A Brisk Jaunt To Motivic Homotopy

Transcription

1 A Brisk Jaunt To Motivic Homotopy Kyle Ferendo April 2015 The invariants employed by algebraic topology to study topological spaces arise from certain qualities of the category of topological spaces. Homotopy theory seeks to abstract these qualities so that mathematicians can recognize other categories which lend themselves to study via the tools of algebraic topology, and also, perhaps less serendipitously but more often, engineer variations of existing categories to which the tools of algebraic topology may be applied. Motivic homotopy theory is an example of the latter endeavor: to elucidate invariants of schemes (e.g. Chow groups) by articulating the homotopy-theoretic context from which they arise. The purpose of this exposition is to build upon the fundamentals of algebraic topology and of category theory in order to introduce some of the techniques of homotopy theory and to convey some first applications of homotopy theory to algebraic geometry. A principal setting for homotopy theory is a model category, which is a category together with some extra structure satisfying five particular axioms. After introducing model categories and exploring some of their properties, we discuss simplicial sets. Simplicial sets have the same homotopy theory as topological spaces in a sense we will eventually make precise but are much more manageable in several respects, and appear in certain applications of homotopy theory, including motivic homotopy theory. We will emphasize the relationship among the model category structure of simplicial sets, spectra, and cohomology theories. Finally, we discuss motivic homotopy theory itself, focusing on motivic cohomology. By the time we reach this point, we hope for many of the constructions to be echoes of familiar motifs. The exposition and development throughout anticipates the recapitulation of the themes of homotopy theory in the section on motivic homotopy theory. This write-up surveys much material. Whenever feasible, we provide explicit constructions of the machinery at hand, but we often refer to some of the sources listed below for thorough proofs that our constructions actually have the desired properties. It is worth clarifying what knowledge we assume beforehand. This includes, essentially for the sake of motivation, point-set topology. We also assume familiarity with basic algebraic topology, including homotopy groups and the nature of (co)homology. We present the connection between cohomology theories and spectra, with particular attention to singular cohomology, but this exposition will be difficult to follow without prior exposure. We also assume familiarity with basic category theory, but devote some energy to introducing 1

2 several more advanced topics. In particular, we assume familiarity with some of the simplest concepts most likely to appear in other fields, such as duality, limits, adjunctions, (symmetric) monoidal and closed monoidal categories, functor categories, the Yoneda embedding, and the definition of enriched categories and functors. We do not assume familiarity with somewhat less widespread notions, such as ends and copowers. A note on notation: we denote terminal objects with the symbol, initial objects with 0. When the two coincide (in pointed categories), we use. When we present a directed graph and call it a category, what we really mean is the free category generated by this graph. In commutative diagrams, we ll indicate fibrations with two-headed arrows:, cofibrations with arrows with tails:, and weak equivalences with arrows labeled with tildes:. For C a (possibly enriched) category and objects C, D in C, we denote the hom-set/object from c to d with C(c, d). When C is a closed monoidal category we use an underline instead: C(c, d). We notate adjoint functors (L R) : C D to indicate that L : C D is a left adjoint with right adjoint R : D D. Lastly, before beginning the exposition proper, I thank Professor Michael Ching, who introduced me to the fascinating field of homotopy theory and whose remarkable generosity with his time and his energy has allowed me to pursue the interest that he kindled. I am also grateful to the authors of [Dun+07], Motivic Homotopy Theory B. I. Dundas, M. Levine, P.A. Østvær, O. Röndigs, and V. Voevodsky whose book functioned as a very useful skeleton for this course (particularly Dundas s portion, which, despite his intention, served as a challenging and engaging introduction to homotopy theory). Other useful references have included [Hat02], Allen Hatcher s Algebraic Topology, particularly for homology and an introduction to topological spectra; [Dug01], Dan Dugger s paper Universal Homotopy Theories for clarifications concerning methods of generating and localizing model categories; [Kel05], G. M. Kelly s Basic Concepts of Enriched Category Theory for some results which unify, generalize, and elucidate several constructions from simplicial sets; [Rie11a], Emily Riehl s A Leisurely Introduction to Simplicial Sets, which itself contributed more enriched category theory than may have been expected; and [Hov99], Mark Hovey s Model Categories, when more technical detail regarding model categories was needed. 1 Model Categories We begin our jaunt by laying out some of the technical definitions of model categories. Although we will not provide a rigorous explanation of the uses of these definitions until the following subsection, we will give an informal explanation of the motivations for the definitions as we lay them out. Definition 1.1. A model category is a category C equipped with the following extra structure: three subcategories (the objects of which are the objects of C) called cofibrations, weak equivalences, and fibrations, as well as two 2

3 functors α, β : C D1 C D2 where D 1 = and D 2 = and satisfying the following axioms: 1. C is complete and cocomplete. 2. The 2-out-of-3 axiom: for any two composable morphisms f and g, if gf is a weak equivalence and either f or g is as well, then all three are weak equivalences. Because weak equivalences are a subcategory, we already know that if f and g are weak equivalences, then gf is as well. 3. The retract of a cofibration, weak equivalence, or fibration is also a cofibration, weak equivalence, or fibration, respectively. If the reader is more familiar with retracts of objects than retracts of morphisms, this is a retract of objects in the functor category C D1. A (co)fibration is called acyclic if it is also a weak equivalence. A morphism p has the right lifting property with respect to a morphism i (and i has the left lifting property with respect to p) if, for any commuting outer square of the following form, there exists a lift d which makes the following diagram commute: i d 4. All fibrations have the right lifting property with respect to all acyclic cofibrations, and all cofibrations have the left lifting property with respect to all acyclic fibrations. 5. The final axiom tells us that we can always factor any morphism into an acyclic cofibration followed by a fibration or a cofibration followed by an acyclic fibration. Here is a picture: p A α 1(f) f C B α 2(f) A β 1(f) f C B β 2(f) There s a little bit more structure we want these factorizations to have: we would like the factorizations to be functorial. This means first of all that if we have a commutative square (i.e. a morphism in C D1 ) A f B D g E 3

4 then applying either factorization to both arrows on opposite sides of the square induces an arrow in the same direction between the intermediate objects in the factorization in the following commutative diagram: α 1(f) f A C B α 1(g) α 2(f) D F E g α 2(g) and moreover, the factorization, being functorial, respects composition in C D1. Lastly, the fact which all of our diagrams have indicated is true: that α sends morphisms to an acyclic cofibration followed by a fibration, and β sends morphisms to a cofibration followed by an acyclic fibration. Two more bits of terminology: an object X of a model category category C is called fibrant if X is a fibration and cofibrant if 0 X is a cofibration. Note that the properties of fibrations and cofibrations are dual (as the names suggest), and that the properties of weak equivalences are self-dual. Thus, whenever we prove a theorem about (acyclic) fibrations, we will also have a theorem about (acyclic) cofibrations. Some of the motivation for these definitions can be explained already. The weak equivalences will model topological weak equivalences (the continuous functions which induce isomorphisms of homotopy groups). The 2-out-of-3 axiom makes sense in this light, since the composition of an isomorphism with something that is not an isomorphism can never yield an isomorphism. And the right lifting property of fibrations begins to align with the topological definition of a (Serre) fibration when we note that the inclusion X {0} X I is a weak equivalence and, when X is an n-dimensional disk, also an example of what we find we should call a cofibration. Although the necessity of some details of model categories may remain murky at the moment, the idea is to articulate a setting in which we will be able to build a new homotopy category from an existing category in which objects which were only weakly equivalent become isomorphic and homotopic morphisms are identified. But if we take too naïve an approach to this task, certain pathologies may arise (i.e. the resulting category may not be locally small). By insisting on the correct additional structure, abstracted from properties of categories in which we already understand what homotopy means, we can avoid these pathologies. Before continuing to the next definition, which will introduce the correct notion of a morphism of model categories, let us familiarize ourselves with the axioms (and also discover their usefulness!) by proving a couple of propositions, beginning with the fact that the converse of the lifting axiom is true. 4

5 Lemma 1.2. A morphism p is a fibration if (and only if, according to the lifting axiom) it has the right lifting property with respect to every acyclic cofibration. A morphism is an acyclic fibration (if and) only if it has the right lifting property with respect to every cofibration. Proof. Let p be a morphism that has the right lifting property with respect to every acyclic cofibration. We use α to give a factorization of p (a commutative triangle) and expand this triangle into a commutative square along the identity arrow for the domain of p. Then because α 1 (p) is an acyclic cofibration, we use our hypothesis on p to produce a lift. This is the first diagram below; the second diagram is equivalent to the first, but has been rearranged to indicate that p really is a retract of α 2 (p). Because α 2 (p) is a fibration, the retract axiom tells us that p is a fibration as well. A α 1(p) A p α 1(p) A B A p α 2(p) p B α 2(p) C C C C The second half of the proof is dual, and uses β instead of α. This is a crucial lemma which is ubiquitous in the following. And just as we take for granted that theorems will yield dual theorems, we will also assume that we can exchange theorems about fibrations for theorems about acyclic fibrations by exchanging acyclic cofibrations for cofibrations in our proofs. Lemma 1.3. Given a pushout square A C f B i i f B A C in which i is a cofibration, i is also a cofibration. Proof. For p an arbitrary acyclic fibration, consider the following diagram: i f A B D C B A C E f i d d We need to show that i has the left lifting property with respect to p; that is, that there is a lift in the diagonal of the square on the right. Because i is a cofibration, it has the left lifting property with respect to p, so we have a lift d through the rectangle defined by i, p, gf, and hf. But then we have g h p 5

6 a morphism gf = di which factors both through f and i, so by the universal property of pushouts, this morphism also factors through B A C, and the resulting morphism, d, gives us the required lift in the square on the right. We next address the question of morphisms between model categories. Definition 1.4. A left Quillen functor is a functor between model categories which sends cofibrations to cofibrations and acyclic cofibrations to acyclic cofibrations. A right Quillen functor is dual. A Quillen pair is an adjunction between model categories in which the left adjoint, called a left Quillen functor, sends cofibrations to cofibrations and acyclic cofibrations to acyclic cofibrations, and in which the right adjoint, called right Quillen functor, has the dual property. Proposition 1.5. The right adjoint of a left Quillen functor is a right Quillen functor. Hence, an adjunction in which the left adjoint is a left Quillen functor is automatically a Quillen pair. Proof. Let F G : C D be a Quillen pair with F a left Quillen functor. Recall that this means that the following diagram commutes up to natural isomorphism (not necessarily the identity) C op D id G C op C F op id Hom C D op D Set Hom D and denote this natural isomorphism φ : D(F (A), B) = C(A, G(B)) for all objects (A, B) of C op D. We would like to show that for any fibration of D p : D 1 D 2, G(p) has the right lifting property with respect to all acyclic cofibrations in C and is therefore also a fibration. To do this, we use the natural isomorphism between hom-functors φ to find a corresponding lift in D and to move this lift back into C. Let i : C 1 C 2 be an arbitrary acyclic cofibration in C and consider the following diagrams: a F (C 1 ) D 1 φ(a) C 1 G(D 1 ) F (i) d p i φ(d) G(p) F (C 2 ) D 2 b φ(b) C 2 G(D 2 ) We can see that p has the right lifting property with respect to F (i) (which is an acyclic cofibration because F is a left Quillen functor) and the diagram on the left (in D) induces the diagram on the right (in C, and commutative because φ is a natural isomorphism), which gives us the lifts we need to confirm that G(p) is a fibration. 6

7 Lemma 1.6. Left Quillen functors send weak equivalences between cofibrant objects to weak equivalences. This is known as the Ken Brown Lemma. Proof. Given a weak equivalence of cofibrant objects f : A B in the domain of a left Quillen functor F : C D, we construct the diagram below. We deduce some properties of the arrows appearing therein before passing to the codomain of F. A i 1 β1(f id) β2(f id) A B C B B i 2 Denote g = β 1 (f id) and h = β 2 (f id). Because the coproduct inclusions i 1 and i 2 are equivalently the pushout over 0, Lemma 1.3 tells us that these inclusions are cofibrations. Note that hgi 1 = f is a weak equivalence by hypothesis and h is a weak equivalence by the factorization axiom, so the 2-out-of-3 axiom tells us that gi 1 is a weak equivalence. Similarly, gi 2 is a weak equivalence because all identity arrows are weak equivalences. Being compositions of cofibrations, gi 1 and gi 2 are both acyclic cofibrations, so F sends them both to acyclic cofibrations, and in particular, weak equivalences. Because F (id B ) = F (h)f (gi 2 ) is also weak equivalence, the 2-out-of-3 axiom tells us that F (h) is a weak equivalence. But F (f) = F (h)f (gi 1 ) and the composition of weak equivalences is a weak equivalence, so F (f) is a weak equivalence. Now, a very brief comment on the homotopy part of homotopy theory. Definition 1.7. The homotopy category HoC of a model category C is a category satisfying the property that any functor from C sending all weak equivalences to isomorphisms factors uniquely up to natural transformation through HoC. If model categories were only defined to be categories with a distinguished subcategory of weak equivalences, a homotopy category may fail to exist. Because of the constraints we place on model categories, it is true that every model category has a homotopy category. We will not prove this fact. Morphisms in the homotopy category can be described as zig-zags: normal compositions of morphisms in C as well as compositions in which a weak equivalence is composed after another morphism when its codomain is equal to the codomain of the preceding morphism. Thus, weak equivalences are given formal inverses (subject to the relations already existing in the category and the obvious requirement that the composition of a weak equivalence and its inverse be the identity). There are some useful technical constructions of the homotopy category of a model category which we will not elaborate here. However, we will comment on the fact that the homotopy category of the full subcategory of fibrant and cofibrant objects is equivalent to the normal homotopy category, as we can observe by using α to factor morphisms to the terminal object through a fibrant object weakly equivalent to the object with which we began, and then using a 7

8 similar process involving β. As a consequence of Ken Brown s lemma, Quillen pairs induce adjunctions between the homotopy categories of a model category (the functors in this adunction of homotopy category are called the total derived functors of the Quillen pair); for more details, see subsection in [Hov99]. When the derived functors of a Quillen pair are an equivalence of homotopy categories, the Quillen pair is called a Quillen equivalence, and the model categories it connects are said to be Quillen equivalent. Note that just as topological spaces may have isomorphic homotopy groups but be homotopy inequivalent, model categories may have equivalent homotopy categories but be Quillen inequivalent, and also that the Quillen equivalence of model categories is in fact the equivalence relation generated by the relation of Quillen pairs inducing equivalences, so two model categories may be Quillen equivalent even if there is no Quillen equivalence between them (but because both categories are Quillen equivalent to a third). When two categories are Quillen equivalent, they are understood to be alternative presentations for the same homotopy theory. 2 Simplicial Sets (and some useful category theory) In simplest possible terms, simplicial sets are important because there is a model category of simplicial sets which is Quillen equivalent to the model category of topological spaces, but which possesses much nicer category-theoretic properties, so that we do not need to trouble ourselves with such tiresome properties as e.g. compactly generated spaces in order for the category to be Cartesian closed. On a more profound level, both simplicial sets and topological spaces are of fundamental importance because they model the (, 1)-category of -groupoids, over which every (, 1)-category (i.e. homotopy theory) is homotopically enriched. This is a fact which will prove somewhat relevant to our penultimate section, in which delve a bit more deeply into the theory of model categories. Definition 2.1. Let (sometimes called the simplex category) denote the category of (non-zero) finite ordinals (i.e. the skeleton of the category of finite totally ordered non-empty sets) with morphisms order-preserving functions. A simplicial set is a functor op Set and a morphism of simplicial sets is a natural transformation, so we have the category of simplicial sets S = Set op. We can elucidate the structure further. Denote by [n] the ordered set {0,..., n} (note that [n] has cardinality n + 1). Observe that is generated by the following morphisms: { d i : [n 1] [n] d i j (j) = j + 1 { s i : [n + 1] [n] s i j (j) = j 1 j < i j i j i j > i 8

9 so that any morphism φ : [m] [n] with {i 1 < < i k } = [n] φ([m]) and {j 1 < < j l } = {j [m] φ(j) = φ(j + 1)}. Then we have the unique factorization φ = d ik d i1 s j1 s j l. Of course, this factorization and its uniqueness arise from the existence of certain relations on the generating morphisms, but the factorization is a more compelling fact than the relations, so we leave the interested reader to compute the relations. Within the structure of a simplicial set, there is a geometric significance to these generating morphisms. For a simplicial set X use the notation X([n]) = X n and d i = X(d i ) : X n+1 X n and similarly for s i. Then the elements of X n are called n-simplices, the d i face maps, and the s i degeneracy maps. Given an n-simplex a, d i (a) identifies the n 1-simplex which is the i-th face of a. The s i are called degeneracy maps because s i (a) identifies the degenerate n + 1-simplex with vertex j of s i (a) equal to vertex j of a for j i and vertex j of s i (a) equal to vertex j 1 of a for j > i. Thus vertex i of a is repeated in s i (a). The elements of X n not in the image of any s i are called non-degenerate n-simplices, and a simplicial set is generated by its non-degenerate simplices and the face maps between them. The category of pointed simplicial sets is S = Set op = /S where this last expression indicates an undercategory. This means, a priori, that pointed simplicial sets are pointed in each dimension. But because the structure morphisms between dimensions must respect these basepoints, and because there is only one morphism X 0 X n, corresponding to the unique morphism [n] [0], we can see that the basepoint in each dimension n 1 is a degenerate simplex determined completely by the choice of basepoint in X 0. We d now like to discuss several properties of the category of (pointed) simplicial sets in particular, the relationship with (pointed) topological spaces and the closed monoidal structure. Both of these relationships are expressed in terms of adjunctions, and so we develop some general (and quite powerful) machinery in describing adjunctions from such categories which will prove useful later as well. We begin with a few definitions which may lie beyond the scope of the most elementary category theory which one encounters in other fields. These include ends, freely enriched categories, and copowers. Definitions 2.2. Although ends have a somewhat more general definition in terms of extranatural transformations, for our purposes, it suffices to say that the end of a functor F : C op C D is a limit of the form, when it exists: lim f morc f:a b F (a, a) F (b, b) F (id,f) F (f op,id) F (a, b) Here, the diagram under consideration is not the disjoint union of such diagrams, but the one in which F (a, a) occurs on the left only once for each object a obc, but in which F (a, b) occurs on the right once for each morphism in C(a, b). When 9

10 A copower (sometimes called a tensor ) by an object a obv, if it exists, is the endofunctor (defined up to natural isomorphism) a : C C on a V- enriched category C satisfying the natural isomorphism C(a, y) = V(a, C(, y)). When V = Set, we can see that this simply means taking a coproduct indexed by the set a, and when C = V, the copower corresponds to the monoidal product with a. More generally, copowers can be understood as weighted colimits, but weighted colimits are beyond the scope of this exposition. When a V-category C has copowers for all objects of V, we say that it is copowered over V. Next we need to discuss some structure on the category of presheaves Pre V [C] from a small category C to a complete, cocomplete, closed monoidal category (V,, I V ). We describe the relevant constructions, but we refer the reader to Kelly s Basic Concepts of Enriched Category Theory for verifications that the constructions we use have the stated properties. Define the free V-category of C to be the category with the objects of C and hom-objects C V (c, d) = C(c, d) I V, where the last expression indicates a copower of I V over Set. Denote the enriched Yoneda embedding Y C,V : C V Pre V [C]. Recall that this is defined as Y C,V : c C V (, c). Enrich Pre V [C] over V by defining hom-objects Pre V [C](X, Y ) = V(X(c), Y (c)) c obc op With these preliminaries in place, we are ready to give the following general and quite powerful construction. Theorem 2.3. Let C again be a small category, D be a copowered V-category, and Pre V [C] be as before. Then there is a way of describing every enriched adjunction (L R) : Pre V [C] D. Freely enrich C over V and let F : C D be any enriched functor. Then define the functor R : D Pre V [C] by R : d (D(F ( ), d) : C op V) Define L : Pre V [C] D by the following coend (where continues to denote a copower): L : X c obc X(c) F (c) Only slightly more abstractly, L is the left Kan extension of F through the enriched Yoneda embedding, but Kan extensions, like weighted colimits, are beyond the scope of this exposition. We claim but do not prove that L and R are adjoint, and that every adjunction from Pre V [C] has this form. For a proof, see [Rie11a]. As a result of this theorem, there is a nice, canonical way of constructing a closed monoidal structure on Pre V [C] whenever we know beforehand how we d like the monoidal product to be defined. The construction tells us that we are interested in a functor Pre V [C] Pre V [C] C, and we choose this functor to be defined by X (X Y( ) : C Pre V [C]) 10

11 Then it is possible to check that the left adjoint determined by each object X is the monoidal product with X, and the right adjoint gives us the desired hom-object with domain X. Definition 2.4. More concretely, in the case of S, we define the monoidal product object-wise (which, in general, means we have X Y : c X(c) Y (c), where the second monoidal product is the one we are given for V). Since Set is cartesian closed, we wind up with S(X, Y ) = S(X [ ], Y ), where [ ] denotes the Yoneda embedding of the simplex category. In the case of S, the enriched Yoneda embedding is naturally isomorphic to the functor [ ] +, the composition of [ ] followed by the free pointed simplicial set functor S S induced by the free pointed set functor Set Set, both of which add a disjoint basepoint and both of which are left adjoint to the corresponding forgetful functors. The evident monoidal product in the pointed case is the smash product X Y = (X Y )/(X Y ) the pointed hom-objects are S (X, Y ) = S (X [ ] +, Y ). Although the closed monoidal structure for (pointed) simplicial sets is interesting will prove important later (and is one of several features which make the category of simplicial sets preferable to that of (all) topological spaces, which is not closed monoidal), at the moment, we are more interested in establishing a connection between simplicial sets and topological spaces which will help us to think about simplicial sets are spatial objects with tractable, combinatorial encodings. To this end, we indicate an adjunction formed in the manner described above. Definition 2.5. The functors ( Sing) : S Top which interest us are induced by the functor from the simplex category to Top which sends [n] to the topological n-simplex n and, loosely speaking, which sends a morphism g : [m] [n] to the function m n acting on the vertices of m as g acts on the elements of [m] and either collapsing the faces completely or keeping them rigid. Intuitively, the realization functor that is the resulting left adjoint creates a copy of the topological n-simplex n for each n-simplex in a simplicial set X and glues these (possibly very many) topological simplices together according to the face and degeneracy maps of X in order to create a topological space (specifcally, a CW-complex) modeled on the blueprint provided by X. On the other hand, the singular simplicial set functor gathers up all of the information it can about a topological space in terms of continuous maps from simplices in each dimension to that space. In the sense that these functors do not define an equivalence of categories, simplicial sets are like a lossy compression format (in analogy to how mp3 or jpg files compress audio and visual information) for topological spaces, but for the purposes of homotopy theory, they capture the most vital information and store it in a relatively manageable form. Finally, we conclude our section by defining a model category structure for simplicial sets. 11

12 Definition 2.6. A morphism f mors is a weak equivalence if and only if f is a weak equivalence of topological spaces (that is, π 0 ( f ) is a bijection, and f induces isomorphisms for all homotopy groups when restricted to each path-component of its domain). Define the k th -horn Λ k [n] [n] to be the sub-simplicial set generated by the same non-degenerate simplices as [n] but excluding the unique non-degenerate n-simplex and the k th non-degenerate n 1-simplex. More precisely, recall that [n] is the functor represented by n. Thus the simplices we wish to exclude correspond to morphisms in, and these are id : [n] [n] and d k : [n 1] [n]. A morphism of S is a fibration if it has the right lifting property with respect to all horn inclusions. A morphism of S is an acyclic cofibration if it has the left lifting property with respect to all fibrations. Define [n] [n] to be the simplicial set generated by all the same nondegenerate simplices as [n] excluding only the unique non-degenerate n-simplex corresponding to the identity morphism. A morphism of S is an acyclic fibration if it has the right lifting property with respect to all inclusions [n] [n]. A morphism of S is a cofibration if it has the left lifting property with respect to all acyclic fibrations. The weak equivalences, cofibrations, and fibrations on pointed simplicial sets are defined to correspond exactly to these classes of morphisms on the underlying simplicial sets (that is, a morphism of pointed simplicial sets is e.g. a weak equivalence if its image under the forgetful functor to simplicial sets is a weak equivalence). The proof that this definition actually satisfies the axioms of a model category can be found, amongst other places, in [MP12], sections 17.5 and 17.6 (the definition there is non-identical but equivalent to the one given here). It turns out that all monomorphisms of simplicial sets are cofibrations, and so all simplicial sets are cofibrant. Because the model category structure is generated by a set of cofibrations and a set of acyclic cofibrations, it is called cofibrantly generated. Cofibrantly generated model categories play an important role in the theory of model categories, but we do not discuss them any further in this exposition. 3 Spectra and Cohomology We assume that the reader is familiar with cohomology and its axioms; an exposition on these topics may be found in [Hat02] in section 3.1. The remarkable fact we discuss in this section is the fact that cohomology theories correspond to spatial objects called spectra. Definition 3.1. Suppose we have a pointed model category M and a functor called suspension. We neglect to give the general meaning of suspension, but we define it in the cases we need, and it is discussed in greater generality in [Hov99] in chapter 6. A spectrum is defined to be a sequence {X i } i N of objects in M along with for each i N a morphism ΣX i X i+1. A morphism of 12

13 spectra f : X Y is a sequence of morphisms f i : X i Y i that commute with suspension: ΣX i X i+1 Σf i f i+1 ΣY i Y i+1 For pointed simplicial sets, we define the n-dimensional simplicial sphere to be a quotient of simplicial sets previously discussed: S n = [n]/ [n], with the basepoint defined to be the image of [n] under the quotient. Note that S n has two non-degenerate simplices, one in dimension 0 and one in dimension n. Note that S 0 X = X and that S m S n = S m+n. We define suspension for pointed simplicial sets to be the smash product with S 1. Since the suspension of pointed simplicial sets is defined by a monoidal product, and since S is closed monoidal, suspension has a right adjoint: ΩX = S (S 1, X). The adjunction gives us a redefinition of spectra, in which we use morphisms X i ΩX i+1. There is a model category structure on spectra; we do not define it here, but it can be found (though not justified) in [Dun+07] in section Σ and Ω form a Quillen self-equivalence on this category, where we define these functors simply by applying them to each object and morphism in the sequences defining spectra. In this model category, we have weak equivalences between the spectra {ΣK i } i N {K i+1 } i N. The fibrant objects in this model category are the spectra for which the morphisms K i ΩK i+1 are weak equivalences in the underlying pointed model category (which, at the moment, is psim). But observe that ΩX (where X is a fibrant, pointed simplicial set) is a group object up to homotopy essentially because S 1 is a cogroup object up to homotopy, and similarly, Ω 2 X is an abelian group object up to homotopy. The reasoning here is just the same as that which allows us to define homotopy groups of topological spaces (and note that an easy, if inelegant, way of defining a fibrant-replacement functor for simplicial sets is by composing the singular simplicial set functor after the topological realization functor). Note that for fibrant spectra, each component space is an infinite loop space, since K 0 ΩK 1 Ω 2 K 2, and therefore certainly an abelian group object up to homotopy. All of this is building up to the fact that it is possible to define a generalized cohomology theory using the abelian groups h n (X) = [X, K n ]. Of course, there are more axioms to check, but this is the content of section 4.3 of [Hat02], and we do not reproduce it here. A remarkable fact is that all generalized cohomology theories are represented by spectra, and this is proven in section 4.E of [Hat02] (and known as the Brown Representability Theorem). And as one might expect (given the close relationship between homology and cohomology), spectra can be used to represent homology theories as well. Put succinctly, we define h n (X) = colim [S n, X K 0 ] [S n+1, X ΣK 0 ] [S n+1, X K 1 ] 13

14 If this seems reminiscent of stable homotopy, it ought to, since stable homotopy is a itself a homology theory. We are interested in the most ubiquitous cohomology theories, singular cohomology, and the spectra representing them (uniquely only up to weak equivalence), the Eilenberg-MacLane spectra. Our interest derives from the fact that the principal accomplishment of motivic homotopy theory has been the elucidation of certain properties of cohomology theories for schemes (and, more narrowly, fields). We present further below the homotopy-theoretic construction of one cohomology theory in particular (named, quite unevocatively, motivic cohomology ) which is analogous to singular cohomology for topological spaces and simplicial sets, which we discuss now. Definition 3.2. Recall that an ordinary cohomology theory (that is, one which obeys the dimension axiom, requiring that h n ( ) = 0 for n > 0) is completely determined (at least on finite CW-complexes) by the abelian group A = h 0 ( ), that this group is called the group of coefficients for the cohomology theory, and that the cohomology groups of a space X with coefficients A are notated H (X; A). There are various ways of defining and computing such theories, some of which apply only to certain sub-classes of spaces, but one definition, which applies to all spaces, is called singular cohomology (with coefficients). Put very tersely, this definition uses a composition of functors ( Top op S op Ab op) op Ch+ (Ab) op coch + (Ab) Ab where these functors the singular simplicial set functor, the free abelian group functor, the alternating sum of boundary maps of simplicial abelian sets, the dual functor from chain complexes to cochain complexes defined by homming into the group of coefficients, and finally the n-th cohomology functor. We will not dwell any longer on the definition of singular cohomology theories because we are more interested in the spectra representing them. Definition 3.3. Now we move in the direction of constructing the Eilenberg- MacLane spectrum for a given abelian group A. First of all, this spectrum is composed, at each level n, of an Eilenberg-MacLane space K(A, n). The defining feature of an Eilenberg-MacLane space is that { A if m = n π m (K(n, A)) = 0 if m n for n 1. This definition applies to both topological spaces and fibrant simplicial sets (the homotopy groups of which are defined, in general, by topological realization; but this definition agrees with homs in S for fibrant pointed simplicial sets). A useful fact about Eilenberg-MacLane spaces is that for all abelian groups A, they exist and are unique up to weak equivalence. This proof can be found in section 4.2 of [Hat02]. Another useful fact, a consequence of the preceding, is that we always have K(n 1, A) ΩK(n, A). This is an immediate consequence of the adjunction S (S m, K(n, A)) = S (S m 1, ΩK(n, A)). This fact indicates that Eilenberg-MacLane spectra are, in fact, fibrant as spectra. 14

15 Construction 3.4. Now we explicitly construct Eilenberg-MacLane spaces. It is not immediately clear from the definition that we should be able to hope for a sequence of functors K(, n) : Ab S, but in fact, our construction is functorial (that is, morphisms of abelian groups actually do induce morphisism of Eilenberg-MacLane spaces). We construct our functors as compositions Ab Ch + (Ab) Ab op S We need to define each functor in this composition. The first functor Ab Ch + (Ab) simply sends an abelian group A to the chain complex which is A in degree n and 0 everywhere else. The last functor is the forgetful functor which remembers identity elements as basepoints. To define the middle functor, we use the construction above in which we define an enriched (over Ab) adjoint pair (L R) : Ab op Ch + (Ab) by specifying a functor F : Ch + (Ab). In this case, the resulting adjoint pair is an equivalence of categories known as the Dold-Kan correspondence. Our work now lies in defining F : Ch + (Ab). This functor is itself a composition S Ab op Ch + (Ab) The first arrow is the Yoneda embedding and the second is the free abelian group functor. The composition the first two functors, then, is in fact the enriched Yoneda embedding for the free Ab-enrichment of. The last arrow in this composition is really what the rest of the construction boils down to. It is called the Moore complex. Given a simplicial abelian group B (in other words, an object of Ab op ; the convention is that a presheaf from to the category of foos is called a simplicial foo ), we define the Moore complex M(B) to be the chain complex concentrated in non-negative degree which, in degree m, is M(B) m = m 1 i=0 ker(d i ) where this intersection is a subgroup of B m, so the face maps in question are those from dimenion m to m 1. The degree-decreasing map of the chain complex is the restriction m = d m M(B)m. There s a simplicial identity which makes verification that this defines a chain complex simple: for i < j we have d i d j = d j 1 d i. Thus we have m 1 m = d m 1 d m 1, but M(B) m ker(d m 1 ). Our definition diverges from our trend of giving purely category-theoretic definitions, and it may be unclear how the definition of Moore complexes is functorial, but to see that this is the case, we merely recall that kernels are limits, and so are intersections (that is, the intersection of several kernels is exactly the same as their pullback). Limits of limits are limits, and limits are functorial. 4 Constructing New Model Categories We next explore methods for generating a free model category from a small category and for modifying (localizing, in the jargon) the model structure in 15

16 a useful way analogous to introducing relations in an algebraic object. These techniques are critical, because they allow us to apply the methods of homotopy theory to new contexts (namely, when we re interested in a small category, such as the category of smooth schemes of finite type over a field). The exposition in this section will address the techniques most important to motivic homotopy theory, but more complete references include [Dug01] and [PSH]. The following definition is a specific instance of a much more general phenomenon called the projective model structure on a category of functors from a small category to a cofibrantly generated model category. Because of the specificity of our application, we do not define cofibrantly generated model categories here. Our definition does make reference to the model category of pointed simplicial sets; for a discussion of this particular and very important category, see the following section of this paper. However, because this construction is a particular instance of a more general phenomenon, its definition is rather formal, and should be comprehensible even without much familiarity with the pointed simplicial sets. Construction 4.1. Given a small category C, the universal pointed model category built from C, denoted U C, is the functor category S C, where S is the model category of pointed simplicial sets. The weak equivalences of U C are the natural transformations which are objectwise weak equivalences, and similarly, the fibrations are the natural transformations which are ojbectwise fibrations (that is, every morphism of C in the natural transformation defining a morphism in U C is a weak equivalence or fibration, respectively). The cofibrations are the morphisms with the left lifting property with respect to the acyclic fibrations. We embed C in U C by composing the following functors: the Yoneda embedding C Set Cop, the functor Set Cop S Cop induced by the functor sending each set to the discrete simplicial set built off of it, and the functor S Cop S Cop = U C induced by the functor sending each simplicial set to the pointed simplicial set freely generated from it by adjoining a disjoint basepoint (think about it: this is the enriched Yoneda embedding of the freely enriched category construction which appeared earlier). By abuse of notation, we will sometimes speak of objects of C as though they are objects of U C; in those cases, we are invoking this embedding. A universal pointed model category really is a universal object (up to homotopy) in the appropriate category of pointed model categories; see Dugger s paper for an elaboration on this fact. Universal pointed model categories give us a way of generating free model categories from a given small category. Aside from the universal property of this construction, it provides a way of freely adding homotopy colimits to a category in a manner analogous to how the category of set-valued presheaves freely adds colimits to a category (see [Rie11b] for an introduction to homotopy colimits). But if we d like to add relations to one of these free model categories, how can this be accomplished? The following construction provides an answer, in that it allows us to add a collection of morphisms to the weak equivalences of 16

17 a model category in a universal way by limiting the morphisms we consider fibrations. Construction 4.2. Given a model category C and a collection of morphisms T, the left Bousfield localization of C at T, denoted C loc(t ), is a new model category structure, if it exists, on the same underlying category defined in the following way. We call a fibrant object A obc T -local if for all morphisms t T, HoC(t, A) is an isomorphism. A morphism f of C is called a T -local weak equivalence if HoC(t, A) is an isomorphism whenever A is T -local. Then we define a new model category structure with the same cofibrations (and therefore the same acyclic fibrations), more weak equivalences: the T -local weak equivalences, and fewer fibrations: the morphisms with the right lifting property with respect to cofibrations that are also T -local weak equivalences. The intuition here is that fibrations are meant to detect weak equivalences, and so if we want to add T to our collection of weak equivalences, we need to restrict our definition of fibrant objects to those which perceive morphisms of T as such. This, in turn, gives us a way of describing the other morphisms which must necessarily be weak equivalences if those of T are. The dual construction is called right Bousfield localization. Some foreshadowing: if we would like a particular object A of C to behave as an interval, then we would like the projection A X X to be a weak equivalences in U C. In those cases, we can use Bousfield localization to make it so. 5 Motivic Homotopy Theory Motivic homotopy theory is the homotopy theory of schemes. For a scheme Q, we develop a homotopy theory for smooth schemes of finite type over Q, and notate this category Sm/Q. There is not an obvious notion of a weak equivalence within this category, and so we follow Dugger to produce a free pointed model category generated by Sm/Q (we want this category to be pointed in order to facilitate the construction of its stabilization, the category of motivic spectra, where cohomology theories live). We localize the free (that is, projective) model category U Sm/Q of pointed simplicial presheaves (called motivic spaces) on Sm/Q twice, first with respect to the Nisnevich topology (so that morphisms from the homotopy colimits of the images of hypercovers to the images of the schemes they cover become weak equivalenes), and then with respect to the projections X A 1 Q X, where were use A1 Q to denote the motivic space that is the image of the 1-dimensional affine space under the enriched Yoneda embedding into motivic spaces. Recall that we define the 1-dimensional affine space over a scheme Q to be A 1 Q = (Spec Z[t]) Q, where the projection onto Q gives the required morphism to Q. We denote the model category resulting from these localizations M Q. Next, observe that because Sm/Q is small, M Q has a canonical closed monoidal structure as described more generally in Theorem 2.3. This allows us 17

18 to define the M Q -enriched category of M Q -enriched functors from fm Q, the subcategory of presheaves all of whose values are in finite simplicial sets, M Q. This category serves as a category of spectra for motivic spaces, as described more generally in [DRØ]. References [Dug01] [Dun+07] [DRØ] [Hat02] [Hov99] [Kel05] [MP12] [PSH] [Rie11a] [Rie11b] Daniel Dugger. Universal Homotopy Theories. In: Advances in Mathematics 164 (1 Dec. 2001), pp doi: /aima B.I. Dundas et al. Motivic Homotopy Theory. New York: Springer, isbn: Bjorn Ian Dundas, Oliver Röndigs, and Paul Arne Østvær. Enriched Functors and Stable Homotopy Theory. In: Documenta Mathematica (). Allen Hatcher. Algebraic Topology. New York: Cambridge University Press, isbn: Mark Hovey. Model Categories. Mathematical Surveys and Monographs 63. Providence: American Mathematical Society, isbn: G.M. Kelly. Basic Concepts of Enriched Category Theory. London Mathematical Society Lecture Notes Series 64. New York: Cambridge University Press, isbn: J.P. May and K. Ponto. More Concise Algebraic Topology. Chicago Lectures in Mathematics. Chicago: The University of Chicago Press, isbn: title = Philip S. Hirschhorn. Emily Riehl. A Leisurely Introduction to Simplicial Sets url: Emily Riehl. Homotopy (Limits and) Colimits url: http: // 18