index i0 such that Ui0 GGA S is completely decomposed at x. Since a Zariski covering {Ui ,GGA S} only contains open embeddings Ui ,GGA S and any point in Ui has the same residual field as its image in S, Zariski covers are Nisnevich covers. Moreover, any Nisnevich cover is in particular an étale cover (that is in addition completely decomposed). To show that this generates a Grothendieck topology on Sm/S, observe first that clearly ∼ =

an isomorphism of schemes {S 0 GGA S} is Nisnevich covering. Moreover, since the property of being an étale morphism is closed under pullbacks along any other morphism and the pullback of a completely decomposed morphism is again a completely decomposed morphism by the argument above, the Nisnevich covers are closed by change of basis. Finally, suppose gij

fi

that {Ui GGA S}i is a Nisnevich covering, and that {Vij GGA Ui }j are Nisnevich coverings for all i. We need to see that {Vij GGA Ui GGA S}i,j gives a Nisnevich covering. First, the composition of étale morphisms of finite type is again an étale morphism of finite type. Also, given any point x ∈ S, there is a u ∈ Ui0 such that fi0 (u) = x and such that ∼ =

k(x) GGA k(u) is an isomorphism. Similarly, there is a v ∈ Vi0 j0 such that gi0 j0 (v) = u ∼ =

and again k(u) GGA k(v) is an isomorphism. Therefore, this same v ∈ Vi0 j0 satisfies ∼ =

∼ =

fi0 ◦ gi0 j0 (v) = x and k(x) GGA k(u) GGA k(v), therefore {Vij GGA Ui GGA S}i,j is a Nisnevich covering. We may now invoke the machinery of section 3.3 to get local model structures on MS . Both the projective and the injective global structures may be used, so we can always use whichever suits best the situation. Recall that the local model is the left Bousfield localization of the global model, with respect either to local weak equivalences, or only to the hypercovers from the site (which are acyclic fibrations). The cofibrations in the local model will be the same as in the global model, and the weak equivalences are the local weak equivalences17 . The fibrations are identified, by Corollary 3.3.11, with the simplicial presheaves Which are the ones inducing isomorphisms on all sheaves of homotopy groups, or, since the topos MS has enough points, the ones inducing isomorphisms on all stalks. 17

125


F that are fibrant in the global model, and which admit descent for all hypercovers (or any class of hypercovers which contains a dense set). More precisely, the local injective model MS,loc.inj has • cofibrations : the monomorphisms of simplicial presheaves; • weak equivalences : the local weak equivalences; • fibrations : the morphisms having the right lifting property with respect to (local) acyclic cofibrations. This model still has the nice property that every object is cofibrant, while fibrant objects are the objects that were fibrant in the global injective model and that satisfy descent for all hypercovers (Corollary 3.3.11), i.e., the fibrant objects of the global injective model that are (Nisnevich) sheaves up to homotopy. The local projective model MS,loc.proj has • cofibrations : the same as in MS,proj ; • weak equivalences : the local weak equivalences; • fibrations : the morphisms having the right lifting property with respect to (local) acyclic cofibrations. Since the cofibrations are the same, the representables rX are still cofibrant, while fibrant objects are the objects that were fibrant in the global projective model and that satisfy descent for all hypercovers (Corollary 3.3.11). Thas is, the fibrant objects are the objectwise Kan complexes that are (Nisnevich) sheaves up to homotopy. Since for any smooth scheme X ∈ Sm/S, its embedding rX ∈ MS,loc.proj is a sheaf in the étale topology, then it also is in the coarser Nisnevich topology. Therefore the local projective model has the advantage that, for any scheme X, its image rX ∈ MS,loc.proj is both cofibrant and fibrant. Moreover, both local models are left proper, simplicial and cellular since the left Bousfield localization preserves these properties (Theorem 2.3.1). In fact, they are also right proper, see for example Lemma 1.7 in [Bla01] for the projective version.

3.5

Unstable Motivic Homotopy Theory

We now have two local model structures on the category of motivic spaces MS , where the local models reflect some aspects of the ’geometry’ of the initial category Sm/S of smooth schemes of finite type over S. Indeed, the localization process from the global model structures to the local model structures may be seen as a way of correcting how colimits are computed in MS to give them a geometric meaning, as illustrated in the previous section 3.4. However, is this localization enough to give an interesting homotopy theory for schemes ? Thanks to the property that any representable rX is cofibrant and fibrant in the local projective model structure on MS , the answer is that the category of motivic spaces does not help us identify schemes. Proposition 3.5.1. The Yoneda embedding from the category of smooth schemes Sm/S to the homotopy category of motivic spaces Ho(MS ), is fully faithful. 126


Proof. The idea is to use the local projective model, and the functor that embeds schemes in the homotopy category of simplical presheaves is the composite r

Sm/S ,GGA MS,loc.proj GGA Ho(MS,loc.proj ). Since in this model, for any schemes X, Y ∈ Sm/S, their images rX, rY ∈ MS,loc.proj is both cofibrant and fibrant, the hom-set in the homotopy category is the quotient of the hom-set in the model category by the homotopy relation Ho(MS )(rX, rY ) = MS,loc.proj (rX, rY )/ ∼ . Moreover, since ∆[0] ,GGA ∆[1] is an acyclic cofibration of simplicial sets, then rX ⊗ ∆[0] ∼ = rX GA rX ⊗ ∆[1] is a cofibration of simplicial presheaves. Therefore, by the 2-out-of-3 property the unique morphism id ⊗! ∼ rX ∼ rX ⊗ ∆[1] GGGA rX ⊗ ∗ = = rX ⊗ ∆[0]

is also a weak equivalence. Moreover, the cofibration ∂∆[1] ,GGA ∆[1] induces a cofibration rX

a

∼ rX ⊗ ∂∆[1] GA rX ⊗ ∆[1], rX =

and thus rX ⊗ ∆[1] is a cylinder object for rX rX

`

rX

rX ∼ rX ⊗ ∆[1].

By Corollary 1.2.6 in [Hov99], since rX is cofibrant and rY is fibrant, for any homotopic GGA rY , there is a homotopy through any cylinder object, so in particmaps f ∼ g : rX GGA ular through rX ⊗ ∆[1]. However, since rX and rY are discrete, there are only constant homotopies and thus f ∼ g =⇒ f = g. This shows that Sm/S embeds fully faithfully in Ho(MS ) by the chain of isomorphisms Sm/S(X, Y ) ∼ = MS (rX, rY ) ∼ = MS,loc.proj (rX, rY )/ ∼ = Ho(MS,loc.proj )(rX, rY ).

The local model structure on motivic spaces MS are therefore no enough for studying the category of schemes Sm/S. To try to adjust these model structures, we can apply the same construction to Grothendieck sites that are more geometric than Sm/S, and see what goes out of it. There are two important examples, which give respectively a homotopy theory of topological spaces and a homotopy theory of real manifolds, and which lead both to the same conclusion : there still is a missing ingredient. These examples are the Example 5.6 in [Dug01c] for the topological spaces, and the example of real manifolds is given in Section 8 of the same article. Given a small category C, recall that U C is the homotopy theory built from C, as defined in section 3.2. Observe 127


that starting with C = ∗, tautologically U C gives sSet, the homotopy theory of simplicial sets. However, simplicial sets are usually seen as ’built out of ∆’, so a more interesting example would be by starting with the category of simplices ∆. Proposition 3.2.2 gives a Quillen adjunction in the diagram r

∆ n

7→

∆n

U∆

a

Top, which commutes only up to a natural transformation. It turns out that the adjoint pair GGA U ∆ DG G ⊥ G Top is not exactly a Quillen equivalence, the only missing ingredient is that there is no information in U ∆ which says that the representables ∆[n] are contractible. Indeed, if we denote the set A = {∆[n] GGA ∗}, the Quillen pair descends to a Quillen equivalence of model categories GGA U ∆/A DG G ⊥ G Top. Another interesting example which gives a similar conclusion is by starting with the category GGA ManR of real manifolds. Similarly, there is a Quillen pair U ManR DG G ⊥ G Top, which becomes a Quillen equivalence of model categories, after localizing with respect to the maps π A = {R × M GGA M } GGA U ManR /A DG G ⊥ G Top. The moral of the story is that the this universal construction needs to know something about the interval ∆[1], or the real line R being contractible. A similar process is applied in the category of motivic spaces, where an interesting homotopy theory arises where the role of the interval or the real line is played by the affine line A1 . In some sense one could interpret the situation as the following. Endowing the category of motivic spaces with a local model structure is just a way to set up a good category that reflects the geometry of schemes. Identifying A1 as an interval and localizing to make it behave like an interval is a way to set up a place in which one can in addition do homotopy theory of schemes. In the two previous examples, the fact that the interval was contractible automatically ∼ forces I × F GGA F into weak equivalences. A direct approach in the motivic setting would be to apply a (left) Bousfield localization with respect to all those morphisms rA1 × F GGA F . However, this is in general not a proper class and therefore existential problem for this localization may occur. A first step would be to declare the morphism rA1 GGA ∗, which can be identified with the morphism rA1 × ∗ GGA ∗, to be a weak equivalence, and then see which maps are forced to become weak equivalences in the localized category. Recall that what is denoted here id

by ∗ is the representable simplicial presheaf associated to the object S GGA S ∈ Sm/S, the terminal object of the category of schemes. When we consider Sm/k, the category of 128


smooth schemes of finite type over a field k, this terminal object is the identity between the scheme with one point Spec k = {∗}. Observe first that any morphism ∗ GGA A1 becomes a weak equivalence in the localized category by the two out of three property of the diagram ∼

rA1

∗

∗.

id As a consequence, any morphism of schemes ∗ GGA A1 ∈ Sm/S will turn into a weak ∼ equivalence ∗ GGA rA1 in the category of motivic spaces MS . Note that when the initial category of schemes is Sm/k, such a map Spec k GGA A1 corresponds exactly to a k-rational point of the affine line A1 . Moreover, for any fibrant simplicial presheaf F ∈ MS , one also get that the canonical projection ∼

rA1 × F GGA F has to be a weak equivalence. Indeed, since the local models are right proper, the pullback rA1 × F

F

y rA1

fib. ∼

∗

∼

implies that the projection A1 × F GGA F has to be a weak equivalence too. More generally, if rA1 is fibrant in some local model structure on MS , and if F is a fibrant replacement functor in this model, for any simplicial presheaf G ∈ MS , consider the following glueing of pullback squares rA1 × G

G

y

∼

rA1 × F(G)

F(G)

y fib. rA1

∼ fib.

∼

∗.

The morphism rA1 × F(G) GGA F(G) is a weak equivalence by right properness, and also ∼ a fibration since rA1 GGA ∗ is. Therefore, the vertical morphism rA1 × G GGA rA1 × F(G) is also a weak equivalence, again by right properness, and the upper square becomes

129


rA1 × G ∼

G

y

∼ ∼

rA1 × F(G)

F(G). ∼

By the two out of three property, the canonical projection rA1 ×G GGA G is then also forced to be a weak equivalence. We can now use the local projective model structure MS,loc.proj in which the representable rA1 is fibrant. This implies that all the canonical projections rA1 × F GGA F would become weak equivalences if we localize the local projective model structure with respect to a map ∗ GGA rA1 . Definition (Motivic unstable model category). Given a local model structure on MS , the left Bousfield localization at the morphism rA1 GGA ∗ gives an unstable motivic model 1 1 category, denoted here by MSA , or just by M A when there is no confusion about the base scheme. Since the injective and the projective local model structures share the same weak equivalences, the canonical projections rA1 × F GGA F are weak equivalences in any of the unstable motivic category. By formally applying a left Bousfield localization, we proved the following theorem. Theorem 3.5.2. The left Bousfield localization of the local projective or local injective model structure on the category of motivic spaces MS at the canonical projections rA1 × F GGA F 1 gives the unstable motivic model structure MSA . Therefore, the cofibrations in the unstable motivic model are the same as in the local models, which are the same as in the global model. In particular, the representable functors rX, for any scheme X ∈ Sm/S, are cofibrant in both models, and any motivic space 1 F ∈ MSA is cofibrant in the injective model. The weak equivalences in this model are called A1 -weak equivalences, and are given by the Bousfield localization. Recall that for two motivic spaces F, G ∈ MS , there is a notion of simplicial mapping space given by Map(F, G) : n G [ GA MS (F ⊗ ∆[n], G). Call S = {rA1 ×F GGA F } the class of natural projections, where we can use either notation rA1 × F = A1 ⊗ F , and let’s work in the local projective model MS,loc.proj . A fibrant motivic space G ∈ MS,loc.proj is called A1 -local (or S-local) if for any such projection, the induced map ∼ Map(F, G) GGGA Map(F × rA1 , G) ∈ sSet is a weak equivalence of simplicial sets. The A1 -local objects are exactly the fibrant objects A1 in the Bousfield localization MS,loc.proj . Moreover, a morphism of motivic space G GGA H is called an A1 -weak equivalence, if for every A1 -local object F , the induced map ∼

Map(H, F ) GGGA Map(G, F ) 130

∈ sSet


is a weak equivalence of simplicial sets. By the general machinery, the A1 -weak equivalences are exactly the weak equivalences in the Bousfield localization.

131

Index λ-sequence, 50 I-cell complex, 52 ˘ Cech complex, 110 acyclic cofibration, 37 acyclic fibration, 37 associativity coherence, 12 associator, 11 augmentation map, 32 augmented simplex category, 32 augmented simplicial object, 32, 113 bifibrant, 37 closed symmetric monoidal category, 13 cocylinder object, 43 cofibrant, 37 cofibrant replacement, 44 cofibrantly generated model category, 54 cofibration, 36 combinatorial model category, 62, 63 compact object, 59 completely decomposed, 124 copower, 19 cosimplicial object, 30 cosimplicial resolution, 95 coskeleton, 31, 112 cotensor, 18 cotensored category, 18 covering family, 21 covering sieve, 21 cylinder object, 42 derived functor, 46 descent condition, 118 effective monomorphism, 60

enriched category, 11, 16 fibrant, 37 fibrant replacement, 45 fibration, 36 flasque model structure, 89 generalized cover, 107 geometric realization, 10 global injective model structure, 81 global projective model structure, 81 Grothendieck pretopology, 21 Grothendieck topology, 21 homotopy category, 42 homotopy function complex, 71 hypercompletion, 104 hypercover, 107 indiscrete topology, 26 internal-hom, 13–15 left Bousfield localization, 73 left derived functor, 46 left homotopy, 43 left Kan extension, 9 left lifting property, 36 left localization, 72 left proper, 63 left Quillen functor, 47 local acyclic fibration, 106 local epimorphism, 107 local equivalence, 73 local fibration, 105 local injective model structure, 103 local lifting, 105

132

INDEX

local model structure, 101 local object, 73 local weak equivalence, 102 locally fibrant, 106 locally finitely presentable, 120 locally finitely presentable category, 62 locally presentable category, 62 mapping space, 65 model category, 36 monoidal category, 11 motivic space, 120 Nisnevich covering, 125 Nisnevich topology, 124 power, 19 presheaf, 24, 29 proper, 63

small object argument, 49 small presentation, 99 small site, 21 symmetric monoidal category, 12 symmetry coherence, 12 tensor, 18 tensor product, 11 tensored category, 18 total left derived functor, 47 total right derived functor, 47 transfinite composition, 50 truncation functor, 31, 112 unit coherence, 12 unstable motivic model category, 130 weak equivalence, 36 weak factorization system, 39

Quillen adjunction, 47 Quillen equivalence, 48 Quillen pair, 47 recognition theorem, 55 relative I-cell complex, 52 retract, 34 retraction, 35 right derived functor, 46 right homotopy, 44 right Kan extension, 9 right lifting property, 36 right localization, 72 right proper, 63 right Quillen functor, 47 sheaf, 25 sheafification functor, 27 simplicial category, 65 simplicial object, 30 simplicial presheaf, 19, 67 singular functor, 10 site, 21 skeleton, 31, 112 small object, 50 133

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