Categorical homotopy theory J.F. Jardine* December 6, 2003 Abstract. This paper is an exposition and extension of the ideas and meth- ods of Cisinksi, set at the level of A-presheaves on a small Grothendieck site, where A is an arbitrary test category in the sense of Grothendieck. The homo- topy theory for the category of simplicial presheaves and all of its localizati* *ons can be modelled by A-presheaves in the sense that there is a corresponding model structure for A-presheaves with an equivalent homotopy category. The theory specializes, for example, to the homotopy theories of cubical sets, cubi- cal presheaves, and gives a cubical model for motivic homotopy theory. The applications of Cisinski's ideas are explained in some detail for cubical sets. Introduction Traditionally, categorical homotopy theory is a combination of a small collecti* *on of simple ideas and definitions, with a rather subtle skill set. In broad outline, one associates to each small category C a simplicial set BC, variously called its nerve or classifying space, whose n-simplices are stri* *ngs of composable arrow of length n in C. This is a functorial construction: given a functor f : C ! D, applying f to strings of arrows of length n in C produces a corresponding string in D, and one obtains an induced simplicial set map f* : BC ! BD. The classifying space functor C 7! BC preserves products, and it is almost a tautology that if n = {0, . .,.n} is a finite ordinal number, thought of as a poset and hence as a small category, then Bn is the standard n-simplex n. It follows that any natural transformation C x 1 ! D of functors f, g : C ! D induces a simplicial homotopy BC x 1 ! BD between the induced simplicial set maps f*, g* : BC ! BD. Thus, if C and D are equivalent categories or if a functor C ! D has an adjoint, then the associated classifying spaces are homotopy equivalent. A further consequence is that if a category C has either an initial or terminal object, then the classifying space BC is contractible. The subtlety of the theory lies in the analysis of the homotopy fibres of the map f* : BC ! BD induced by a functor f : C ! D. Every object d 2 D has an associated comma category f # d whose objects consist of all morphisms ____________________________* This research was supported by NSERC. 1 f(c) ! d in D; the morphisms of this category are commutative diagrams f(c)_____f(ff)__//AF (c0) AAA ---- ø AAA -- tau0 A__""-- d where ff : c ! c0 is a morphism of C. There is an obvious forgetful functor f # c ! C which takes the diagram above to the morphism ff in C, and any morphism fi : d ! d0of D induces a functor fi* : f # d ! f # d0by composition with fi. It is a basic observation of Quillen that the forgetful functors f # d ! C assemble to define a homotopy equivalence holim---!B(f # d) ! BC. d2D Quillen's Theorem B asserts that if all induced maps fi* : B(f # d) ! B(f # d0) are weak equivalences, then all diagrams of simplicial set maps B(f # d)_____//_BC | | | |f* fflffl| fflffl| B(D # d)_____//BD are homotopy cartesian. It follows, in this case, that B(f # d) is weakly equiv- alent to the homotopy fibre of f* : BC ! BD over the vertex corresponding to the object d. Quillen's Theorem A says that if all of the simplicial sets B(f # d) are wea* *kly equivalent to a point then the map f* : BC ! BD is a weak equivalence. This result is a consequence of Theorem B, but is most effectively proved by a comparison of homotopy colimits _ part of the appeal of the result lies in the simplicity of that proof. All of this has been known since the early 1970s, when Quillen [17] introduc* *ed these concepts and results as a foundation for his description of higher algebr* *aic K-theory. This set of techniques is still fundamental for algebraic K-theory, a* *nd Theorem B is now one of the most important theorems in the foundations of homotopy theory, although recognizing when it can be applied can be something of a black art. It is a matter of taste whether or not the homotopy theory of simplicial sheaves and presheaves is categorical homotopy theory as such. The main tech- niques and results of the theory are geometric in the sense that they come from ordinary homotopy theory, although they are expressed in the categorical con- text of sheaves and presheaves on a Grothendieck site and derive much of their power in applications from holding at that level of generality. The homotopy theory of simplicial presheaves is a direct extension of the homotopy theory of 2 simplicial sets. The development of these theories and their applications was initiated in the 1980s [12], [7], [19] and continues to the present [16], [8]. * *The homotopy theory of stacks [13], [9] is a vital and important subindustry of this work. Thomason's work on the model structure for the category of small categories [18] is also necessarily part of this historical narrative, but its impact has * *so far been rather muted. It is strongly related to but not necessary for the ideas exposed in this paper. In my view, the thesis of Denis-Charles Cisinski [3] represents the next leap forward for the subject. Cisinski's thesis is primarily concerned with the proof of some conjectures of Grothendieck concerning diagram categories that model homotopy theory, but the techniques that he has developed are arguably more important than the conjectures themselves. The theory begins with Grothendieck's concept of a test category A and the corresponding category of A-sets, which consists of contravariant set-valued functors X : Aop ! Set (ie. presheaves) on A. In general, if A is a test category, then the corresponding category of A-sets is a model for the standard homotopy category. The category of simplicial sets, which are contravariant functors X : op ! Set on the category of ordinal numbers is a standard example. The category of abstract hypercubes (here called the box category) is also a test category, and the corresponding category of -sets, or cubical s* *ets, is another model for the standard homotopy category. The product of two test categories is a test category, so that the general theory shows that, for examp* *le, bisimplicial sets, bicubical sets, and simplicial cubical sets all give models * *for the homotopy category. So, when is a small category A a test category? Each object a of the category A determines a representable functor a = hom (a, ), and there is a cell catego* *ry iA X for each A-set X: the objects of iA X are the A-set morphisms a ! X (or elements of X(a), a 2 A) and the morphisms of the cell category are the diagrams a _____________//B b BB """ BBB """ B__B~~"" X of A-set morphisms, which can be interpreted as incidence relations in the A-set X. If Y is a simplicial set, then the corresponding cell category i Y is the u* *sual simplex category of Y , which has often been denoted in the literature (see, for example, [6]) by # Y . The functor X 7! iA X has a right adjoint C 7! i*AC, and we say that the small category A is a test category if the space BA is contractible, and the canonical functor ffl : iA i*AC ! C is aspherical in the sense that all spa* *ces B(ffl # c) are contractible for all small categories C. Now we can be more precise about the homotopy theory: if A is a test category, then there is a closed model structure on the category of A-sets with 3 cofibrations defined to be inclusions of diagrams and for which the weak equiv- alences are those A-set maps f : X ! Y such that the induced simplicial set map f* : BiA X ! BiA Y is a weak equivalence. Then it is relatively easy to show that the functor X ! BiA X induces an equivalence Ho(A - Set) ' Ho(S) between the homotopy category of A-sets and the homotopy category of simpli- cial sets. One can go further, and formally invert a set S of cofibrations in t* *he model structure of A-sets to produce a Bousfield localization of the homotopy category of A-sets in an essentially standard way. These model structures are given by Cisinski in his thesis [3]. Grothendieck introduced the notion of test category, and he knew that A- sets would model the ordinary homotopy category for all test categories A _ indeed, the equivalence of homotopy categories is just a formal consequence of the definition of test category. Grothendieck also introduced the study of good classes of functors between small categories, which could potentially serve as classes of weak equivalences for homotopy theories. He called such classes üf ndamental localisers", and the terminology persists in [3]. These classes are called "weak equivalence classes" in this paper. A weak equivalence class is a class W of functors between small categories which satis* *fy the conditions that one would expect: informally speaking, the class satisfies * *the analog of the closed model axiom CM2 (the two out of three axiom), contains all strong deformation retractions, and contains the functor C ! * if C has a terminal object. The öt tal spaceö f a functor f : C ! D is a formal homotopy colimit of the comma categories f # c in such a theory. The standard features of categorical homotopy theory imply that the class W1 of all functors C ! D such that the induced map BC ! BD is a weak equivalence of simplicial sets satisfies the requirements for a weak equivalence class of functors. Grothendieck made two conjectures about these objects: Conjecture A. Suppose that W is a weak equivalence class, and that f : C ! D is a functor such that f* : BC ! BD is a weak equivalence of simplicial sets. Then f is a member of W. In other words, W1 is the smallest weak equivalence class. Conjecture B. Suppose that W is a weak equivalence class and that A is a (local) test category. Then the class of all maps f : X ! Y of A-sets such that the functor iA X ! iA Y is a member of W is the class of weak equivalences for a model structure on the category of A-sets for which the cofibrations are the monomorphisms. In [3], Cisinski proves the first conjecture in its entirety and the second conjecture in the case where W is generated over W1 by a set of functors. Con- jecture A, at least so far, appears to be much more important for applications than Conjecture B. 4 This paper was written to express this collection of ideas and their proofs in something like standard homotopy theoretic language and notation, and to begin to describe their applications. I was initially attracted to Cisinksi's thesis as a result of my own work on the homotopy theory of cubical sets [10]. I knew that there was a model struc- ture on the category of cubical sets whose associated homotopy category was equivalent to that of simplicial sets. The cofibrations are the monomorphisms and the weak equivalences are those maps which induce weak equivalences of topological realizations. The verification of this model structure was achieved with some bounded cofibration tricks from localization theory, and the equiva- lence of homotopy categories depended on a cubical set excision theorem which arose from a somewhat involved subdivision argument. Cisinski displays the same model structure on cubical sets as an example of his theory, and then the equivalence of homotopy categories arises from formal nonsense, since the box category is a test category. He also proves much more, namely that the model structure on the category of cubical sets is proper and that the fibrations are analogs of Kan fibrations. The techniques of [10] cannot begin to reach these last results, and their proofs involve some of the most delicate aspects of Cisinski's work. These incl* *ude an internal description of homotopy colimits in a cofibrantly generated model structure and a general notion of regularity, which amounts the assertion that an A-set X is a homotopy colimit of its cells. Regularity holds in contexts, like cubical sets, where an A-set can be constructed inductively by attaching cells. The subtlety of the theory for cubical sets is this: properness and the identification of fibrations are proved by displaying three ostensibly different model structures for the category of cubical sets, which are then shown to be identical as a result of Grothendieck's Conjecture A and regularity. The main results for cubical sets are proved in the final section of this pa* *per: properness is proved in Theorem 85, and Theorem 88 gives the good classifica- tion of cubical set fibrations. After the fact, cubical set excision (Theorem 9* *1) turns out to be a direct consequence of the formal techniques displayed here, along with the excision theorem for simplicial sets [11]. Much of the rest of the paper is an exposition of the basic theory. I have chosen to display that theory in terms of presheaves of A-sets, here called A- presheaves, on an arbitrary small Grothendieck site C, with a view to displaying potential applications. Except for questions related to regularity and heirarch* *ies of weak equivalence classes, there is essentially nothing special about the hom* *o- topy theory of A-sets: it is a special case of a homotopy theory of A-presheave* *s. That homotopy theory arises in part from a "Swiss army knife" result (The- orem 46 in Section 3) which establishes a model structure for the category of A-presheaves in which some set S of monomorphisms become weak equiva- lences, and which depends on a suitable theory of intervals on the category of A-presheaves. An interval theory is expressed here as a monoidal action : (A - Pre(C)) x ! A - Pre(C) of the box category on the category of A-presheaves, satisfying a list of expec* *ted 5 properties, and the purpose of which is to define some notion of naive homotopy of morphisms. Examples of such theories arise here either from taking iterated products X x Ixn with objects I having two distinct global sections, or from Kan's tensor product operation [14] in the particular context of cubical sets. The affine line A1 and the global sections 0, 1 : * ! A1 generate an interval theory in the motivic context. The construction of the resulting ( , S)-model structure on the category of A-presheaves follows the general outlines that one finds in localization theory, except that one is not localizing another model structure to construct it. It is general nonsense that an injective replacement of a map or object can always be constructed, and then one defines a weak equiivalence to be a map f : X ! Y which induces an isomorphism ß(Y, Z) ~= ß(X, Z) of naive homotopy classes (defined by intervals) for all injective objects Z. It is one of the innovation* *s of Cisinksi's thesis that naive homotopy equivalences alone can be used to prove a bounded cofibration property (Lemma 38), and then Theorem 46 comes out in the usual way, modulo some fussing with pushouts of trivial cofibrations (Lemma 42). This model structure is proper if the set S of cofibrations is decently behaved and the interval theory is defined by an actual interval I (Theorem 47), and in particular specializes to a proper model structure on the category of cubical sets. One of the interesting aspects of Theorem 46 and Theorem 47 is that the set S can be empty, so that there is always a "primitive" model structure defined by an interval theory, and this model structure is proper. Theorem 71 of Section 5 says that any localized model structure on the category of simplicial presheaves induces a model structure on the category of A-presheaves, in such a way that the associated homotopy categories are equivalent. This holds over any small Grothendieck site and for any test catego* *ry A. The level of generality of this result (and of Theorem 46) is perhaps the only real innovation of the present paper. The theorem says that all simplicial presheaf homotopy theories (including the motivic homotopy theories) have A- models for any test category A. Theorem 71 specializes to the existence of a model structure on the category of A-sets with homotopy category equivalent to any localized homotopy theory of simplicial sets, and as such reduces the proof of of Grothendieck's Conjectu* *re B to the simplicial set case. This result does not follow from the Swiss army knife Theorem 46 _ it is a subsidiary structure, but the model structures that these results generate coincide in a wide variety of interesting cases, such as cubical sets. One of the morals of this stream of ideas, and this is perhaps a bit ironic, is that cubical sets are everywhere. The definition and formal properties of the box category and the category - Set of cubical sets are summarized in Section 2 of this paper. The basic properties of test categories are treated in Section 1. The proof of the assertion that the box category is a test category turns out to be a bit subtle. In fact, the category of cubical sets seems to be the delicate case throughout the theory. It is a rather disconcerting fact that products misbehave very badly in the homotopy theory of cubical sets: in particular that the product 1 x 1 of a pair of copies of the standard interval 6 in cubical sets has the homotopy type of a circle (Remark 22). This forces one to be careful with interval theories everywhere, and prompts the discussion of aspherical A-sets. Section 4 contains a general discussion of homotopy colimits, internally de- fined nerves, and the relation with the Grothendieck construction in an ( , S)- model structure on a category of A-sets. Homotopy colimits are defined inter- nally, by taking colimits of projective cofibrant resolutions, which resolutions exist since the ambient model structures are cofibrantly generated. From this point of view, the internal nerve BhC of a small category C is the homotopy colimit for the diagram which assigns a point to each object of C. This, of course, generalizes the observation that the ordinary nerve BC is the homotopy colimit of a diagram of points in the category of simplicial sets. The standard properties of the ordinary nerve construction also hold for the internal nerve. In particular, there is a weak equivalence holim---!Bh(f # d) ! BhC d2D for any functor f : C ! D, which, in turn, means that the internal nerve of the Grothendieck construction models a homotopy colimit in this sense. Section 6 contains an exposition of the basic aspects of the theory of weak equivalence classes of functors, along with proofs of Conjecture A (Corollary 8* *0) and the case of Conjecture B corresponding to test categories and ä ccessible" weak equivalence classes (Theorem 81). As one might expect, Grothendieck's Conjecture B can be proved with a localization argument, in the presence of Conjecture A, but that is not the way that it is done here. I prefer instead to follow Cisinski's lead in using an omnibus result (ie. Theorem 71) which subsumes all localization arguments. Theorem 46 has a similar flavour. There is yet another striking innovation of Cisinski which is displayed in Section 6: the cell category functor X 7! iA X preserves homotopy cocarte- sian diagrams in striking generality (Corollary 78). This was certainly not well known, even for simplex categories, and it is a central feature of this theory. I would like to thank Denis-Charles Cisinski for a series of helpful discuss* *ions that we had during the meeting öH motopy Theory and its Applications" held in London, Canada in September, 2003. This meeting was supported by the Fields Institute, and I would like to thank the Institute for making that meeti* *ng possible. 7 Contents 1 Homotopy theory of categories 9 2 Cubical sets: basic properties 18 3 Fundamental model structures 26 4 Homotopy colimits 44 5 Homotopy theories for test categories 58 6 Weak equivalence classes of functors 63 7 Homotopy theory of cubical sets 70 8 1 Homotopy theory of categories Suppose that X is a simplicial set. The simplex category i X = # X has objects consisting of all simplices n ! X and morphisms consisting of commutative triangles of simplicial set maps nC____________//_ m CC --- CCC --- CC!! ""-- X Write cat for the category of small categories, and consider the functors i B S ____//_cat___//S Say that a functor f : C ! D between small categories is a weak equivalence if the induced map f* : BC ! BD is a weak equivalence of simplicial sets. For each simplicial set X there is a functor QX : i X ! S which takes an object oe : n ! X to the simplicial set n. Then it is well known that the maps oe : n ! X define a natural weak equivalence fX : holim---!QX ! X, and that the canonical projection ßX : holim---!QX ! B(i X) is also a natural weak equivalence. It follows that the nerve functor B and the simplex category functor induce an equivalence of categories Ho(cat) ' Ho(S) after formally inverting the weak equivalences in cat and S respectively. Suppose that A is a small category, and write A - Set(written A^in [3]) for the category of set-valued contravariant functors defined on A; these functors will be called A-sets. Write a = hom ( , a) for the representable contravariant functor associated to an object a 2 A. The A-set a will often be called the standard a-cell. Similarly, if X is an A-set, the elements of set X(a) will be called the a-cells of X. The a-cells of X are classified by A-set maps a ! X, by the usual Yoneda lemma argument. Suppose that X is an A-set, and write iA X for the category whose objects are the natural transformations a ! X and whose morphisms are the com- mutative triangles a _____________//B b BB """ BBB """ B__B~~"" X The assignment X 7! iA X is functorial in X, and defines a functor iA : A - Set ! cat. The category iA X will often be called the cell category of X. 9 Say that a map f : X ! Y is a weak equivalence of A-sets if the induced map f* : B(iA X) ! B(iA Y ) is a weak equivalence of simplicial sets, or equivalent* *ly if the induced functor f* : iA X ! iA Y is a weak equivalence in cat. According to these definitions, the functor iA induces a üf nctor" iA* : Ho(A - Set) ! Ho(cat). A basic question of Grothendieck (üP rsuing stacks") is the following: when is iA* an equivalence of categories? The functor iA : A - Set ! cat has a right adjoint i*A: cat ! A - Set which is defined by i*A(C)(a) = hom (A # a, C). This follows from the fact that every A-set (being a contravariant functor) is a colimit of representables. More explicitly, the natural map hom(iA X, C) ! hom (X, i*A(C)) is easy to describe: if oe : a ! X is an element of X(a) and f : iA X ! C is a functor, then the composite functor A # a ~=iA a oe*-!iA X f-!C is an element f*(oe) 2 i*AC(a). An A-set morphism g : X ! i*AC is determined by functors g(oe) : A # a ! C, one for each element oe : a ! X, which make the obvious diagrams of functors commute. Given such a g, define a functor g* : iA X ! C by associating to an object oe : a ! X the object g(oe)(1a) 2 C. One can show that these two natural maps are inverse to each other, and there is a corresponding bijection hom (iA X, C) ~=hom (X, i*AC), so that that i*Ais right adjoint to iA . Note that the category iA i*AC has objects all functors f : A # a ! C and has morphisms given by all commutative diagrams A # a______`*_____//A # b EE zz EEE zzz f EE""E__zgzz C where ` : a ! b is a morphism of A. The adjunction map ffl : iA i*AC ! C is the functor which associates to each functor f : A # a ! C the object f(1a) 2 C. 10 Lemma 1. There is an isomorphism of categories ffl # c ~=iA i*A(C # c) for all categories C. Proof. An object of the category iA i*A(C # c) is a functor f : A # a ! C # c, and a morphism of this category is a commutative diagram A # a_______`*______//_A # b HH ww HHH www f HH##HH--wwgww C # c as above. A functor f : A # a ! C # c can be identified uniquely with a pair (f0, f00) consisting of a functor f0 : A # a ! C and a morphism f00: f0(1a) ! c. This identification induces the required isomorphism of categories, since an object of ffl # c consists of a functor f : A # a ! C and a morphism_ f(1a) ! c. |__| The essential idea is to come up with conditions on A so that the adjunction maps ffl : iA i*A(C) ! C are weak equivalences for all categories C. Observe th* *at if all counit maps ffl are weak equivalences then all unit maps j : X ! i*AiA X are weak equivalences of A-sets, by a triangle identity. It also follows easily* * that a functor f : C ! D is a weak equivalence if and only if f* : i*AC ! i*AD is a weak equivalence of A-sets in this case. A functor f : C ! D is said to be aspherical if the simplicial set B(f # d) * *is weakly equivalent to a point for all d 2 D. If f is aspherical then it is a weak equivalence by Quillen's Theorem A. Say that a category A is aspherical if the canonical map ß : A ! * is aspherical: in view of the fact that ß # * ~=A, A is aspherical if and only if A is weakly equivalent to a point. Say that a map f : X ! Y of A-sets is aspherical if the induced functor f* : iA X ! iA Y is aspherical. In general, there is an isomorphism f* # ( a ! Y ) ~=iA ( a xG F ) (1) so that f : X ! Y is aspherical if and only if all pullbacks a xG F are weakly equivalent to a point. Every aspherical map of A-sets is a weak equivalence, by Quillen's Theorem A. The class of aspherical maps of A-sets is closed under pullback. From this point of view, an A-set F is aspherical if the map F ! * is as- pherical. This means precisely that the induced functor iA F ! A is aspherical. The isomorphism iA (F ) # a ~=iA (F x a) (2) is of central use in analyzing objects of this sort. Say that A is a weak test category if the adjunction map ffl : iA i*A(C) ! C* * is a weak equivalence for all small categories C. 11 It follows from Lemma 1 and Quillen's Theorem A that A is a weak test category if and only if the functor D 7! i*AD takes categories having a terminal object to A-sets which are weakly equivalent to a point. Suppose that the functor C 7! i*A(C) takes aspherical categories to A-sets which are weakly equivalent to a point. Then categories having terminal objects are examples of aspherical categories, so that A is a weak test category. Suppo* *se that A is a weak test category and that C is an aspherical category. Then the adjunction map ffl : iA i*A(C) ! C is a weak equivalence, so that the A-set i*A* *(C) is weakly equivalent to a point. We have proved the following: Lemma 2. The following statements are equivalent: 1)A is a weak test category, ie. all adjunction maps ffl : iA i*A(C) ! C are weak equivalences. 2)if D is a category with terminal object, then the A-set i*A(D) is weakly equivalent to a point 3)if C is aspherical, then the A-set i*A(C) is weakly equivalent to a point. Say that A is local test category if all categories A # a are weak test cate* *gories. Lemma 3. The following are equivalent: 1)A is a local test category; 2)if D is a category with a terminal object, then the A-set i*A(D) is aspher- ical, or equivalently the canonical functor ß : iA i*A(D) ! A is aspherica* *l; 3)if C is an aspherical category, then the A-set i*A(C) is aspherical, or eq* *uiv- alently the canonical functor ß : iA i*A(C) ! A is aspherical; Proof. The A-set i*A(C) is aspherical if and only if all categories iA i*A(C) # a ~=iA#ai*A#a(C) are weakly equivalent to a point. Now use Lemma 2. |___| Say that A is a test category if it is both a local test category and a weak test category. This, however, is not the right definition to use in practice, * *in view of the following: Lemma 4. A category A is a test category if and only if it is a local test cate* *gory and is aspherical. Proof. Suppose that A is a local test category and that A is aspherical. Suppose that D is a category with terminal object. We want to show that the A-set i*A(D) is weakly equivalent to a point. But the functor iA i*A(D) ! A is aspherical by Lemma 3 and A is aspherical, so that the A-set i*A(D) is weakly equivalent to a point. It follows that A is a weak test category as well as a local test catego* *ry. 12 Suppose that A is a test category. Then the functor i*Ahas a left adjoint and therefore preserves terminal objects. The terminal object of the category of A-sets is the one point A-set *, and there is an isomorphism iA (*) ~= A. Since A is a weak test category, the adjunction map ffl : iA i*A(*) ! * is a_we* *ak equivalence. It follows that A is aspherical. |__| Remark 5. One can show by using the argument in the proof of Lemma 4 that if A is a weak test category, then A is aspherical. Example 6. Suppose that A is the category of finite ordinal numbers, so that - Set is the category S of simplicial sets. If C is a category i* (C) is the simplicial set with n-simplices specified by i* (C)n = hom ( # n, C) If D is a category with terminal object t then there is a contracting homotopy h : D x 1 ! D. The functor i* preserves products, so that h induces the composite i* (D) x i* (1) ~=i* (D x 1) h*-!i* (D). There is a natural functor ff : # n ! n. This functor is essentially a last vertex map, and is specified on objects by ff(` : m ! n) = `(m). The particular example ff : i 1 ! 1 of this functor defines a 1-simplex ff : 1 ! i* (1), and there is a composite i* (D) x 1 1xff---!i* (D) x i* (1) ~=i* (D x 1) h*-!i* (D). which gives a contracting homotopy for i* (D). It follows that all maps ffl : i i* (C) ! C are weak equivalences. We know [6, p.236] that every simplicial set X is a homotopy colimit of its simplices in the sense that there is a weak equivalence holim---! n ! X, n!X and that the homotopy colimit is weakly equivalent to B(i X). It follows that the simplicial set i* (C) is naturally weakly equivalent to BC, and there are natural weak equivalences i* (C) ' Bi i* (C) ffl*-!BC. (3) Suppose that D is a category with a terminal object. In order to show that is a local test category (and hence a test category) we must show that the canonical functor ß : i i* (D) ! is aspherical. The isomorphism (2) implies that there is an isomorphism i i* (D) # n ~=i (i* (D) x n). But i* (D) is a contractible simplicial set, so that i* (D) x n is weakly equi* *v- alent to a point. It follows that the category i (i* (D) x n) is aspherical. 13 Lemma 7. Suppose that A and B are small categories and that f : X ! Y is a morphism of (A x B) - Set. If f induces weak equivalences of B-sets X(a, ) ! Y (a, ) for all objects a 2 A, then f is a weak equivalence of (A x B)- sets. Proof. Consider the functors iAxB X ßX--!A x B p-!A where q is a projection. An element of the category a # pßX can be identified with a pair (a fl-!a1, x 2 X(a1, b1)), and a morphism (fl, x) ! (fl0, y) consists of a morphism (a1 `-!a2, b1 ø-!b2) of A x B such that `fl = fl0 and (`, fl)*(y) = x. There is a functor !a : iBX(a, ) ! a # pßX which is defined by sending the object x 2 X(a, b) of iBX(a, ) to the object (a 1a-!a, x 2 X(a, b)) There is a functor fla : a # pßX ! iBX(a, ) which is defined by sending the element fl (a -! a1, x 2 X(a1, b1)), to the element (fl, 1)*(x) 2 X(a, b1). Then fla!a = 1 and the morphisms (fl, 1) : (1a, (fl, 1)*(x)) ! (a, x) define a natural transformation !afla ! 1. The functors !a and fla define a homotopy equivalence BiBX(a, ) ' B(a # pßX ) which is natural in presheaves X. The assumptions therefore imply that the map f : X ! Y induces a weak equivalence B(a # pßX ) f*-!B(a # pßY ) for all objects a 2 A. It follows that f induces a weak equivalence BiAxB (X) ! BiAxB (Y ) of the respective homotopy colimits over A. |___| 14 Lemma 8. Suppose that A is a local test category and that B is a small category. Then the A x B is a local test category. Proof. Suppose that C is a small category with terminal object t. It suffices to show that the object i*AxB#(a,b)(C) is weakly equivalent to a point (see the proof of Lemma 3). There is an isomorphism of categories A x B # (a, b) ~=A # a x B # b. Write A0= A # a and B0= B # b. Then in this notation we must show that the (A0x B0)-set i*A0xB0C is weakly equivalent to a point when we know that the A0-set i*A0C is weakly equivalent to a point. There are identifications i*A0xB0C(a0, b0) = hom (A0# a0x B0# b0, C) = hom (A0# a0, hom (B0 # b0, C)), where hom (B0# b0, C) is the obvious category of functors and natural transfor- mations. This category has a terminal object, namely the functor B0# b0! C which takes all objects to the terminal point. It follows that all A0-sets hom (A0#? x B0# b, C) ~=i*A0hom(B0# b, C) are weakly equivalent to a point. It therefore follows from Lemma 7 that the (A0x B0)-set morphism hom(A0# a0x B0# b0, C) ! * is a weak equivalence. |___| Corollary 9. Suppose that A is a test category and that the small category B is aspherical. Then the product A x B is a test category. Other useful tools include the following: Lemma 10. Suppose that A and B are small categories, and suppose that B is aspherical. Let p* : A - Set! (A x B) - Set which is induced by composition with the projection functor p : A x B ! A. Then a map f : X ! Y is a weak equivalence of of A-sets if and only if the induced map f*; p*X ! p*Y is a weak equivalence of (A x B)-sets. Proof. There is an isomorphism i(AxB)p*X ~=iA X x B. |___| Let q* : S ! (A x ) - Setbe the functor which is defined by composition with the projection A x ! . The functor i : A ! S defined by a 7! B(A # a) induces a functor i* : S ! A - Set where i*X(a) = hom (B(A # a), X). Similarly the functor j : A x ! S defined by (a, n) 7! B(A # a) x n defines a functor j* : S ! (A x ) - Set with j*X(a, n) = hom (B(A # a) x n, X). 15 Lemma 11. Suppose that A is a local test category. Then with the definitions above, there are natural weak equivalences of (A x )-sets p*i*X ! j*X q*X for all simplicial sets X. Proof. The map q*X(a, *) ! j*X(a, *) is the simplicial set map X ! hom (B(A # a), X). The contracting homotopy B(A # a) x 1 ! B(A # a) induces a homotopy equivalence X ! hom (B(A # a), X) for all simplicial sets X. It follows that all maps q*X(a, *) ! j*X(a, *) are weak equivalences of simplicial sets, so that the induced map q*X ! j*X is a weak equivalence of (Ax )-sets for all simplicial sets X by Lemma 7. The map p*i*X ! j*X can be identified in simplicial degree n with the A-set map hom (B(A # a), X) ! hom (B(A # a), hom ( n, X)). The contracting homotopy n x 1 ! n induces a contracting homotopy of hom ( n, X) onto X, and hence induces a contracting i*A(1)-homotopy of j*X(*, n) onto p*i*X(*, n) (we need to know that A is a local test category, so that i*A(1) is aspherical, exactly at this point). The category A is a local test category, so all maps p*i*X(*, n) ! j*X(*, n) are weak equivalences_of A-sets. |__| Corollary 12. The functor i* : S ! A - Set preserves weak equivalences if A is a local test category. Proof. The functor q* preserves weak equivalences by Lemma 7, so that p*i* preserves weak equivalences by Lemma 11. The functor p* reflects weak equiv-_ alences by Lemma 10, so i* preserves weak equivalences as claimed. |__| Lemma 11 admits a more general formulation, which will be of some use later. Suppose that i : A ! cat is an arbitrary functor. Then i induces a functor i* : S ! A - Set which is defined by a 7! hom (Bi(a), X). Then the functor j : A x ! S defined by (a, n) 7! Bi(a) x n induces j* : S ! (A x ) - Set, with j*X(a, n) = hom (Bi(a) x n, X). Then the proof of the following is an abstraction of the proof of Lemma 11: Lemma 13. Suppose that A is a small category. Suppose that all categories i(a) have terminal objects, and that the A-set i* 1 is aspherical. Then with the definitions above, there are natural weak equivalences of (A x )-sets p*i*X ! j*X q*X for all simplicial sets X. 16 Corollary 14. Suppose in addition to the assumptions of Lemma 13 that the category A is aspherical, so that A is a test category. Then the functor i* : S* * ! A - Set preserves and reflects weak equivalences. Proof. The functor q* preserves and reflects weak equivalences by Lemma 7, so that p*i* preserves and reflects weak equivalences by Lemma 13. The functor p* preserves and reflects weak equivalences by Lemma 10, so i* preserves_and_ reflects weak equivalences. |__| Here is a source of local test categories: Lemma 15. 1) Suppose that A is a local test category and that X is an A-set. The the category iA X is a local test category. 2)The category of iA X-sets is equivalent to the category A - Set # X of A-set morphisms Y ! X over X. Proof. Suppose that oe : a ! X is an object of iA X. Then there is an isomorphism of categories iA # oe ~=A # a by the Yoneda Lemma. All categories A # a are weak test categories since A is a local test category. It follows that iA X is a local test category. Suppose that Y : (iA X)op! Set is an iA X-set. There is an A-set ~Ywith G ~Y(a) = Y (oe), oe2X(a) and there is plainly an induced A-set morphism ßY : ~Y! X. The assignment Y 7! ßY is functorial in Y . Conversely, if p : Z ! X is a morphism of A-sets, then the assignment oe 7! p-1(oe) Z(a) for oe : a ! X defines a presheaf p-1 on iA X. These two functors are inverse to each other up to isomorphism, so that there is an equivalence of categories iA X - Set' A - Set# X. |___| Now we have a final corollary of Lemma 10 and 11. The following is a relative version of Corollary 12: Corollary 16. Suppose that A is a local test category and that Y is an A- set. Then the functor S ! A - Set defined by X 7! Y x i*X preserves weak equivalences. Proof. Write i*[A]X = i*X, where i*X(a) = hom (B(A # a), X) as in Lemma 11. Then there is an isomorphism i*[iAYX]~=Y x i*[A]X 17 in the category of A-sets over Y . Lemma 15 says that the category of iA Y - sets is equivalent to the category of A-sets over Y . The functor X 7! i*[iAYX] preserves weak equivalences since iA Y is a local test category by Lemma 15. Finally, the forgetful functor from A-sets over Y to A-sets defined by sending_ the object Z ! Y to Z preserves and reflects weak equivalences. |__| 2 Cubical sets: basic properties Write n_= {1, 2, . .,.n}, and let 1n be the n-fold product of copies of the cat* *egory 1 defined by the ordinal number 1 = {0, 1} of the same name. Write 10 for the category consisting of one object and one morphism. A face functor (d, ffli) : 1m ! 1n is defined by an ordered inclusion d : m_* *! n_ and a set of elements ffli 2 {0, 1}, i 2 n_- m_. The corresponding functor is specified by the diagrams (d,ffli) 1m _____//CC1n CCC pri| diCCC!!fflffl|| 1 where diis the projection prd-1(i)if i is in the image of d, and diis the const* *ant functor at ffli for i 2 n_- m_. A degeneracy functor s = sd : 1n ! 1k is specified by an ordered inclusion d : k_! n_, and the diagram sd 1r _____//AA1k AA |pr prd(i)AAA| i __Afflffl| 1 is required to commute. ~ There is an isomorphism of posets n : 1n -=! P(n_) which is defined by associating to the n-tuple ffl = (ffl1, . .,.ffln) the subset n(ffl) = {i | ff* *li= 1} of the set n_= {1, . .,.n}. Suppose that (d, ffli) : 1m ! 1n is a face functor, and consider the composi* *te poset morphism 1n (d,ffli)---!1m --m! P(n_). Suppose that A = m (d, ffli)(0, . .,.0) and that B = m (d, ffli)(1, . .,.1). * *Write [A, B] for the subposet of P(m_) consisting of all subsets C such that A C * *B. The poset [A, B] is often called the interval between A and B. Then one can show that the face functor (d, ffli) can be identified up to isomorphism with t* *he poset morphism d* : P(n_) ! P(m_) which is defined by C 7! d(C) [ B. The odered inclusion d : n_! m_ determines a bijection n_~=B - A, and that d* induces a poset isomorphism P(n_) ~=[A, B]. An ordered inclusion d : k_! n_can be identified with a subset A n_of order k in the obvious way, and any degeneracy sd : 1n ! 1k sits in a commutative 18 diagram sd k 1n ______//1k_~=_//P(k_) n|~=| ~=|| fflffl| fflffl| P(n_)_____________//P(A) where the morphism P(n_) ! P(A) is defined by C 7! C \ A. Consider the composite functor 1m -(d,ffli)--!1n s-!1k There is a pullback diagram of order preserving functions m___d__//n_OOOO s0|| |s| | | r_____//_k_ d0 and there is a corresponding commutative diagram of face and degeneracy func- tors (d,ffli) 1m _____//1n (4) s0|| s|| fflffl| fflffl| 1r (d0,ffls(i))//_1k The sets of face and degeneracy functors are each closed under composition, and degeneracy functors can be öm ved past" face functors according to the recipe specified above. Lemma 17. Suppose given a commutative diagram 1m __s__//1n s0|| |(d,ffli)| fflffl|0fflffl| 1n (d0,ffl0i)//_1k composed of face functors and degeneracies. Then (d, ffli) = (d0, ffl0i) and s * *= s0. The proof is left to the reader. The box category is the subcategory of the category of small categories which is generated by the face and degeneracy functors. Its objects consist of the categories 1k, k 0, and it follows from Lemma 17 that a morphism ` : 1n ! 1m in can be uniquely written as a composite 1nB______`_____//_1m== BBB ____ sBBB__B___d_ 1k 19 where s is a degeneracy functor and d is a face functor. The pair (i, ffl) consisting of i 2 n_and ffl 2 {0, 1} determines a unique f* *ace functor d(i,ffl): 1n-1 ! 1n, defined by d(i,ffl)(fl1, . .,.fln-1) = (fl1, . .,.iffl, . .,.fln-1). Suppose that i < j. Then there is a commutative diagram of face functors d(i,ffl1)//_n-1 1n-2 1 (5) d(j-1,ffl2)|| |d(j,ffl2)| fflffl| fflffl| 1n-1 d(i,ffl1)//_1n if n 2. If i = j there is a diagram ;_______//1n-1 (6) | | (i,1) | |d fflffl| fflffl| 1n-1 d(i,0)//_1n-1 The degeneracy functor sj : 1n ! 1n-1 is the projection which forgets the jth factor, so that sj(fl1, . .,.fln) = (fl1, . .,.flj-1, flj+1, . .,.fln) Write s1 : 1 ! 10 for the obvious map to the terminal object 10 in the box category . Then there are relations sjsi= sisj+1, if i .j (7) Similarly, sjd(j,ffl)= 1, (8) and there are commutative diagrams (i,ffl) 1n _d____//1n+1 if i < j (9) sj-1|| |sj| fflffl| fflffl| 1n-1 d(i,ffl)//_1n and (i+1,ffl) 1n d____//_1n+1 if i j. (10) sj|| |sj| fflffl| fflffl| 1n-1 d(i,ffl)//_1n 20 Lemma 18. The diagrams (5), (9) and (10) are pullbacks in the box category. Proof. A box morphism ff : 1r ! 1n factors through the face d(i,ffl): 1n-1 ! 1n if and only if the images ff(x) = (ff1(x), . .,.ffn(x)) have the form ffi(x)_= * *ffl for all x 2 1r. |__| A cubical set X is a contravariant set-valued functor X : op! Set. Write Xn = X(1n), and call this set the set of n-cells of X. A morphism f : X ! Y of cubical sets is a natural transformation of functors, and we have a category - Set of cubical sets. The standard n-cell n is the contravariant functor on the box category which is represented by 1n. The cell category i X for a cubical set X is defined as in Section 1: the objects of i X are the morphisms oe : n ! X (equivalently n-cells of X, as n varies), and a morphism is a commutative triangle of cubical set morphisms. The nerve functor restricts to a covariant simplicial set-valued functor !* * S which is defined by 1n 7! B(1n) = ( 1)xn This functor can be used to define a cubical singular functor S : S ! - Set, where S(Y )n = hom S(( 1)xn , Y ). This functor has a left adjoint (called triangulation) X 7! |X|, where |X| = lim-!( 1)xn . n!X Here, the colimit is indexed by members of the cell category i X. There are similarly defined realization and singular functors | | : - Setø Top : S relating cubical sets and topological spaces, and of course realization is left adjoint to the singular functor in that context as well. Example 19. Suppose that C is a small category. The cubical nerve B (C) is the cubical set whose n-cells are all functors of the form 1n ! C, and whose structure maps B (C)n ! B (C)m are induced by precomposition with box category morphisms 1m ! 1n. Observe that there is a natural isomorphism B (C) ~=S(BC), where BC is the standard nerve for the category C in the category of simplicial sets. In a cubical set X, write d(i,ffl)for the function Xn ! Xn-1 which is induced by the functor d(i,ffl), and call this function a face map. Similarly, the deg* *en- eracies sj : Xn ! Xn+1 are the functions which are induced by the functors sj : 1n+1 ! 1n. Say that a cell oe 2 Xn is degenerate if it is the image of some sj, and is non-degenerate otherwise. Define the n-skeleton sknX for a cubical set X to be the subcomplex which is generated by the k-cells Xk for 0 k n. 21 Lemma 20. A map f : sknX ! Y of cubical sets is completely determined by the restrictions f : Xk ! Yk for 0 k n, Proof. We want to show that the maps f : Xk ! Yk extend uniquely to a morphism f* : sknX ! Y . Suppose that z 2 sknXn+1. Then z is degenerate, so that z = six for some x 2 Xn, and it must be that f*(z) = sif(x) if the extension exists. Suppose that z is degenerate in two ways, so that also z = sjy for some i < j and y 2 Xn. Then x = d(i,0)six = d(i,0)sjy = sj-1d(i,0)y, while sjsi(d(i,0)y) = sisj-1(d(i,0)y) = six = sjy. All degeneracies are injective, so that y = sid(i,0)y, and sif(x) = sisj-1d(i,0)f(y) = sjsid(i,0)f(y) = sjf(y). Inductively, the map f* : skn(X)r ! Yr for r = k is completely determined_by the maps for r < k in the same way. |__| It follows that there are pushout diagrams F @ n _____//sk X x2NXn n-1 | | | | F fflffl| fflffl|| n ______//_skX x2NXn n where NXn denotes the non-degenerate part of Xn, and @ n = skn-1 n. In other words, there is a good notion of skeletal decomposition for cubical sets. The object @ n is the subcomplex of the standard n-cell which is generated by the faces d(i,ffl): n-1 ! n. It follows from the fact that the diagram (5) is a pullback in the box category that there is a coequalizer G G n-2 ' n-1 ! @ n (ffl1,ffl2) (i,ffl) 0 i \ X0 = skn-1, where denotes the subcomplex of X which is generated by y. This means that there is a pushout diagram @ n ____//_X0 | | | | fflffl| fflffl| n __y___//X The assumption that x* = y* implies that the dotted arrow lifting exists in the solid arrow pushout diagram |@ n| ____//_|X0|;;_ ____ | x*_____|_ | ______ i*| |fflffl____fflffl| | n|__y*__//|X| making it commute. The map i* is an inclusion which is not surjective, since the solid arrow diagram is a pushout. But the existence of the dotted arrow * *__ forces i* to be surjective. This is a contradiction, so x = y. |* *__| 24 Corollary 25. Suppose that f : X ! Y is a map of cubical sets such that the induced simplicial set map f* : |X| ! |Y | is a monomorphism. Then f is a monomorphism of cubical sets. Finally, an argument similar to the proof of Lemma 24 yields the following: Proposition 26. Suppose that f : X ! Y is a map of cubical sets such that the induced simplicial set map f* : |X| ! |Y | is an isomorphism. Then f is an isomorphism of cubical sets. The canonical forgetful functor ß : i X ! for a cubical set X specializes to a forgetful functor ß : i B C ! where B C is the cubical nerve for a small category C _ see Example 19. Lemma 27. Suppose that C has a terminal object t. Then the functor i B C ! is aspherical. In particular, the cubical set B (C) is aspherical, and the cat* *egory i B (C) is weakly equivalent to a point. Proof. We must show that all categories i (B (C)x n) (see (1)) are aspherical. The objects of the category i (B (C) x n) consist of pairs of functors (f : 1k ! C, 1k oe-!1n), and morphisms are defined in the obvious way. The category C has terminal object t, so there are natural diagrams 1kC CCfC d(k+1,0)||CCC fflffl|!!Cf* 1k+1_____//COO==- -- d(k+1,1)||t--- |-- 1k Suppose that s : 1k+1 ! 1k is the degeneracy defined by projection onto the first k factors. Then the assignment (f : 1k ! C, 1k oe-!1n) 7! (f*, oe . s) defines a functor h : i (B (C) x n) ! i (B (C) x n), and the coface maps d(k+1,0)define a homotopy d(k+1,0): (f, oe) ! (f*, oe . s) from the identity on i B (C) to h. The coface maps d(k+1,1)define a homotopy from the endofunc- tor (f, oe) 7! (t, oe) to the functor h. It follows that the category i (B _C_* *x n) is equivalent to i n, and hence is aspherical. |__| 25 Lemma 28. The box category is a test category. Proof. Suppose that D is a category with terminal object. By Lemma 3, in order to show that is a local test category, we must show that all cell categories i (i* D x n) are aspherical. Every poset 1n has a terminal object tn = (1, . .,.1). There is a functor # 1n ! 1n which is defined by sending an object ` : 1m ! 1n to `(tm ). This functor is natural in morphisms of the box category , and induces a cubical set map ff : B C ! i* C which is natural in small categories C. We know from Lemma 27 that the cubical set B (1) is aspherical. Let h : D x 1 ! D be the contracting homotopy for the category D, and consider the induced composite i* D x B (1) 1xff---!i* D x i* (1) ~=i* (D x 1) ! i* (D) Then the projection i* DxB (1) ! i* D is aspherical since B (1) is aspherical. The displayed homotopy also implies that the projection i* D x n ! n is a weak equivalence of cubical sets. Thus, the box category is a local test category. The category is also plainly aspherical because it has a terminal object, so Lemma 4 shows that_it is a test category. |__| Remark 29. Lemma 28 and its proof are part of a general yoga. Suppose that i : A ! catis a functor which is defined on a small category A. Then the A-set i*(C) is defined for a small category C by a 7! hom (i(a), C). Suppose that the following conditions hold: 1)all categories i(a) have terminal objects. 2)if D has a terminal object, then the A-set i*D is aspherical. 3)the category A is aspherical. Then A is a test category. The argument is the same as that given for Lemma 28. This argument appears in the context of the discussion of aspherical functors in [3]. Note that the cubical nerve B C is i*C for the inclusion functor i : * * ! cat, as in Lemma 13. Remark 30. Lemma 27 implies that B 1 is aspherical, and the proof of Lemma 28 implies that I = i* 1 is aspherical, but the cubical set 1 is not aspherical by Remark 22. 3 Fundamental model structures Suppose throughout this section that A is a small category and that C is a small Grothendieck site. We shall write A - Pre(C) for the category of A-presheaves 26 (or presheaves of A-sets) on the site C. Let S denote a set of monomorphisms in the category A - Pre(C). The set S can be empty. The box category is a monoidal category with multiplication : x ! induced by the product functor (1n, 1m ) 7! 1n+m , and with unit object the terminal object * = 10. Note that a monoidal functor ! M taking values in a monoidal category M is completely determined by the image of the maps 0, 1 : * ! 1 in M, so that monoidal functors ! M can be identified with interval objects in M. An interval theory in the category of A-presheaves is a coherent action : A - PreC x ! A - Pre(C) of the box category on the category of A-presheaves, written as (X, 1n) 7! X n, and which is subject to the following conditions: DH0 The map ; ! ; 1 is an isomorphism. DH1 The functor X 7! X 1 preserves monomorphisms and filtered colimits. DH2 For every monomorphism i : X ! Y and every coface d : n-1 ! n the square i 1 X n-1 ____//_Y n-1 1 d || |1|d fflffl| fflffl| X n ______//_i/1/_Y n is a pullback. DH3 For 1 i n the square ; ________//_X n-1 | | (i,0) | |d fflffl| fflffl| X n-1 _d(i,1)//_X n is a pullback. DH4 There is a cardinal number i such that |X n| < ~ if |X| < ~, for all cardinals ~ > i. 27 Write X K = lim-!X n n!K and define a cubical function space hom (X, Y ) for A-sets X and Y by hom (X, Y )n = hom (X n, Y ). Then there is a natural bijection hom(X K, Y ) ~=hom (K, hom (X, Y )) relating morphisms in A-presheaves to cubical set homomorphisms. It follows that the assignment K 7! X K preserves colimits in cubical sets K. Lemma 31. The cubical set inclusion @ n n induces a natural inclusion X @ n ! X n. Proof. The axiom DH2 implies that all squares 1 d(i,ffl1) X n-2 _____//X n-1 1 d(j-1,ffl2)|| 1|d(j,ffl2)| fflffl| fflffl| X n-11_d(i,ffl1)//_X n are pullbacks for i < j. In effect,this diagram is isomorphic to the diagram (1 d(i,ffl2)) 1 (X j-2) n-j _______//(X j-1) n-j 1 d(1,ffl2)|| 1|d(1,ffl2)| fflffl| fflffl| (X j-2) n-j+1 _____//(X j-1) n-j+1 (1 d(i,ffl1)) 1 In the presence of axiom DH3 (which takes care of the intersections of faces not covered by instances of the square above), it follows that the canonical map X @ n ! [(i,ffl)X n-1 is an isomorphism, by comparison of coverings. |___| It follows that any cubical set inclusion K L induces a monomorphism X K ! X L. Example 32. If I is any A-presheaf equipped with a monomorphism (d0, d1) : * t * ! I then the assignment (X, 1n) 7! X x Ixn defines a coherent action I : A - Pre(C) x ! A - Pre(C) of the box category on the category of A-presheaves, and this action satisfies the conditions DH0 - DH3. Note that DH3 follows from the condition that (d0, d1) is a monomorphism. The axiom DH4 is satisfied by any infinite cardinal i such that |I| < i. 28 Example 33. The assignment (X, Y ) 7! X Y defines a monoidal structure on the category of cubical sets, and. this monoidal structure induces a coherent action : - Setx ! - Set of the box category on the category of cubical sets, given by (X, 1n) 7! X n by the obvious restriction of structure. Of the axioms, DH2 is the hardest to prove in this example, but one can verify it by recalling that the "triangulation" functor | | : -Set ! S preserv* *es and reflects monics and isomorphisms, and preserves pullbacks provided that one of the maps being pulled back is a monomorphism. One also needs to know that |X Y | ~=|X| x |Y |. It is easy to show that |X n| = |X| if X has infinite cardinality. Remark 34. Axiom DH2 implies that if i : X ! Y is an inclusion of A- presheaves, then the diagram X @ n ____//_X n i 1|| |i|1 fflffl| fflffl| Y @ n _____//Y n is a pullback. It follows that the canonical map (Y @ n) [X @ n (X n) ! (Y n) [ (X @ n) is an isomorphism. One can then show that, if j : K ! L is a monomorphism of cubical sets, then the induced map (Y K) [(X K) (X L) ! Y L is a monomorphism, so that the map (Y K) [(X K) (X L) ! (Y K) [ (X L) is an isomorphism. Define the class of anodyne ( , S)-cofibrations (or just anodyne cofibration* *s) in the category of A-presheaves to be the saturation of the set consisting of a* *ll inclusions (Y n) [ (LU a un(i,ffl)) ! LU a n (12) arising from all subobjects Y LU a, together with all inclusions (A n) [ (B @ n) ! B n (13) arising from all monomorphisms f : A ! B in the set S. The set ( , S) consisting of all maps of the form (12) and all maps of the form (13) is a set of generators for class of ( , S)-anodyne cofibrations. Note that condition (12) implies that any inclusion C ! D of A-presheaves induces an anodyne S-cofibration (C n) [ (D un(i,ffl)) ! D n. (14) 29 Lemma 35. If j : C ! D is an anodyne ( , S)-cofibration, then the induced map (D @ 1) [ (C 1) ! D 1 is an anodyne ( , S)-cofibration. Proof. We have already seen in Remark 34 that the map (D @ 1) [ (C 1) ! D 1 is a monomorphism. It is enough to prove the statement of the Lemma for maps of the form (12) and (13). The map (((Y n)[(LU a un(i,ffl))) 1)[((LU a n) @ 1) ! (LU a n) 1 can be identified up to isomorphism with the map (Y ( n 1)) [ (LU a ((un(i,ffl) 1) [ ( n @ 1))) ! LU a ( n 1) which is isomorphic to the map (Y n+1) [ (LU a un+1(i,ffl)) ! LU a n+1 by a cubical set isomorphism (11). Similarly, the map (((A n) [ (B @ n)) 1) [ ((B n) @ 1) ! (B n) 1 is isomorphic to the map (A n+1) [ (B @ n+1) ! B n+1. |___| Say that an A-set map p : X ! Y is injective if it has the right lifting property with respect to all anodyne cofibrations. An A-set X is said to be injective if the map X ! * is injective. A naive homotopy is a map X 1 ! Y . Note that naive homotopy of maps f : X ! Z is an equivalence relation if Z is injective, by extension arguments along anodyne cofibrations of the form X un(i,ffl)! X n. Say that a map g : X ! Y is an ( , S)-equivalence (or just an equivalence) if it induces an isomorphism ~= g* : ß(Y, Z) -! ß(X, Z) in naive homotopy classes for all injective objects Z. A cofibration is a monomorphism. An ( , S)-fibration (or a fibration) is a map which has the right lifting property with respect to all maps which are simultaneously cofibrations and ( , S)-equivalences, aka. trivial cofibrations. 30 Lemma 36. All anodyne cofibrations are equivalences. Proof. If i : C ! D is an anodyne cofibration, and f : C ! Z is a map where Z is injective, then the dotted lifting exists in the diagram f C _____//Z>>___ ____ i||________ fflffl|____ D so that the function i* : ß(D, Z) ! ß(C, Z) is surjective. If g1, g2 : D ! Z are maps which become homotopic when restricted to C, then g1, g2 and the homotopy define a map (C 1) [ (D @ 1) ! Z which extends to a map D 1 ! Z by Lemma 35, so g1 and g2 are homotopic. __ Thus, the function i* is injective as well as surjective. |_* *_| Here is a general set of tricks that applies to any set T of monomorphisms g : C ! D of A-presheaves. Suppose that ff is a cardinal such that ff > i where i is a cardinal which satisfies DH4. Suppose further that ff > |C| where C is the underlying small site, and that ff > |A| where A is our choice of local test category. Suppose t* *hat ff > |D| for all morphisms g : C ! D appearing in the set T and that ff > |T |. Choose a cardinal ~ such that ~ > 2ff. Suppose that f : X ! Y is a morphism of A-presheaves. Define a functorial system of factorizations is X _____//EEEs(f) EEE |fs f EEE""Efflffl|| Y of the map f indexed on all ordinal numbers s < ~ as follows: 1)Given the factorization (fs, is) define the factorization (fs+1, is+1) by * *re- quiring that the diagram F ffD // DC ______ Es(f) g*|| || F fflffl| fflffl| D D _____//Es+1(f) is a pushout, where the disjoint union is indexed over all diagrams D of 31 the form ffD C _____//Es(f) g || fs|| fflffl| fflffl| D __fiD_//_Y with C ! D in the set T . Then the map is+1 is the composite X is-!Es(f) g*-!Es+1(f) 2)If s is a limit ordinal, set Es(f) = lim-!t 2ff. Then the natural map lim-!L(Xt) ! L(limXt) t ff, and let Ffl(X) denote the filt* *ered system of subobjects of X having cardinality less than fl. Then the natural map lim-!L(Y ) ! L(X) Y 2Ffl(X) is an isomorphism. 4)If |X| < 2! where ! ~ then |L(X)| < 2!. 5)Suppose that U, V are subobjects of an A-presheaf X. Then the natural map L(U \ V ) ! L(U) \ L(V ) is an isomorphism. 32 Proof. It suffices to prove all of these statements for the functor X ! E1(X). Note as well that E1(X) is defined by the pushout diagram C x hom(C, X) _______//X | | | | fflffl| fflffl| D x hom(C, X) _____//E1(X) Statements 1) and 3) follow, respectively from the fact that the maps lim-!hom(C, Xt) ! hom (C, limXt) t 2ff) and A = A0. Let B = lim-!iAi. Then, by construction, B is 2~-bounded and the restriction of the homotopy K to LB 1 factors through the inclusion j : LB ! LY . The diagram ~j L(B \ X) _____//LX ~i|| |i*| fflffl| fflffl| LB ___j___//_LY 34 is a pullback, and i*oe(LB) LB. It follows that oe restricts to a map oe0 : LB ! L(B \ X). Then ~joe0~i= oej~i= oei*~j= ~j so that oe0~i= 1. Finally, j~ioe0= i*oej by construction, so the restricted hom* *otopy LB 1 ! LB is a homotopy from ~ioe0to the identity. In particular, the_induced map B \ X ! B is an equivalence. |__| We need to know that the class of trivial cofibrations is closed under pusho* *ut, and for that we need to prove the following: Lemma 39. Suppose given a diagram f,g C _____//E i|| fflffl| D where i is a cofibration, and suppose that there is a naive homotopy h : C 1 ! E from f to g. Then the induced map g* : D ! D [gE is an equivalence if and only if f* : D ! D [f E is an equivalence. Proof. There are pushout diagrams d0 h C ________//_C 1_________//_E i|| |i*| i*|| fflffl| fflffl| fflffl| D _d0*_//D [C (C 1)_h0_//D [f E where the top composite is f. There are also pushout diagrams C 1 _____h_______//_E | | | | fflffl| h0 fflffl| D [C (C 1) ________//D [f E j|| j*|| fflffl| fflffl| D 1 ___h*___//_(D 1) [h E The maps d0*, j and j* are anodyne cofibrations. Thus f* = h0.d0*is equivalent to h0 and h0 is equivalent to h*, so f* is an equivalence if and only if h* is * *an equivalence. A similar analysis holds for the induced map g* : D ! D [g E. Thus f* is __ an equivalence if and only if g* is an equivalence. |__| 35 Lemma 40. Suppose that i : C ! D is a trivial cofibration. Then the cofibra- tion (C 1) [ (D @ 1) ! D 1 is an equivalence. Proof. Consider the diagram C @ 1 ____//_D @ 1____//LD @ 1_____//L(LD @ 1) | | | | | | | | fflffl| fflffl| fflffl| fflffl| C 1 ______//D 1_____//_LD 1_____//L(LD 1) Then there is an induced diagram (C 1) [ (D @ 1)____//_(C 1) [ L(LD @ 1) | | | | fflffl| fflffl| D 1 _________________//L(LD 1) in which the horizontal maps are equivalences. There is a factorization 0 C __i__//AAD0 AAA |p i AA__Afflffl|| D where i0is anodyne and p is both injective and an equivalence. In the induced diagram (C 1) [ L(LD0 @ 1) _____//(C 1) [ L(LD @ 1) | | | | fflffl| fflffl| L(LD0 1) ________________//L(LD 1) the top horizontal map is induced by the homotopy equivalence L(LD0 @ 1) ! L(LD @ 1) an equivalence by Lemma 39, and the bottom horizontal map is a homotopy equivalence (see the Remark following this proof). The left hand vertical map is an equivalence by comparison with the map (C 1) [ (D0 @ 1) ! D0 1 and Lemma 35. |___| 36 Remark 41. The map LD0 K ! LD K is an ( , S)-equivalence if D0! D is an ( , S) equivalence, for any cubical set K. To see this note that, if h : X 1 ! X is a homotopy from a map ff : X ! X to the identity on X, then the endpoint maps * ! 1 induce anodyne maps X K ! X 1 K, so the map ff K is an equivalence. It follows that if f : LD0! LD is a homotopy equivalence with homotopy inverse g, then the maps gf K and fg K are equivalences, so that f K is an equivalence. Lemma 42. The class of trivial cofibrations is closed under pushout. Proof. First of all, if j : C ! D is a cofibration and an equivalence, then eve* *ry map ff : C ! Z taking values in an injective object Z extends to a map D ! Z. In effect, there is a homotopy h : C 1 ! Z from ff to a map fi . j for some map fi : D ! Z. Then the lifting H exists in the diagram (h,fi) (C 1) [ (D {1})____//_Z77________ _____ | ________ | ____H_____ fflffl|_____ D 1 since the vertical map is an anodyne cofibration, so ff extends to the morphism H|D {0}. Suppose that the diagram C ____//_C0 j|| |j0| fflffl| fflffl| D ____//_D0 is a pushout, where j is a trivial cofibration. Suppose that the A-presheaf Z is injective. Then every map ff0: C0! Z extends to a map fi0: D0! Z since the diagram is a pushout and j has this extension property. The diagram (C 1) [ (D @ 1)____//(C0 1) [ (D0 @ 1) | | | | fflffl| fflffl| D 1 __________________//D0 1 is a pushout. The left hand vertical map is a trivial cofibration by Lemma 40 and therefore has the left lifting property with respect to the map Z ! * by the argument above. It follows that the induced map j0*: ß(D0, Z) ! ß(C0, Z) is a bijection, so that j0 is an equivalence. |__* *_| 37 Lemma 43. Suppose that the map p : X ! Y is injective and that the object Y is injective. Then p has the right lifting property with respect to all tri* *vial cofibrations, so that p is a fibration. Proof. Suppose given a diagram A __ff_//X (16) i|| |p| fflffl| fflffl| B __fi_//Y where i is a trivial cofibration. Then there is a map ` : B ! X such that ` . i = ff since X is injective. The extension h exists in the diagram (pffprA,(fi,p`)) (A 1) [ (B @ 1)__________//_Y55___________ ______ | ___________ | _____h_______ fflffl|_______ B 1 since the vertical map is a trivial cofibration and Y is injective. Here, prA* * : A 1 ! A is the map induced by the cubical set map 1 ! *. In particular, the diagram (16) is homotopic to the diagram A __ff_//X>>___ ____ i|| `____|p|____ fflffl|_fflffl|___ B _p`__//Y for which the indicated lifting exists, via the diagram ffprA A 1 _____//X ixi|| |p| fflffl| fflffl| B 1 _h___//Y Form the diagram (ffprA,`) (A 1) [ B ____//_X99_____ ____ | _______p| | ________ | fflffl|____ fflffl| B 1 __H____//_Y to show that the required lifting exists for the diagram (16). |_* *__| Lemma 44. Suppose that p : X ! Y is a fibration and an equivalence. Then p has the right lifting property with respect to all cofibrations. 38 Proof. Suppose, first of all, that Y is injective. The map p is a naive homotopy equivalence, so there is a map g : Y ! X and a homotopy h : Y 1 ! Y from p . g to 1Y . The lifting H exists in the diagram g Y _______//_X;;__ 0________ d0|| h_____|p|__ fflffl|___fflffl|__ Y 1 _h___//Y since d0 is an anodyne cofibration and p is injective. Let oe = h0. d1. Then p . oe = 1Y . The map oe is a trivial cofibration. Thus, the lifting exists in the diagram (oe.pr,(1X ,oe.p)) (Y 1) [ (X @ 1)_________________//_X33_______________ _______ | H______________ p| | _______________ | fflffl|____________ fflffl| X 1 ____p_1____//_Y 1pr_//_Y by Lemma 40. It follows that the identity diagram on p : X ! Y is homotopic to the diagram oe.p X _____//X>>___ ____ p|| oe___|p|____ |fflffl_fflffl|____ Y __1__//Y Thus, any diagram A _____//X j|| |p| |fflffl fflffl| B _____//Y is homotopic to a diagram which admits a lifting. It follows that p has the rig* *ht lifting property with respect to all cofibrations. If Y is not injective, form the diagram j X ______//_Z p|| |q| fflffl| fflffl| Y __jY_//L(Y ) where j is an anodyne cofibration and q is injective. Then j is an injective mo* *del for X and the map p is an equivalence, so the injective map q is an equivalence. Then the map q has the right lifting property with respect to all inclusions of A-presheaves by Lemma 43. 39 Factorize the map X ! Y xL(Y )Z as X ____i____//JJW JJJ | JJJ |ß JJ$$fflffl| Y xL(Y )Z where ß has the right lifting property with respect to all cofibrations and i i* *s a cofibration. Write q* for the induced map Y xL(Y )Z ! Y . Then the composite q*ß has the right lifting property with respect to all cofibrations and is ther* *efore a homotopy equivalence and hence an equivalence. The cofibration i is also an equivalence, and it follows that the lifting exists in the diagram 1X X _____//X>>" "" i|| "" |p| fflffl|"fflffl|" Z _q*ß_//Y so that p is a retract of a map which has the right lifting property with_respe* *ct to all cofibrations. |__| Corollary 45. A map p : X ! Y is a fibration and an equivalence if and only if it has the right lifting property with respect to all cofibrations. Proof. Suppose that p has the right lifting property with respect to all cofibr* *a- tions. Then p is a fibration. It is also a homotopy equivalence by a standard_ argument, so it is an equivalence. |__| Theorem 46. Suppose that A is a small category and let C be Grothedieck site. Suppose that the morphism : A - Pre(C) x ! A - Pre(C) is an interval theory for the category of A-presheaves on the site C. Suppose that S is a set of monomorphisms of A-presheaves. Then the category of A- presheaves and the classes of ( , S)-equivalences, ( , S)-fibrations and cofibr* *a- tions together satisfy the axioms for a closed model category. Proof. Corollary 45 and a small object argument based on all inclusions Y LU a together imply that every map f : X ! Y has a factorization f X _____________//BBY>>" BBB """" i BB__B"""p W where i is a cofibration and p is a fibration and an equivalence. 40 Lemma 38 and Lemma 42 together imply that there is a set of trivial cofi- brations A ! B which generates the class of all trivial cofibrations. It follows that every map f : X ! Y has a factorization f X ____________//_@@Y??~ @@ ~~~ j @@__@~~~q Z where j is a trivial cofibration and p is a fibration. We have therefore verified the factorization axiom CM5. The lifting axiom__ CM4 is a consequence of Corollary 45. All other axioms are trivial. |__| Theorem 47. Suppose that A is a small category. Suppose that the interval theory I : A - Pre(C) x ! A - Pre(C) is defined by an interval I in the sense that Z n = Z x Ixn Suppose further that all cofibrations in the set S pull back to weak equivalenc* *es along all fibrations p : X ! Y with Y fibrant. Then the (I, S)-model structure on the category of A-presheaves is proper. Proof. Write W for the class of all maps f : U ! V such that the induced map f* is an equivalence in all diagrams f* U xY X _____//V xY X_____//X | | |p | | | fflffl| fflffl| fflffl| U _____f____//_V_______//Y for all fibrations p with Y fibrant. The class includes all projections K n = K x Ixn ! K. The class W is closed under pushout, and an iterated pushout argument there- fore implies that all projections K un(i,ffl)! K are members of W. It follows that all generating anodyne cofibrations (Y n) [ ( a un(i,ffl)) ! a n are in W. All maps f : A ! B in the set S are in W by assumption. It follows by induction on n using comparisons of pushout diagrams C @ n-1 ____//_C un(i,ffl) | | | | fflffl| fflffl| C n-1 ______//C @ n 41 that all morphisms f 1 : A @ n ! B @ n are in W, and hence that all morphisms f 1 : A K ! B K are in W for all cofibrations f 2 S. The class W is closed under retractions and transfinite compositions as well as pushout, so all anodyne cofibrations are in W. Suppose that p : X ! Y is a fibration with Y fibrant, and consider a diagram X |p| |fflffl A __i__//B_fi_//Y where i is a trivial cofibration. Then there is a diagram X |p| fflffl| A ____i___//B__fi__//_Y<>___ ____ i|| _____|q|___ fflffl|_fflffl|___ V 0_p*ß_//W Every equivalence f : X ! Y has a factorization f = q .j where q is a trivial fibration and j is a trivial cofibration. It follows that every equivalence_pu* *lls back to an equivalence along all fibrations. |__| Example 48. Suppose that C is a small Grothendieck site and let A be the ordinal number category . The category of -presheaves on C is the category s Pre(C) of simplicial presheaves on C, which is well known [7] to have a cofi- brantly generated simplicial model structure for which the weak equivalences are the local weak equivalences and the cofibrations are the monomorphisms. Pick a generating set S of trivial cofibrations for this theory. Let 1 denote * *the interval theory associated to the constant simplicial presheaf on the simplicial set 1 with the inclusions of its vertices. The associated ( 1, S)-model stucture is the standard model structure on s Pre(C). In effect, every injective object for this theory is globally fibrant* * in the usual sense, and the injective model construction j : X ! LX is a local weak equivalence as well as a cofibration. A map f : X ! Y of simplicial presheaves is a local weak equivalence if and only if the induced map LX ! LY of fibrant models is a naive homotopy equivalence, and this is equivalent to f being an ( 1, S)-equivalence. Example 49. If C = *, then the category of simplicial presheaves on C is the category S of simplicial sets, and local weak equivalences specialize to weak equivalences of simplicial sets in the standard sense. The generating set S of trivial cofibrations can be taken to be the set of inclusions nk n, and the interval is 1 with the inclusion of its two vertices. The associated ( 1, S)- model structure on S is the standard model structure. Example 50. Suppose that C is a small Grothendieck site and f : A ! B is a monomorphism of simplicial presheaves on C. One formally inverts f in the homotopy category [6] by enlarging the generating set S of local trivial cofibrations (Example 48) by adjoining the set of cofibrations (Y x B) [ (A x LU n) ! B x LU n arising from all inclusions Y LU n of simplicial presheaves which are freely generated by simplices in sections, where U ranges over the objects of C. The resulting set Sf, together with the interval structure 1, gives the ( 1, Sf)- model structure on s Pre(C). This model structure is the f-local model structure 43 for s Pre(C), since every injective model for the ( 1, Sf)-model structure is a fibrant model for the f-local model structure. Example 51. Suppose that X is a scheme of finite dimension, and let C be the site (Sm|X )Nis of smooth schemes over X with the Nisnevich topology [5], [16], [8]. The motivic model structure for the category of simplicial presheaves on the smooth Nisnevich site (Sm|X )Nis is the f-local theory, where f : * ! A1 is some choice of rational point. It follows from Example 50 that the motivic model structure on s Pre(Sm|X )Nis is the ( 1, Sf)-model structure. One can take a different approach, by specifying the interval theory A1 to be the theory arising from the presheaf represented by the X-scheme A1, with the rational points 0, 1 : * ! A1 as endpoints. Let S be the generating set of trivial cofibrations for the ordinary local model structure on s Pre(Sm|X )Nis as in Example 48. Then the (A1, S)-model structure on s Pre(Sm|S)Nis is the motivic model structure on that category. Example 52. The category -Set is the case C = * of the category -Pre(C) of presheaves on C taking values in the category of cubical sets. There is an interval theory : - Setx ! - Set which is specified by (X, 1n) 7! X n _ see Example 33 in Section 2. In this case, take S = ;. The monomorphisms in the category of cubical sets are generated by all inclusions @ n n (these take the place of the inclusions Y LU a for this theory). It follows that the injective maps in the ( , ;)-model structure * *for cubical sets are those maps p : X ! Y which have the right lifting property with respect to all inclusions un(i,ffl) n. Every weak equivalence f : X ! Y in this model structure induces a weak equivalence f* : |X| ! |Y | of the asso- ciated topological realizations. We shall see later (as a consquence of Theorem 90) that maps which induce weak equivalences of topological realizations are exactly the weak equivalences for this model structure. It will also come from a more sophisticated analysis that the model structure for cubical sets is proper (Theorem 85) and that the fibrations are exactly the injective maps (Theorem 88). 4 Homotopy colimits Suppose that A is a small category, and that : A - Setsx ! A - Sets is an interval theory on the category of A-sets (ie. A-presheaves for which the site C = *). Let S be a set of cofibrations of A-sets. We shall be primarily interested in ( , S)-model structures M on the cate- gory of A-sets for which the following assumptions are satisfied: 44 M1 Every map a ! * is a weak equivalence of M. M2 The projection X x i*A(1) ! X is a weak equivalence of M for all A-sets X. That said, much of what follows does not depend on these assumptions. They will be specifically invoked as needed. We say that the model structure M is regular (or that its class of weak equivalences is regular) if the map holim---! a ! X a!X is a weak equivalence of M for all A-sets X. Homotopy colimits are constructed internally in the model M. This con- struction is "standard" but perhaps not yet widely known, and will be sum- marized here. It will also be related to the internal nerve Bh(C) for a small category C in the model category M. If X : I ! A - Set is a functor defined on a small category I, then one defines the homotopy colimit holim---!i2IX(i) in M by setting holim---!X(i) = limZ(i) i2I -!i where ß : Z ! X is a pointwise trivial fibration and Z is a projective cofibrant I-diagram. To explain, when we say that a map f : X ! Y of I-diagrams has the property P pointwise, we mean that all constitutent maps f : X(i) ! Y (i) of A-sets have the property P. In particular, a map ß : Z ! W is a pointwise trivial fibration if and only if all maps ß : Z(i) ! Y (i) are trivial fibratio* *ns of M. Recall (see, for example, [1]) that, since M is cofibrantly generated, there is a model structure on the category of I-diagrams I ! M for which the weak equivalences and fibrations are defined pointwise. The cofibrations for the the* *ory are called projective cofibrations, and the model structure on the category of * *I- diagrams is called the projective model structure. Observe that if f : Z ! Z0 is a pointwise weak equivalence of projective cofibrant I-diagrams then it has a factorization f Z ____________//_AAZ0>>_ AAA ____ i AA__A___q W where i is a trivial projective cofibration and q is left inverse to a trivial * *pro- jective cofibration. The colimit functor takes trivial projective cofibrations * *i to 45 trivial cofibrations of M; in effect, the colimit functor is left adjoint to th* *e con- stant functor from A - Set to I-diagrams in A-sets, and the constant functor preserves fibrations and trivial fibrations. It follows that the homotopy type in M of the homotopy colimit holim---!iX(i) is independent of the choice of projective cofibrant model ß : Z ! X. It also follows that any pointwise weak equivalence f : X ! Y of I-diagrams induces a weak equivalence f* : holim---!iX(i) ! holim---!iY (i) in M. Example 53. The construction just given specializes to the standard descrip- tion of homotopy colimit for simplicial sets, up to natural weak equivalence. To see this, recall [6, VII.2.3] that the homotopy inverse limit holim---X o* *f a small diagram X : I ! S of Kan complexes can be defined by holim---X = hom (B(I #?), X) where the function complex construction takes place in the category SI of I- diagrams in simplicial sets. . It is also shown in [6] that if all objects X(i) of the diagram X are Kan co* *m- plexes and if j : X ! Z is a (globally) fibrant model for X in the model catego* *ry of I-diagrams with pointwise weak equivalences and pointwise cofibrations, then there is a weak equivalence holim---X ~=lim-Z. It's worth repeating the proof here: the map j induces a weak equivalence j* : holim---X ! holim---Z by comparison of towers of fibrations, and the induced map hom (*, Z) ! hom (B(I #?), Z) is a weak equivalence since the map B(I #?) ! * is a pointwise weak equivalence of I-diagrams (all I-diagrams are cofibrant) and Z is globally fibrant. The homotopy colimit holim---!IX is defined dually, so that there is a natur* *al isomorphism of function spaces hom (holim---!IX, Y ) ~=holim---Iophom(X, Y ) for all simplicial sets Y , as in [2, XII.4.1]. This isomorphism forces holim--* *-!X to be the co-end of the diagrams *x1 B(j # I) x X(i)ff__//B(i # I) x X(i) 1xff*|| fflffl| B(j # I) x X(j) 46 arising from all morphisms ff : i ! j of I. It is then an exercise to show that the object holim---!IX is isomorphic to the diagonal of the bisimplicial set G X(i0), i0!...!in which is the standard description. Now suppose that ß : Z ! X is a projective cofibrant model for the I diagram X. Then I claim that there is a weak equivalence lim-!Z ' holim---!X, where holim---!X has the standard definition [2],[6]. In effect, the map ß induces a weak equivalence ß* : holim---!Z ! holim---!X by standard results about bisimplicial sets. If Y is a Kan complex then the function complex hom (Z, Y ) is a globally fibrant Iop-diagram, by an adjunction argument. There is a commmutative diagram hom (lim-!IZ, Y_)~=__//lim-Iophom(Z, Y ) æ*Z|| |æ*| fflffl| fflffl| hom (holim---!IZ,_Y~)=//_holim---Iophom(Z, Y ) and the map æ* is a weak equivalence since hom (Z, Y ) is a globally fibrant Iop-diagram. The induced map æ*Zis a weak equivalence for all Kan complexes Y , so that the canonical map æZ is a weak equivalence of simplicial sets. Remark 54. The usual model structure on the category S of simplicial sets is the primary example of a regular model structure M on a category of A-sets. In this case, A is the category of finite ordinal numbers _ see Example 49. The fact that a simplicial set Y is a homotopy colimit of its simplices in the sense that the map holim---! n ! Y n!Y is standard, and is usually seen [6, IV.5.2] as a consequence of a result of Qu* *illen which asserts that if f : X ! Y is a map of simplicial sets then the induced map holim---! n xY X ! X n!Y is a weak equivalence. This result is in fact equivalent to regularity for the standard model structure on simplicial sets _ see Corollary 61 below. 47 Lemma 55. 1) Suppose that the diagram A __ff_//X i|| || fflffl| fflffl| B ____//_Y is a pushout in the category of A-sets, where i is a cofibration. Then the canonical map from the homotopy colimit of the diagram B- iA ff-!X to Y is a weak equivalence of M. 2)Suppose that a diagram X0 ! X1 ! . . . indexed by some ordinal number consist of cofibrations. Then the canon- ical map from the homotopy colimit of this diagram to lim-!iXi is a weak equivalence of M. Proof. For part 1) find a factorization >X0>" j""""|p|" "" fflffl| A ___ff//_X where p is a trivial fibration and j is a cofibration. Then the diagram B- iA j-!X0 is projective cofibrant, and one can use a standard patching lemma argument since all A-sets are cofibrant in M. __ For part 2), observe that the given diagram is projective cofibrant. |_* *_| Suppose that f : I ! J is a functor between small categories, and that X : I ! A-Set is a functor on I. One defines the homotopy left Kan extension Lf*X : J ! A - Set by setting Lf*X = f*Z where ß : Z ! X is a pointwise trivial fibration and Z is projective cofibrant. Here f*Z denotes the left Kan extension of Z along f; it is defined for j 2 J by setting f*Z(j) = lim-!Z(i). f(i)!j Note that the functor f* is left adjoint to restriction along the functor f, which is denoted by f*. The restriction functor f* clearly preserves pointwise fibrations and pointwise weak equivalences, so that the functor f* preserves projective cofibrations and trivial projective cofibrations. It follows in part* *icular 48 that the object Lf*X = f*Z is cofibrant, and that the homotopy type of Lf*X in the projective model structure of J-diagrams in M is independent of the choice of cofibrant resolution Z up to pointwise weak equivalence. Once again, if ff : X ! Y is a pointwise equivalence of I-diagrams in M, then the induced map ff* : Lf*X ! Lf*Y of J-diagrams in M. Note that left Kan extension along the functor I ! * is just the colimit, and that left Kan extensions compose up to natural isomorphism. The latter statement means that if I f-!J g-!K are composable functors between small categories, then there is a natural iso- morphism of functors g*f* ~=(gf)*. (17) It follows that if ß : Z ! X is a projective cofibrant resolution of a diagram X : I ! M, then there are identifications Lg*(Lf*X) = g*(Lf*X) = g*(f*Z) ~=(gf)*Z = L(gf)*X (18) where the first identification follows from the fact that Lf*X = f*Z is project* *ive cofibrant. Suppose that C is a small category, and define the internal nerve BhC in M by setting Bh(C) = holim---!* c2C In other words, Bh(C) = lim-!Z(c), c2C where Z ! * is a cofibrant resolution of the functor * : C ! A - Set which takes all objects of C to a point. Any functor f : C ! D induces a map f* : Bh(C) ! Bh(D), albeit some- what non-canonically. Suppose that ßC : ZC ! * and ßD : ZD ! * are projective cofibrant resolutions in the categories of C-diagrams and D-diagrams respectively. Then ßD is a pointwise trivial fibration, so that the restricti* *on f*ßD : f*ZD ! * is a pointwise trivial fibration. It follows that the lifting ` exists in the diagram f*ZD<< xx x`xxx f*ßD|| xx fflffl| ZC __ßC___//* and any two such lifts are homotopic. The composite lim-!ZC (c) ! limf*ZD (c) ! limZD (d) c -!c -!d defines a map Bh(C) ! Bh(D), and this map is well defined in the homotopy category. In this way, the association C 7! Bh(C) defines a functor cat ! Ho (M). 49 Lemma 56. Suppose that the small category D has a terminal object, and that X : D ! A - Set is a functor. Then there is a weak equivalence holim---!X ! X(t) in M. In particular, the map BhD ! * is a weak equivalence of M. Proof. Suppose that t is the terminal object of D and let Z ! X be a projective cofibrant resolution of the D-diagram X. Then there is an isomorphism lim-!Z ~= Z(t), and there is a weak equivalence Z(t) ! X(t) which is part of the structur* *e_ of the projective cofibrant resolution. |__| Lemma 57. Suppose that X is a set which is identified with a discrete category. Then there is a weak equivalence BhX ! X in M. Proof. An X-diagram in A-sets is a collection {Zx} of A-sets Zx indexed by the elements x 2 X, and there is an isomorphism G lim-!{Zx} ~= Zx. x2X If {Zx} ! {*} is a projective cofibrant resolution, then each of the trivial fibrations Zx ! *Fhas a section * ! Zx which is a trivial cofibration. The induced map X ! x Zx is a trivial cofibration of M, so the map G BhX = Zx ! X x is a weak equivalence of M. |___| Lemma 58. Suppose that f : C ! D is a functor between small categories. Then there is a canonical weak equivalence holim---!Bh(f # c) ! Bh(C) d2D in the model structure M. Proof. Consider the functors C f-!D ! * and choose a projective cofibrant resolution ß : Z ! * in the category of C- diagrams. There is an identification lim-!Z(c) ~=lim-!lim-!Z(c) c2C d2Df(c)!d on account of the isomorphism (17). The restriction functor Q* defined by the forgetful functor Q : f # d ! C has a right adjoint Q!defined by Q!F (c) = lim- F (ff) c!c0,ff0:f(c0)!d 50 where the inverse limit is computed over all pairs of diagrams ?c0? f(c0) ~~~| CCC 0 ~~ | | CffC ~~~ | || CCC c@ |fi| f(fi)| !!d== @@@ | | --- @@ | | -- @Øfflffl|Ø fflffl||ff00--- c00 f(c00) The index category has one component for each morphism ! : f(c) ! d in D, and each such component contains an initial object defined by the pair of arrows c 1-!c, f(c) !-!d. It follows that Y Q!F (c) = F (!) !:f(c)!d In particular, the functor Q!preserves pointwise fibrations and pointwise trivi* *al fibrations, and so the restriction functor Q* preserves projective cofibrations* * as well as pointwise trivial fibrations. It follows that lim-!Z(c) = lim-!Q*Z(!) ' Bh(f # d). !:f(c)!d !:f(c)!d for all objects d of D. |___| Lemma 59. Suppose that the functors f : C ! D and g : D ! C define a homotopy equivalence of categories. Then the induced maps f* : BhC ! BhD and g* : BhD ! BhC are weak equivalences of M. Proof. The assumption that the functors f and g define a homotopy equivalence in catmeans that there are natural transformations between both f . g and g . f and the respective identity functors. Suppose that a category E has a terminal object and consider a projection pr : C x E ! C. Then there is an isomorphism pr # c ~= C # c x E for each object c 2 C. The category C # c x E has a terminal object, so that the projection C # c x E ! C # c induces a weak equivalence Bh(C # c x E) ! Bh(C # c) for each c 2 C by Lemma 56. It follows from Lemma 58 that the map Bh(C x E) ! BhC is a weak equivalence. Suppose that the functor h : C x 1 ! D is a homotopy of functors f1, f2 : C ! D. The projection functor pr : C x 1 ! C induces a weak equivalence Bh(C x 1) ! BhC, so that the two canonical inclusions C ! C x 1 induce the 51 same map BhC ! Bh(C x 1) in the homotopy category. It follows that f1 and f2 induce the same map in the homotopy category. The composites fg and gf are both homotopic to identity functors. It follows that the induced functors (fg)* : BhD ! BhD and (gf)* : BhC ! BhC are isomorphisms in the homotopy category Ho(M), so f* is an isomorphism in_the homotopy category. |__| Corollary 60. Suppose that f : X ! Y is an A-set morphism. Then there is a cononical weak equivalence holim---!Bh(iA ( a xY X)) ! Bh(iA X) oe: a!Y in the model structure M. Proof. Apply Lemma 58 to the induced functor f* : iA X ! iA Y , and observe that there is an isomorphism f* # oe ~=iA ( a xY X) for each oe : a ! Y . |___| Corollary 61. Suppose that the model structure M on the category of A-sets satisfies the property M1 and is regular. Suppose that f : X ! Y is a map of A-sets. Then the canonical maps a xY X ! X induce a weak equivalence holim---!( a xY X) ! X. a!Y in the model structure M. Proof. Apply Corollary 60, and observe that the regularity condition and M1 together imply that there are natural weak equivalences Bh(iA (Z)) ' Z for all A-sets Z. |___| Corollary 62. Suppose that the model structure M on the category of A-sets satisfies the conditions M1, M2 and is regular. Then there are natural weak equivalences i*AC holim---! a ! Bh(iA i*AC) ! BhC a!i*AC for all small categories C in the model structure M. Proof. The fibres ffl # c of the functor ffl : iA i*AC ! C have the form ffl # * *c ~= iA i*A(C # c), by Lemma 1. The maps i*A(C # c) ! * are weak equivalences of M by M2, since the contracting homotopy (C # c) x 1 ! (C # c) induces a morphism i*A(C # c) x i*A(1) ! i*A(C # c). Thus, the map ffl induces a weak equivalence BhiA i*AC ! BhC. The other two displayed morphisms are weak equivalences by, respectively, the regularity assumption and a comparison_of homotopy colimits. |__| 52 Suppose given a small diagram F : I ! cattaking values in small categories. Recall that the Grothendieck construction sIF (also denoted by some variant of si2IF (i)) is a category having all pairs (x, i) with i 2 I and x 2 F (i) as ob* *jects. The morphisms of this category are the pairs (f, ff) : (x, i) ! (y, j) such that ff : i ! j is a morphism of I and f : ff*(x) ! y is a morphism of F (j). There are a few things to know about Grothendieck constructions: Lemma 63. Suppose that f : C ! D is a functor between small categories. Then there is a natural homotopy equivalence Q : s (f # d) ! C d2D in the category cat of small categories. Proof. The functor Q is the forgetful functor which sends a pair (f(c) ! d, d) to the object c 2 C. We shall display a functor i : C ! s (f # d) d2D such that the composite Q . i is the identity. We shall also show that there is a natural transformation (or homotopy) from i . Q to the identity functor on sd(f # d). The Grothendieck construction sd(f # d) is isomorphic to a categoryw which has as objects all morphisms fi : f(c) ! d of D, and the morphisms are com- mutative diagrams f(ff) f(c)____//_f(c0) (19) fi|| |fi0| fflffl| fflffl| d____`___//d0 where ff : c ! c0is a morphism of C and ` : d ! d0is a morphism of D. From this point of view, the functor Q is defined by sending the morphism (19) to the arrow ff : c ! c0of C. There is a functor i : C ! sd(f # d) which sends the morphism ff to the diagram f(ff) f(c)____//_f(c0) 1|| |1| fflffl| fflffl| f(c)_f(ff)//_f(c0) The composite Q . i is the identity, and the diagrams f(c)__1__//f(c) 1 || |`| fflffl| fflffl| f(c)__`___//d 53 define a natural transformation i.Q ! 1 of functors from sd(f # d) to itself. * *|___| There is a canonical functor ß : sIF ! I for any diagram F : I ! cat of small categories, which is idefined by ß(x, i) = i. There is a functor fi : F (i) ! ß # i which is defined by the assignment x 7! 1i : ß(x, i) ! i. There is a functor gi : ß # i ! F (i) which is defined by sending the morphism ff : ß(j, y) ! i to ff*(y) 2 F (i). The functors gi are natural in i, and one sees that gi. fi = 1 for all i 2 I. For each object ff : ß(j, y) ! i there is commutative diagram (ff,1) ß(j,Dx)__________//_ß(i, ff*(x)) DDD uuuu ffDDDD uuu1u Düüzz i and the collection of all such diagrams defines a homotopy from the identity on ß # i to fi. gi. We have proved the following: Lemma 64. For any small diagram F : I ! cat there is a natural homotopy equivalence gi: F (i) ! ß # i, where ß : sIF ! I is the canonical functor. Recall that M denotes the ( , S)-model structure on the category of A-sets, where is an interval theory and S is a set of cofibrations of A-sets which become weak equivalences in M. Corollary 65. There is a weak equivalence holim---!BhF (i) ! Bh(s F ) i2I I in M for any small diagram F : I ! cattaking values in small categories. Proof. There is a weak equivalence holim---!Bh(ß # i) ! Bh(s F ) i2I I by Lemma 58. Now use Corollary 59 and Lemma 64 to identify Bh(ß # i) with_ BhF (i). |__| Corollary 66. Suppose that f : F ! G is a natural transformation of I- diagrams of small categories such that each induced map BhF (i) ! BhG(i) is a weak equivalence of M. Then the induced map Bh(s F ) ! Bh(s G) I I is a weak equivalence of M. 54 Suppose that F : I ! A-Set is a small diagram of A-sets. Then i 7! iA F (i) is a diagram of categories. The corresponding Grothendieck construction sIiA F is isomorphic to the category whose objects are all morphisms a ! F (i), and whose morphisms are all commutative diagrams a ___`__//_ b x|| |y| fflffl| fflffl| F (i)ff*_//F (j) where ff : i ! j is a morphism of I. Note that this category also coincides up to isomorphism with the Grothendieck construction s hom ( a, F ) a2A associated to the A-presheaf of categories a 7! hom ( a, F ), where hom ( a, F ) is the category with objects x : a ! F (i) and with morphisms ff : x ! y defined by morphisms ff : i ! j in I such that the diagram a F FF x|| FFyFF fflffl|F## F (i)ff*_//F (j) commutes. The canonical A-set maps F (i) ! lim-!iF induce a functor of A-diagrams of categories : hom ( a, F ) ! lim-!F (i)(a), i where the set lim-!iF (i)(a) has been identified with a discrete category. In g* *en- eral, if X is an A-set which is identified with a presheaf a 7! X(a) taking val* *ues in discrete categories, then the Grothendieck construction saX(a) is isomorphic to the category iA X. It follows that the functor induces a functor _ : s iA F (i) ! iA (limF (i)) i2I -!i Note that the category hom ( a, F ) is isomorphic to the Grothendieck con- struction siF (i)(a) of the functor taking values in discrete categories given * *by i 7! F (i)(a). Lemma 67. The functor _ induces a weak equivalence _* : Bh( s iA F (i)) ! Bh(iA (limF (i))) i2I -!i of M in the following cases: 55 1)I is the category 0_____//2 | | fflffl| 1 and F (0) ! F (1) is a monomorphism. 2)I is an ordinal number poset and all maps F (s) ! F (t) are monomor- phisms. Proof. By Lemma 57 and Corollary 66, it suffices to show that the natural transformation F (i)(a) ! lim-!F (i)(a) i of I-diagrams in discrete categories induces a weak equivalence Bh(s F (i)(a)) ! Bh(limF (i)(a)) ~=limF (i)(a) i -!i -!i in both cases under consideration. We know from Corollary 65 that there is an equivalence holim---!BhF (i)(a) ! Bh(s F (i)(a)), i2I i and each BhF (i)(a) is equivalent to the discrete A-set F (i)(a) by Lemma 57. Finally, the canonical map holim---!F (i)(a) ! limF (i)(a) i2I -!i2I is a weak equivalence in cases 1) and 2), by Lemma 55. |___| Remark 68. Lemma 58 suggests a way to avoid the problem of the non- functoriality of the assignment C 7! BhC. Suppose given a small diagram C : I ! cat of small categories, and form the Grothendieck construction sIC. Let ß : sIC ! I be the canonical functor, and suppose that Z ! * is a projec- tive cofibrant resolution of the point over siCi. Then the restriction Qi*Z ! * is a projective cofibrant resolution of the point over ß # i, so that lim-!Qi*Z represents Bh(ß # i) and thus has the homotopy type of BhCi. The diagram i 7! lim-!Qi*Z is functorial in i and thus represents a diagram i 7! Bh(Ci) up to weak equiva- lence. If ff : J ! I is a functor and C is the same I-diagram of small categories then there is an induced commutative diagram of functors sjCff(j)ff__//siCi ß|| |ß| fflffl| |fflffl J ___ff___//_I 56 Choose a cofibrant resolution Z ! * over siCi as above and choose a cofibrant resolution Z0 ! * over sjCff(j). Choose also a map `ff: Z0 ! ff*Z. Then the maps ß # j ! ß # ff(j) induce natural weak equivalences lim-!Qj*Z0! lim-!Qff(j)*Z, so the Bh construction is insensitive to the "change of universes" given by re- striction along ff : J ! I. We shall need a more precise approach to regularity in applications. Say that an A-set is regular in M if the map holim---! a ! X a!X is a weak equivalence of M. From this point of view, the model structure M is regular if and only if all A-sets are regular in M. Lemma 69. 1) Suppose that the diagram X1 _____//X3 i || || fflffl| fflffl| X2 _____//X4 is a pushout and that i is a cofibration. Then if X1, X2 and X3 are regular in M so is X4. 2)If X0 ! X1 ! ... is a totally ordered system of cofibrations between objects which are regu* *lar in M, then lim-!Xi is regular in M. Proof. The diagram BhiA X1 _____//BhiA X3 | | | | fflffl| fflffl| BhiA X2 _____//BhiA X4 is homotopy cocartesian in M: this follows from Corollary 65 and Lemma 67. It follows that the corresponding diagram of homotopy colimits holim---!_a___//holim---! a a!X1 a!X3 | | | | fflffl| fflffl| holim---!_a___//holim---! a a!X2 a!X4 57 is also homotopy cocartesian, and then the map holim---! a ! X4 a!X4 is a weak equivalence of M by a patching lemma argument. __ The statement 2) has a similar proof. |__| 5 Homotopy theories for test categories Suppose that A is a test category, and let the interval I = i*A(1) define an interval theory I : A - Pre(C) x ! A - Pre(C) on the category of A-presheaves on a small Grothendieck site C. Let 1 denote the interval theory on the category s Pre(C) which is asso- ciated to the simplicial set 1 and its inclusions of vertices (see Examples 48, 50). Suppose that S is a set of cofibrations of simplicial presheaves such that* * the class of weak equivalences for the associated (S, 1)-model structure on s Pre(* *C) contains all ordinary local equivalences. Say that a map f : X ! Y of A-presheaves is an S-equivalence if the induced map i* iA (X) ! i* iA (Y ) is a ( 1, S)-equivalence of simplicial presheaves. Since there are natural weak equivalences of simplicial sets i* (C) ' Bi i* (C) ffl*-!BC for any small category C, one sees that f : X ! Y is an S-equivalence of A-presheaves if and only if the induced map BiA X ! BiA Y is an ( 1, S)- equivalence of simplicial presheaves. It is a consequence of Lemma 2 that the maps * i*Ai i* iA (X) iAffl--!i*AiA (X)- jX (20) are S-equivalences for all A-presheaves X. Similarly, for each simplicial presh* *eaf Y the natural morphisms * (ffl) j i* iA i*Ai (Y ) i---!i* i (Y )- Y (21) are local weak equivalences of simplicial presheaves. Choose an infinite cardinal i such that |i*A(1)| < i. Choose a cardinal ff such that ff > i and ff is larger than |C| and |A|. Suppose that ff > |D| for a* *ll morphisms C ! D in the set of cofibrations of simplicial presheaves S and that ff > |S|. Finally choose a cardinal ~ such that ~ > 2ff. The öb unded cofibration" statement Lemma 38 says in the case at hand that, given a diagram X i|| fflffl| A _____//Y 58 of cofibrations of simplicial presheaves such that i is an ( 1, S)-equivalence and |A| < 2~, there is a subobject B Y with A B such that |B| < 2~ and B \ X ! B is an ( 1, S)-equivalence. We shall prove the corresponding statement for cofibrations and S-equivalences in the category of A-presheaves, subject to the choices of cardinals made above. Lemma 70. Suppose given a diagram X i|| fflffl| A _____//Y of cofibrations of A-presheaves such that i is an S-equivalence and |A| < 2~. Then there is a suboject B Y with A B such that |B| < 2~ and B \ X ! B is an S-equivalence. Proof. The induced diagram i* iA X |i*| fflffl| i* iA A_____//i* iA Y of cofibrations of simplicial presheaves satisfies the conditions of Lemma 38. Thus, there is a subobject ff : B1 i* iA Y with |B1| < 2~ such that i* iA A B1 and such that the restricted map B1 \ i* iA X ! B1 is an equivalence of simplicial presheaves. Write C1 = iA A [ iA B1 for the smallest subobject of iA Y which contains i A and the image of the adjoint map ff* : i B1 ! iA Y in the category of presheaves of categories. The presheaf of categories C1 is 2~-bounded in the sense that its presheaves of morphisms and arrows both have cardinality bounded above by 2~. The subobject C1 iA Y is contained in a (smallest) subobject C2 which is a sieve in the sense that whenever a ! X(U) is an object of C2(U) and ` : b ! a is a morphism of A, then the morphism _______`_______//_a bE EE yyy EEE yoeyy E""E __yy X(U) 59 is in D1(U). The subobject D1 is 2~-bounded. Furthermore, there is a subobject A1 Y such that i A1 = D1. In effect, A1(U)(a) = { oe(1a) | oe : a ! Y (U) is an object of D1(U)}. Note that |A1| < 2~. We have therefore found a 2~-bounded subobject A1 Y such that A A1, B1 i*Ai A1, and such that the cofibration i* iA A ! i* iA Y has a factoriza- tion i* iA A B1 i* iA A1 ! i* iA Y Continue inductively to produce families of subobjects Bi Bi+1 i* iA Y and subobjects A Ai Ai+1 Y such that i* iA Ai Bi+1 i* iA Ai+1. where i < fl and fl is a cardinal with 2ff< fl. Write B = lim-!Ai. The functor i* iA preserves filtered colimits of size fl * *by the assumptions on the cardinal fl, as well as monomorphisms and pullbacks. It follows that the induced map i* iA (B \ X) ! i* iA B is a filtered colimit of the maps Bi\ i* iA (X) ! i* iA Bi and is therefore a trivial cofibration of simplicial presheaves. Note as_well t* *hat |B| < 2~ by construction. |__| Theorem 71. Suppose that A is a test category and let C be a small Grothen- dieck site. Suppose that S is a set of cofibrations of simplicial presheaves on C such that the class of all weak equivalences in the resulting ( 1, S)-model structure on the category of simplicial presheaves contains all local equivalen* *ces. Then there is model structure on the category of A - Pre(C) for which the weak equivalences are the S-equivalences and the cofibrations are the monomorphisms. There is an equivalence Ho(s Pre(C))( 1,S)' Ho(A - Pre(C))S of the associated homotopy categories. Proof. Say that a map p : X ! Y of A-presheaves is an S-fibration if it has the right lifting property with respect to all maps which are cofibrations and S-equivalences. 60 Choose a cardinal ~ as in the preamble to statement of Lemma 70. Let TS be the set of all cofibrations C ! D of A-presheaves which are S-equivalences and such that |D| < 2~. It follows from Lemma 67 that if the diagram A _____//X i|| i*|| fflffl| fflffl| B ____//_Y is a pushout diagram of A-presheaves with i a cofibration, then the induced diagram i* iA A_____//i* iA X | | | | fflffl| fflffl| i* iA B_____//i* iA Y is a homotopy co-cartesian diagram of simplicial presheaves. The functor X 7! i* iA X preserves filtered colimits indexed over sufficiently large infinite or* *dinals fl. It is a standard consequence of Lemma 70 that a small object argument of size fl produces a factorization j X _____//BBZ BBB |p f BB__Bfflffl|| Y for every map f : X ! Y of A-presheaves, where p is an S-fibration and j is a filtered colimit of size fl of pushouts of coproducts of maps appearing in TS. The map j is a cofibration and an S-equivalence. The codiagonal r : X t X ! X has a factorization X t XJ_______r________//X;; JJ xxx JJJ xx (i0,i1)%%JJJxxprx X x I where (i0, i1) is a cofibration and pr is a weak equivalence, since I = i*A(1) is aspherical. It follows (since all A-presheaves are cofibrant) that each of t* *he maps i0 and i1 is an S-equivalence as well as a cofibration. Suppose that a map p : X ! Y has the right lifting property with respect to all cofibrations. Then there are commutative diagrams ;_____//_X?? | oe~~~p|~ | ~~ | fflffl|~fflffl|~ Y __1_//_Y 61 and (1,oep) X t X ____//_X;;w ww (i0,i1)||wwww |p| fflffl|ww fflffl| X x I__p.pr//_Y It follows that the induced map p* : i* iA (X) ! i* iA (Y ) is a ( 1, S)-equiva- lence of simplicial presheaves, so that p is an S-equivalence of A-presheaves. Conversely, suppose that p : X ! Y is a fibration and an S-equivalence. Then p has a factorization p X _____________//BBY>>" BBB """" j BB__B"""q W where j is a cofibration and q has the right lifting property with respect to a* *ll cofibrations _ this follows from a transfinite small object argument based on the inclusions Y LU a. But then q is an S-equivalence so j is a cofibration and an S-equivalence, and there is a commutative diagram X __1__//X>>" "" j|| r""" |p| fflffl|"fflffl|" W __q__//Y so that p is a retract of q. The map p therefore has the right lifting property with respect to all cofibrations. We have shown that a map p : X ! Y is a trivial fibration if and only if it has the right lifting property with respect to the set of inclusions Y LU a. It follows that every map f : X ! Y has a factorization f = q . j where j is a cofibration and q is an S-fibration and an S-equivalence. The factorization axiom CM5 and the lifting axiom CM4 have therefore both been established. The rest of the closed model axioms are trivial to verif* *y. The demonstration of the equivalence of homotopy categories Ho(s Pre(C))( 1,S)' Ho(A - Pre(C))S uses the weak equivalences displayed in (20) and (21). |___| Say that the model structure on the category of A-presheaves given by The- orem 71 is the S-model structure. Example 72. Suppose that S is a generating set for the class of locally trivial cofibrations of simplicial presheaves, as in Example 48. Let A be an arbitrary test category. Then the S-model structure on the category on the category of 62 A-presheaves gives a homotopy category which is equivalent to the homotopy category of the standard model structure on simplicial presheaves. This result specializes to the case C = *, giving a model structure on the c* *at- egory of A-sets with associated homotopy category equivalent to the homotopy category of simplicial sets. This homotopy category is therefore equivalent to the standard homotopy theory of topological spaces and continuous maps. This result applies in particular to cubical sets, bisimplicial sets, cubical simpli* *cial sets ... In the broader context, we obtain sensible homotopy theories of cubical presheaves, bisimplicial presheaves, and so on, all of which have homotopy cat- egories equivalent to the homotopy category of simplicial presheaves. Example 73. All localized simplicial presheaf homotopy theories (Example 50) have analogues over any test category, by Theorem 71. In particular, the motivic homotopy theory of simplicial presheaves on the smooth Nisnevich site (Sm|X )Nis on a scheme X (Example 51) has an equivalent counterpart over any test category. Thus, for example, all motivic homotopy types have cubical and bisimplicial representatives. 6 Weak equivalence classes of functors A weak equivalence class is a class W of functors between small categories such that the following conditions are satisfied: LF1 The class W is weakly saturated in the sense that the following hold: a) Every identity morphism is in W. b) Given functors C f-!D g-!E if any two of f, g and g . f are in W, then so is the third. c) Given functors A i-!B r-!A such that r . i = 1, if i . r is a member of W then r is a member of W. LF2 If C has a terminal object, then the functor C ! * is in W. LF3 Given a commutative triangle of functors A _____u______//_@@B~ @@ ~~~ ff@@ØØ@""~fi~~ C if all induced functors ff # c ! fi # c are in W then the functor u is in * *W. 63 A weak equivalence class is called a fundamental localiser in [3]; the terminol* *ogy was introduced by Grothendieck. Example 74. Let W1 denote the class of all functors f : C ! D such that the induced map f* : BC ! BD is a weak equivalence of simplicial sets. Since there is a natural weak equivalence BC ' i* C, we could equally well specify the members of W1 to be those functors f : C ! D which induce weak equivalences i* C ! i* D. The class W1 is a weak equivalence class of functors in the sense described above. The proof of LF3 uses the fact that if ß : D ! C is a functor, then there is a weak equivalence holim---!B(ß # c) ! BD c2C This is an old result of Quillen. Alternatively, it follows from Lemma 58. Remark 75. Consider the projection functor pr : C x D ! C where D has a terminal object. For each c 2 C, the induced functor pr # c ! C # c may be identified up to isomorphism with the projection (C # c) x D ! C # c. The categories (C # c) x D and C # c both have terminal objects, so the projection (C # c) x D ! C # c is in W. It follows that the projection pr : C x D ! C is W-aspherical and hence is a member of the weak equivalence class W. It follows that, if h : C x1 ! D is a homotopy (aka. natural transformation) between functors f, g : C ! D, then f is a member of the class W if and only if g is a member of W. Lemma 76. Suppose that f C0 _____//C2 i|| fflffl| C1 is a diagram of functors of small categories. Then if i is in W then so is the canonical map j : C2 ! siCi Proof. It suffices to assume that i is the identity functor. In effect, there i* *s a map of diagrams f C1 ooi__C0 _____//C2 1|| i|| |1| fflffl| fflffl| |fflffl C1 oo1__C1 __f__//C2 such that all the (vertical) transition functors are members of W. It follows that the induced functor on Grothendieck constructions is a member of W, by LF3. Suppose that i : C0 ! C1 is the identity functor. The canonical functor siCi ! C1 [C0 C2 can be indentified with a functor r : siCi ! C2 which is 64 specified by the assignments (x, 2) 7! x, (y, 0) 7! f(y) and (y, 1) 7! f(y). The canonical functor j : C2 ! siCiis specified by x 7! (x, 2), so obviously r . j * *= 1. The sets of morphisms (f(y), 2) (y, 0) ! (y, 1) (f(y), 2) (y, 0) ! (y, 0) (x, 2) (x, 2) ! (x, 2) specify a string of homotopies from the identity on siCi to the composite j . r. It follows that the composite j . r is a member of W, so that the morphisms_ r and j are members of W by LF1. |__| Note that there is an isomorphism G sCi~= Ci i i for all diagrams indexed by discrete categories. It follows that the class W is closed under small disjoint unions. In what follows suppose that A is a fixed choice of test category. Lemma 77. 1) Suppose given a diagram of A-sets X0 _____//X2 i || fflffl| X1 where the map i is a monomorphism. Then the induced map siA Xi! iA (X1 [X0 X2) i is in W. 2)Suppose given a diagram Y in A - Set which is indexed by some ordinal number ff and such that all morphisms Yi ! Yj are monomorphisms. Then the induced map s iA Yi! iA (limY (i)) i -!i is a member of W. Proof. According to the method of proof of Lemma 67, it suffices to prove part 1) in the case where all Xi are sets (ie. discrete A-sets) and X1 [X0 X2 is a singleton set. Then the pushout diagram has one of the forms ;_____//; X0 ____//_* | | ~ | || | | = | | fflffl|fflffl| fflffl|fflffl| *_____//* X1 ____//_* 65 In either case, there is a canonical functor * ! siXi which is a member of W, by Lemma 76. For 2) it suffices again to assume that all A-sets Yi are discrete. Given y 2 lim-!Yithere is a smallest i < ff such that y 2 Yi, and the fibre of the fu* *nctor siYi ! lim-!Yi over y is isomorphic to the subcategory of ff consisting of all t such that i t. This fibre has an initial object and is therefore W-aspherical. This is true of all fibres, and the fibres coincide with the corresponding comm* *a __ categories since lim-!Yiis discrete, so that the functor siYi! lim-!Yiis in W. * * |__| The argument for the proof of part 2) Lemma 77 came from [15]. The following is now a direct consequence of Lemma 76 and Lemma 77: Corollary 78. Suppose given a pushout diagram of A-sets f X0 ________//_X2 i|| i*|| fflffl| fflffl| X1 _f*_//_X1 [X0 X2 where i is a monomorphism. Then if the functor iA X0 ! iA X1 induced by i is a member of W then the functor iA X2 ! iA (X1 [X0 X2) induced i* is in the class W. The class of weak equivalences in the ordinary model structure on the cat- egory of simplicial sets has a very strong relationship with the collection of * *all weak equivalence classes of functors. Weak equivalences of simplicial sets form an initial theory, according to the following result: Theorem 79. Suppose that W is a weak equivalence class of functors. Suppose that f : X ! Y is a weak equivalence of simplicial sets. Then the induced functor i X ! i Y of simplex categories is a member of W. Proof. First of all, note that i ( n) ~= # n and therefore has a terminal object, so that i n is W-aspherical. All maps of simplices n ! m therefore induce functors i n ! i n which are members of W. Suppose that 0 s0 < s1 < . .<.sr n and let n be the subcomplex of the boundary @ n which is generated by the faces dsj: n-1 ! n. Then there is a pushout diagram sr-1 n-1d__// n | | | | fflffl| fflffl| n-1 _______dsr__// n in which the vertical maps are inclusions (see [6, p.218]). Note that if a face is missing from n then a face is missing from n-1. 66 Thus, one can use Corollary 78 and Lemma 77 to show that the induced functor i n ! i n is a member of W provided that some face is missing from n. It follows, in particular, that all inclusions nk n induce functors i nk! i n which are members of W. It suffices to show that every trivial cofibration i : A ! B induces a funct* *or i A ! i B which is a member of W, by a standard factorization argument. If i : A ! B is a trivial cofibration, it has a factorization j A ____//_@@X @@ p| i@@__@fflffl|| B where p is a Kan fibration and j is a filtered colimit of pushouts of disjoint unions of inclusions nk n. It follows from Corollary 78 and Lemma 77 that the induced functor j* : i A ! i X is a member of W. The fibration p is a weak equivalence, so the lifting oe exists in the diagram j A _____//X>>~ ~~ i|| oe~~ |p| fflffl|~fflffl|~ B __1__//B From the commutative diagram A __1__//A__1__//A j || i|| |j| fflffl|fflffl| fflffl| X __p__//B__oe//_X we see that the composite oe.p induces a functor i X ! i X which is a member of the class W. It follows from LF1 that oe and hence i induce functors_which are members of W. |__| The following result, which in other words asserts that W1 is the minimal weak equivalence class (see Example 74), is Grothendieck's Conjecture A. The first proof of this result appeared in Cisinski's thesis [3]. Corollary 80. Suppose that W is a weak equivalence class of functors, and that f : C ! D is a functor between small categories such that the induced map f* : BC ! BD is a weak equivalence of simplicial sets. Then f is a member of W. 67 Proof. If the induced map BC ! BD is a weak equivalence, then the map i* C ! i* D is a weak equivalence, since there is a natural weak equivalence BC ' i* C. The natural map ffl : i i* C ! C is a member of W by LF3. Theorem 79 implies that the induced map i i* C ! i i* D is a member of W. It therefore follows from the commutativity of the diagram i i* C_____//i i* D ffl|| |ffl| fflffl| fflffl| C ____f____//_D that the functor f is in the class W. |___| The following is a special case of Grothendieck's Conjecture B. This result was first proved by Cisinski [3], and the proof given here is essentially his. Theorem 81. Suppose that W(T ) is the smallest weak equivalence class con- taining a set of functors T , and that A is a test category. Then the class of * *all maps f : X ! Y of A-sets such that the functor iA X ! iA Y is a member of W(T ) is the class of weak equivalences for a model structure on the category of A-sets for which the cofibrations are the monomorphisms. Proof. Suppose, first of all, that A is the category of ordinal numbers, so that the A-set category is the category of simplicial sets. It is enough to establis* *h the result in this case, since the general statement is then a consequence of Theor* *em 71. The class W1 of functors C ! D which induce ordinary weak equivalences BC ! BD is contained in W(T ) by Corollary 80. Each functor f : C ! D in the set T induces a simplicial set map f* : BC ! BD, which can be replaced by a cofibration i(f) : BC ! Y up to weak equivalence. Let S denote the union of the set of all cofibrations i(f), f 2 T , along with the set of all an* *odyne extensions nk n. The ( 1, S)-model structure on simplicial sets is the localization of the standard model on S at the set of cofibrations i(f), f 2 T . I claim that the class W0 of all functors f : C ! D such that f* : BC ! BD is a weak equivalence in the ( 1, S) model structure is the weak equivalence class W(T ). Note that W0 is a weak equivalence class which contains all elements of T , so that W(T ) W0. All simplicial set maps i(f) : BC ! Y are weakly equivalent to maps f* : BC ! BD induced by generators f : C ! D of T , and the functors i BC ! i BD are equivalent to the functors f : C ! D on account of the natural weak equivalences (3). It follows that all ( 1, S)-weak equivalences X ! Y induce functors i X ! i Y which are members of W(T ). If the functor g : E ! F is a member of W0 then the functor i BE ! i BF is a member of W(T ), and there are weak equivalences i BC ' i i* C ffl-!C 68 for all small categories C which are members of W(S) by Corollary 80. |___| A map f : X ! Y of A-sets is said to be a simplicial weak equivalence if the induced map BiA X ! BiA Y is a weak equivalence of simplicial sets. Recall further (Theorem 71, Example 72) that the weak equivalences, so defined, are the weak equivalences for a model structure Ms on the category of A-sets. This model structure satisfies the conditions M1 and M2 of Section 4: in effect, M1 is satisfied since iA a = A # a has a terminal object, and M2 is satisfied sin* *ce the object i*A(1) is aspherical by Lemma 3. Theorem 82. Suppose that A is a test category. Suppose that M is an ( , S)- model structure on the category of A-sets which satisfies conditions M1 and M2 and is regular. Then every weak equivalence of Ms is a weak equivalence of M. Proof. The class F (M) of all functors f : C ! D which induce a weak equiva- lence BhC ! BhD of M is a weak equivalence class. In particular, the axiom LF1 follows from the model axioms for M, the axiom LF2 follows from Lemma 56, and LF3 is a consequence of Lemma 58. If g : C ! D is a functor such that BC ! BD is a weak equivalence of simplicial sets, then the induced map BhC ! BhD is a weak equivalence of M by Corollary 80. If f : X ! Y is a weak equivalence of Ms, then BiA X ! BiA Y is a weak equivalence of simplical sets. Thus, BhiA X ! BhiA Y is a weak equivalence of M by the previous paragraphs, so that f : X ! Y is a weak equivalence of_M by the regularity assumption. |__| Lemma 83. Suppose that A is a test category. Suppose that Y is a fibrant object in the model structure Ms on the category of A-sets. Then the functor X 7! X x Y preserves weak equivalences. Proof. Let i* : S ! A - Set be the functor which is defined by i*X(a) = hom (B(A # a), X), as in the preamble to Lemma 11, and recall that i* is right adjoint to the func* *tor Z 7! BiA Z. Then the canonical morphism j : Z ! i*BiA Z is isomorphic to the map j : Z ! i*AiA Z, and is therefore a weak equivalence of Ms. The functor Z 7! BiA Z preserves trivial cofibrations, so that the functor i* prese* *rves fibrations. Let j : BiA Y ! Z be a trivial cofibration with Z fibrant in the simplicial set category. Then the composite *j X x Y -1xj-!X x i*BiA Y -1xi---!i*Z is the product of the identity on X with a homotopy equivalence Y ! i*Z of fibrant objects. It follows that Y may be replaced by i*Z. The functor Z 7! X x i*Z preserves weak equivalences of simplicial sets Z by Corollary 16. It follows that the simplicial set Z may be replaced up to weak equivalence by the nerve BC of a small category C. 69 Observe that i*BC = i*AC. Write ß for the composite iA (X x i*AC) ! iA i*AC ffl-!C which is induced by the projection Xxi*Ac ! i*AC. Then there are isomorphisms ß # c ~=iA X x (ffl # c) ~=iA X x iA i*A(C # c) by Lemma 1. The functor X 7! X x i*A(D) preserves weak equivalences if the category D has a terminal object, since i*AD is aspherical. Also, there is a natural weak equivalence holim---!B(ß # c) ! BiA (X x i*AC). c2C It follows that the functor X 7! X x i*AC preserves weak equivalences. |__* *_| 7 Homotopy theory of cubical sets Let the object I = i* (1) define an interval theory for the category - Set of cubical sets. Let S be the set of vertex maps * ! n of the standard n-cells. Then there is an (I, S)-model structure on the category of cubical sets, as a result of Theorem 71. We shall say that the model structure Ms on the category of cubical sets is the standard structure. This is the model structure on - Set whose weak equivalences are those maps f : X ! Y which induce weak equivalences Bi X ! Bi Y of simplicial sets. A priori, the standard and the (I, S)-model structures on the category of cubical sets are potentially distinct, but we have the following result: Theorem 84. The class of weak equivalences of the (I, S)-model structure on the category of cubical sets coincides with the class of weak equivalences of t* *he standard model structure Ms on - Set so the two model structures coincide. Proof. Every weak equivalence of the (I, S)-model structure is a weak equiva- lence of Ms. The (I, S)-model structure on - Set is constructed to satisfy the axioms M1 and M2. Thus, according to Theorem 82, we only need to show that the S-local primitive model structure on the category of cubical sets is regular. This, however, is a consequence of Lemma 69, together with the observation that the cofibrations of the category cubical sets are generated by the inclusi* *ons @ n n, provided we can show that all maps holim---! k ! n k! n are (I, S)-equivalences. 70 We know that n ! * is an (I, S)-equivalence, by construction. It follows that the map holim---! k ! Bhi n k! n is an (I, S)-equivalence. But finally, the category i n ~= # 1xn has a terminal object, so the cubical set map Bhi n ! * is an (I, S)-equiva-_ lence by Lemma 56. |__| Theorem 85. The standard model structure Ms on the category of cubical sets is proper. Proof. On account of Theorem 47, it is enough to show that all vertex maps * ! n pull back to weak equivalences along all fibrations p : X ! Y for which the base Y is fibrant. Suppose given a diagram X p|| fflffl| *__v__// n_ff_//_Y The map v is an anodyne cofibration for the (I, S)-structure and Y is fibrant, so there is a map x : * ! Y and a naive homotopy n x I ! Y from ff to the composite n ! * x-!Y. The standard anodyne cofibrations d0, d1 : U ! U xI pull back to weak equiva- lences along p (see the argument for Theorem 47), so it follows that the pullba* *ck along p of the composite * v-! n ff-!Y may be replaced by the pullback of the composite * v-! n ! * x-!Y. Let F be the fibre of p over the vertex x. Then there are pullback diagrams v* npr F ____//_F x _____//F____//_X | | | p| | | | | fflffl| fflffl| fflffl| fflffl| *____v__// n_______//*__x_//_Y Then the map v* is a weak equivalence by Lemma 83. |___| 71 Suppose that k is a fixed standard cell in the category of cubical sets. Recall from Lemma 4 and Lemma 15 the associated cell category i k is a test category, and that the category of i k-sets can be identified with the catego* *ry of ( - Set) # k of cubical sets ø : X ! k. The tensor product pairing for the category of cubical sets determines an interval theory (ø, 1n) 7! ø n where oe n is the composite X n pr-!X ø-! k. Theorem 46 determines an ( , ;)-model structure on the category of cubical sets over k. Lemma 86. Suppose that A is the test category i k, and that M is the corresponding ( , ;)-model structure on the category ( - Set) # k of A- sets. Then every weak equivalence of the standard model structure Ms is a weak equivalence of M. Proof. All vertex maps * ! n ! k are trivial cofibrations, so that all mor- phisms n ! m ! k are weak equivalences of M. In particular, the map n ! k 1-! k to the terminal object is an equivalence of M, so that the condition M1 is verified for this model structure. In the picture k -' holim---! k -' holim---! r ! k r! n r! n of cubical sets over k, the indicated maps are weak equivalences of M, so that the map holim---! r ! n r! n is also an equivalence of M, for all standard cells n ! k of ( - Set) # k. The inclusions in this category are generated by morphisms of the form @ n n ! k, and it follows from Lemma 70 that the model structure M is regular. The object i*A(1) can be identified with the projection k x i* (1) ! k in the category of cubical sets over k. The object i* (1) has a naive contracting homotopy h given by the composite i* (1) 1 ! i* (1) x i* (1) ~=i* (1 x 1) ! i* (1) where the last map in the string is induced by a contraction 1 x 1 ! 1 onto the terminal object. 72 If K ! k is a cubical set over k then the cubical set over k corresponding to the presheaf K x i*A(1) is the composite map K x i* (1) pr-!K ! k and it's not hard to see that h induces a homotopy (K x i* (1)) 1 ! K x (i* (1) 1) 1xh---!K x i* (1) over k in the model structure M. It follows that the projection map K x i* (1) ! K is a natural weak equivalence of M, giving the condition M2._ The Lemma is now a consequence of Theorem 82. |__| Recall that a map of cubical sets f : X ! Y is said to be an injective fibra* *tion if it has the right lifting property with respect to all inclusions un(i,ffl) * * n. A fibration of cubical sets, in the standard theory, is a map which has the right lifting property with respect to all trivial cofibrations A B. Every fibration is an injective fibration. Lemma 87. Every injective fibration f : X ! k of cubical sets is a fibration. Proof. Note that a map g X _____________//_BY BB """ BBB """ B__B~~"" k is a weak equivalence in the (standard) sense that BiA X ! BiA Y is a weak equivalence of simplicial sets if and only if the map g : X ! Y of cubical sets is a standard weak equivalence. This follows from the fact that there is an isomorphism iA (X ! k) ~=i X for A = i k. Thus, the diagram is a cofibration (respectively fibration) if a* *nd only if the map g : X ! Y is a cofibration (respectively fibration) of cubical sets. The standard and ( , ;)-model structures for the category of cubical sets X ! k coincide, by Lemma 86, since every anodyne weak equivalence is a standard weak equivalence. The two theories therefore have the same fibrant objects. In particular, every injective object of ( - Set) # k is fibrant_for* * the standard theory, by Lemma 43. |__| Theorem 88. Every injective fibration of cubical sets is a fibration. Proof. Suppose we know that if a map q : V ! W is an injective fibration and a standard weak equivalence, then it is a trivial fibration. 73 The first thing that this implies is that the standard and ( , ;)-model stru* *c- tures on the category of cubical sets coincide. To see this, observe that if f : X ! Y is a standard weak equivalence then f has a factorization f X ____________//_@@Y??~ @@ ~~~ j @@__@~~~p Z where j is an ( , ;)-anodyne cofibration (hence a standard weak equivalence) and p is an injective fibration. But then p is a standard weak equivalence, and hence has the right lifting property with respect to all inclusions by our assumption, and thus is a trivial fibration for the ( , S)-model structure. Thu* *s, both model structures have the same weak equivalences as well as the same cofibrations. Now suppose that f : X ! Y is an injective fibration, and form the diagram j X ______//_U f|| |p| fflffl| fflffl| Y _jY__//L(Y ) where the horizontal maps are trivial cofibrations, L(Y ) is fibrant and p is an injective fibration (this can be done in the ( , ;)-model structure). Then Lemma 43 implies that p is a fibration. It follows that the induced map p* : Y xL(Y )U ! Y is a fibration. The map X ! Y xL(Y )U has a factorization j X _________//JJW JJJ | JJJ |q JJ$$fflffl| Y xL(Y )U where j is an ( , ;)-anodyne cofibration and q is an injective fibration. Then the map q is also a weak equivalence, and so it is a trivial fibration by our assumption. One sees easily that f is a retract of the composite p*q, and so f is a fibration. Suppose now that the cubical set map q : V ! W is an injective fibration and a weak equivalence. Then in all diagrams m xW V __ø*_// n xW V_____//V q*|| |q*| q|| fflffl| |fflffl fflffl| m _____ø_____// n___oe__//W 74 the maps labelled q* are fibrations (Lemma 87) and the map ø* is a weak equiv- alence since the standard model structure for cubical sets is proper (Theorem 85). It is therefore a consequence of Quillen's Theorem B that all diagrams of simplicial set maps Bi ( n xW V )_____//Bi V | | | | |fflffl fflffl| Bi ( n)__oe*___//Bi W are homotopy cartesian: recall that there are isomorphisms i ( n xW V ) ~=f* # oe for all cells oe : n ! W of W . The map Bi (V ) ! Bi (W ) is a weak equivalence by assumption, so that all maps Bi ( n xW V ) ! * are weak equivalences. In particular, the map q : V ! W is aspherical, and so all induced maps n xW V ! n are trivial fibrations. It follows that q : V ! W has the right lifting property with respect to all inclusions @ n n, and is_ therefore a trivial fibration, as claimed. |__| Recall from Section 2 that the triangulation |X| of a cubical set X is the simplicial set defined by |X| = lim-!B(1n). n!X The cells oe : n ! X of a cubical set X induce simplicial set maps B(1n) ~=| n| oe*-!|X|, and these maps together determine a map fX : holim---!| n| ! |X| oe: n!X in the obvious way. Observe that the canonical map ßX : holim---!| n| ! Bi X oe: n!X is a weak equivalence for all cubical sets X. This is a consequence of the fact that all triangulations | n| are contractible simplicial sets. Proposition 89. The map fX : holim---!| n| ! |X| oe: n!X is a weak equivalence of simplicial sets. 75 Proof. The existence of the natural weak equivalence ßX and Lemma 77 together imply that the functor X 7! holim---!oe| n| takes all pushout diagrams F n x2NXn @ _____//skn-1X | | | | F fflffl| fflffl| x2NXn n ______//_sknX to homotopy cocartesian diagrams. The realization functor X 7! |X| has the same property. It suffices, therefore, to check that fX is a weak equivalence_f* *or all cubical sets X = n, but this is elementary. |__| As a consequence, the standard description of weak equivalence of cubi- cal sets given here, via the functor X 7! Bi X coincides with the geometric description of weak equivalence defined by the functor X 7! |X|. The stan- dard model structure Ms for cubical sets therefore coincides with the geometric model structure given in [10]. The triangulation functor | | : - Set! S also preserves and reflects weak equivalences of cubical sets. The right adjoint S : S ! - Setof the triangulation functor is defined by S(X)n = hom (B(1n), X). This functor is also (see Lemma 13) the functor i* : S ! - Set the functor induced by the inclusion i : ! cat. It is plainly the case that all of the categories 1n have terminal objects, and we know from Proposition 26 that the cubical set i*( 1) = B (1) is aspherical. It follows from Corollary 14 and Lemma 28 that the functor S preserves and reflects weak equivalences of simplicial sets. Theorem 90. The triangulation functor | | and its right adjoint S induce an adjoint equivalence of homotopy categories Ho( - Set) ' Ho(S). The adjunction maps j : X ! S|X| and ffl : |SY | ! Y are natural weak equiva- lences. Proof. There are natural weak equivalences i* i X ' Bi X ' |X| for all simplicial sets X: the first comes from (3) and the second is a consequ* *ence of Proposition 89. The functor i* i induces an equivalence i* i : Ho( - Set) '-!Ho(S) by Theorem 71 (Example 72). This functor is, in particular, fully faithful. It follows that the triangulation functor | | induces a fully faithful functor | |* * on the 76 level of homotopy categories. The functor S also preserves weak equivalences, and therefore induces a functor S : Ho(S) ! Ho( - Set) which is right adjoint to | |. From the collection of pictures ~= [Y, Z]______//[|Y9|,9|Z|] rrr j*|| r~rrr fflffl|=rrr [Y, S|Z|] one sees that composition with the natural cubical set morphism j : Z ! S|Z| is an isomorphism for all maps Y ! Z in the homotopy category. It follows that j is an isomorphism in Ho( - Set), and hence that j is a weak equivalence of cubical sets _ see [6, I.1.14]. It follows that Sffl is a weak equivalence * *for all natural simplicial set maps ffl : |S(Y )| ! Y . The functor S reflects wea* *k __ equivalences, so all canonical maps ffl are weak equivalences of simplicial set* *s. |__| At the risk of adding a final bit of notational confusion, I shall define the topological realization |X| of a cubical set X by setting |X| = lim-!|B(1n)|, n!X where |B(1n)| is the topological realization of the simplicial set B(1n). The object |B(1n)| is, in other words an ordinary topological hypercube. The topo- logical realization functor has a right adjoint S : Top ! - Set which is defined for a topological space Y by S (Y )n = hom (|B(1n)|, Y ). Write S : Top ! S for the ordinary singular functor taking values in simplicial sets. The topological realization of a cubical set X is naturally is* *o- morphic to the topological realization of the triangulation |X| 2 S, so there is a corresponding natural isomorphism S (Y ) ~=S(S (Y ) relating the right adjoints. The following result is the excision statement for cubical sets: Theorem 91. Suppose that a topological space Y is covered by open subsets U1 and U2. Then the canonical map S U1 [ S U2 ! S Y is a weak equivalence of cubical sets. 77 Proof. The idea of proof is to show that the induced map of triangulations |S U1| [ |S U2| ~=|S U1 [ S U2| ! |S Y | is a weak equivalence of simplicial sets. There is a natural isomorphism S Z ~= S(S Z) for all topological spaces Z, and it follows from Theorem 90 that there is a natural weak equivalence |S Z| ~=|S(S Z)| ffl-!S Z, which will be denoted by ffl. It follows that there is a commutative diagram |S U1| [ |S U2|__//|S Y | ffl*|| |ffl| fflffl| fflffl| S U1 [ S U2______//S Y in which the vertical maps are weak equivalences of simplicial sets by a patchi* *ng lemma argument. The map S U1 [ S U2 ! S Y is a weak equivalence of simplicial sets, by excision for simplicial sets_[11, * *Th. 20]. |__| Theorems 90 and 91 also appear in [10]. In particular, Theorem 91 appears as Theorem 27 in that paper, and is the central device given there for establishing the Theorem 90. The proof of Theorem 91 which is given in [10] is a direct (and somewhat dirty) subdivision argument. 78 References [1]B. Blander, Local projective model structures on simplicial presheaves, K-T* *heory 24(3) (2001), 283-301. [2]A.K. Bousfield and D.M. 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