CLOSED CLASSES WOJCIECH CHACHOLSKI 1. Introduction A non empty class C of connected spaces is said to be a closed class if it is closed under weak equivalences and pointed homotopy colimits. A closed class can be characterized as a non empty class of connected spaces which is closed under weak equivalences and is closed under certain simple operations: arbitrary wedges, homotopy push-outs and homotopy sequential colimits. The notion of a closed class was introduced by E. Dror Farjoun [6]. Two important constructions give rise to examples of closed classes. The first one is the Bousfield-Dror periodization functor PA [2]. The class of tho* *se spaces X, such that PA X is weakly contractible, forms a closed class. By looki* *ng just at the properties of this class we can prove, for example, that PA X is weakly equivalent to PA X (see [2], [4]). The second construction is E. Dror Farjoun's colocalization functor CWA . The class of those spaces X, for which there exists a space Y , such that X is weakly equivalent to CWA Y , forms a closed class. This class is denoted by C(A) and is called the class of A-cellul* *ar spaces. By looking just at the properties of the class C(A) we can prove, for example, that CWA X is weakly equivalent to CWA X (see [4], [6]). We say that a closed class C is closed under extensions by fibrations, if for every fibration sequence (Z ! E ! B), such that Z and B belong to C, E belongs to C. A closed class C is closed under extensions by fibrations if and only if for every diagram F : I ! C , such that the classifying space BI belongs to C, the unpointed homotopy colimit hocolimIF belongs to C. The purpose of this paper is to understand to what extent a closed class is closed under extensions by fibrations and under taking unpointed homotopy colimits. We start with proving a theorem that, in particular, implies: o Let F : I ! Spaces? be a pointed diagram, such that the classifying space BI belongs to C. If for every i 2 I, F (i) belongs to C, then so does the unpointed homotopy colimit hocolimIF . o Let (Z ! E ! B) be a fibration sequence with a section. If Z and B belong to C, then so does E. o Let F : I ! C and G : I ! C be diagrams and : F ! G be a natural transformation. If hocolimIF belongs to C, then so does hocolimIG. 1 2 WOJCIECH CHACHOLSKI Surprisingly these and many other results are the consequences of just one statement, see theorem 5.1. We continue with investigating the properties of a base space B (respec- tively of the classifying space BI), which will guarantee that a closed class C* * is closed under extensions by fibrations with base B (respectively C is closed un- der taking the unpointed homotopy colimit of diagrams F : I ! C ). We study the following class: D(C) = {BI | ifF : I ! C is a diagram, then hocolimIF 2 C} The main result of this paper is: Theorem. The class D(C) is a closed class and it is closed under extensions by fibrations. Using this theorem, we can characterize the class D(C) as follows: D(C) = {B | ifZ ! E ! B is a fibration sequence and Z 2 C; then E 2 C} Since D(C) is a closed class, it is closed under weak equivalences. This is a very non trivial fact itself. It is obvious that ? belongs to D(C). What is not clear at all is that for any diagram F : I ! C over a contractible category I, the homotopy colimit hocolimIF belongs to C. As a result we get a new characterization of a closed class: A non empty class C of connected spaces is a closed class if and only if it is closed under weak equivalences and for any unpointed diagram F : I ! C over a contractible category I, the homotopy colimit hocolimIF belongs to C. The techniques that are used to prove the main theorem involve studying the homotopy fiber of a map f : X ! Y through inverse images of simplices in Y (a simplicial analogues of point inverse images). We prove, roughly, that if point inverse images belong to a closed class, then so does the homotopy fiber, see corollary 7.9. One consequence of this is that if the point inverse images * *of f are acyclic with respect to some homology theory, then so is the homotopy fiber of f . Techniques, we have introduced, and properties of D(C) are applied to prove a generalization of a theorem of E. Dror Farjoun (see 9.1 and [7, theorem I]): Theorem. Let : E ! B be a natural transformation between unpointed diagrams E : I ! Spaces and B : I ! Spaces . If for every i 2 I, the homo- topy fiber F ib(i : E(i) ! B(i)) belongs to a closed class C, then so does F ib( : hocolimIE ! hocolimIB ). CLOSED CLASSES 3 2. Notation The symbol denotes the simplicial category [9, x2], in which the objects are the ordered sets [n] = {0; 1; : :;:n}, and morphisms are weakly mono- tone maps of sets. The morphisms of are generated by codegeneracy maps si : [n]! [n + 1] (i = 0; 1; : :;:n) and coface maps di : [n - 1]! [n], subject to well-known cosimplicial identities (see [3]). A simplicial set is a functor K : op ! Sets , where Sets denotes the category of sets [9, x2]. The set K([* *n]) is usually denoted by Kn . A map between two simplicial sets is by definition a natural transformation of functors. A simplicial set K can be interpreted as a collection of sets Kn together with face maps di : Kn ! Kn-1 and degeneracy maps si : Kn+1 ! Kn (i = 1; 2; : :;:n) which satisfy the duals of the cosim- plicial identities (see [3]). For description of haw to do homotopy theory with simplicial sets see [3] and [9]. If oe 2 Kn , then oe is called an n-dimensional simplex of K. The dimensi* *on of oe will be denoted by dim(oe). The object [n] is the standard n-simplex, given by [n]k = mor ([k]; [n]) (see [3]). There is a distinguish n-dimensional simplex o 2 [n]n , the one that comes from the identity map [n] ! [n]. It is easy to check that for any simplicial set K, the assignment f ! f (o ) gives a bijection between the set of maps [n] ! K and Kn . If oe 2 Kn , we will denote the corresponding map by oe : [n] ! K . A pointed simplicial set is a pair (K; k), where k is a chosen simplex of dimension zero in K. We will refer to this 0-dimensional simplex as the basepoint of (K; k). A map between pointed simplicial sets is a map of simplici* *al sets which preserves the basepoints. We will use the following notation for some categories which frequently occur: o Spaces denotes the category of simplicial sets. o cSpaces denotes the category of connected simplicial sets. o Spaces? denotes the category of pointed simplicial sets. o cSpaces? denotes the category of pointed and connected simplicial sets. If I is a small category, the nerve of I, denoted by N (I), is the simpli* *cial set whose n-simplices are n-tuples (i0 ! i1 ! . .i.n) of composable morphisms in I (see [3]). If C is a category and K is an object in C, then by C=K we denote the category of objects of C over K [8, 1x6]. This is the category whose objects are morphisms f : X ! K in C and maps from f : X ! K to g : Y ! K are those morphisms h : X ! Y in C such that gOh = f . 3. The homotopy colimit In this section we describe the notion of a diagram indexed by a simplici* *al set and define the homotopy colimit of such a diagram. The particular form of the homotopy colimit which is going to be used was introduced by E. Dror 4 WOJCIECH CHACHOLSKI Farjoun [7]. Various properties of the homotopy colimit are also listed in the appendix. The reference for the proofs is [5]. The motivation for these constructions comes from the fact that any map f : X ! K of simplicial sets, can be reconstructed (up to a weak equivalence) from the homotopy colimit of a diagram indexed essentially by the range of K. The constituents of this diagram are the analogues (in the simplicial category) of the point inverse images of f . The ability to build a map in a homotopically meaningful way from its range and its point inverse images is going to be explored in the following sections. 3.1. Definition. Let K : op ! Sets be a simplicial set. The category asso- ciated to K, sometimes called the transport category of K or Grothendieck construction on K, is the category whose objects are pairs ([n]; oe) where [n] is an object of op and oe 2 Kn . A morphism ([n]; oe) ! ([m]; o ) is a map ' : [m] ! [n] in (or ' : [n]! [m] in op) such that K(')(oe) = o . To avoid introducing to many notation we will denote the category asso- ciated to a simplicial set K by the same symbol K, and speak of "functors with domain K". One can think about the category K as having as objects the sim- plices of K and morphisms generated by the arrows di : oe ! dioesi: oe ! sioe, subject to some relations that come from the simplicial structure of K. The morphisms si : oe ! sioeis called the degeneracy morphisms and di : oe ! dioe the boundary morphisms. The notion of the category associated to a simplicial set can be used to define the subdivision of K (see [11 ]): 3.2. Definition. If K is a simplicial set, the subdivision of K is the nerve (see section 2) of the category associated to K. The subdivision of K will be denoted by sdK. 3.3. Example. The category associated to [0] = ? is isomorphic to op. Diagrams over ? are simplicial spaces. If K is a simplicial set, a functor F : K ! Spaces over the category as* *soci- ated to K will be called a diagram with shape K. Diagrams are our main object of interest. As example 3.3 suggests, the category associated to a simplicial s* *et can be quite complicated. Since we care about diagrams, in order to simplify the situation, we will distinguish the class of bounded diagram which are tech- nically more manageable. It turns out that all the examples of diagrams, we are going to consider, are bounded. 3.4. Definition. F : K ! Spaces is a bounded diagram if for any degeneracy morphism si : oe ! sioe, F (oe) = F (sioe) and F (si) = idF (oe) 3.5. Example. A bounded diagram with shape ? is determined by the value on the only zero dimensional simplex of ?. The category of bounded diagrams with shape ? is equivalent to the category of simplicial sets. CLOSED CLASSES 5 3.6. Example. The category of bounded diagrams over [1] is equivalent to the category of diagrams of the form A B ! C, so called push-out diagrams. Out of a bounded diagram F : [1] ! Spaces , a push-out diagram can be extracted in the following way: F (0) F (0; 1) ! F (1) 3.7. Example. Let f : X ! K be a map. For every simplex oe 2 K, let Ff(oe) be the space that fits into the following pull-back square: Ff(oe) --- - ! X ?? ? y ?yf [dim(oe)] ---oe-! K Roughly speaking Ff(oe) is the inverse image in X of the simplex oe of K. This construction clearly defines a functor Ff : Kop ! Spaces . Out of Ff we can built a bounded diagram df : sdK ! Spaces such that for an n-dimensional simplex v = oe0 '0!oe1 '1!::: 'n-1!oenin sdK: df (v) = Ff(oen ) ( df (di : v ! div) = id: Ff(oen )! Ff(oen ) ifi < n Ff('n-1 ) : Ff(oen )! Ff(oen-1 ) ifi = n The construction d is natural. If f : X ! K , g : Y ! K and h : X ! Y are maps such that gOh = f , then there is a natural transformation df ! dg. In par- ticular there is a natural transformation : df ! d(id) induced by f : X ! K itself. It turns out that: colimsdK : colimsdK df ! colimsdK d(id) = (f : X ! K ) We will see in example 3.12 that the diagram df has nice homotopic properties with respect to the map f . If f : X ! K is a fibration, then the functor Ff : Kop ! Spaces has * *the property that for every morphism ', Ff(') is a weak equivalence. In this case Ff(oe) is weakly equivalent to the homotopy fiber of f . It is clear from the definition that the diagram df : sdK ! Spaces inherits the same properties. This motivates the following definition: 3.8. Definition. F : K ! Spaces is a good diagram if it is a bounded diagram and for every morphism ' 2 K, F (') is a weak equivalence. The construction d defines a functor: d : Spaces=K ! {bounded diagrams over sdK} (f : X ! K ) 7! (df : sdK ! Spaces ) such that fibrations are carried out to good diagrams. 6 WOJCIECH CHACHOLSKI We will now introduce a construction which will allow us to recover, up to homotopy, a map f from the diagram df . One can think about this construction as sort of an "inverse" of d: 3.9. Definition. The homotopyIcolimit is the following functor: : {diagrams over K} ! Spaces I K i F j . F = oe2K [dim(oe)] x F (oe) ~ K where ~ is an equivalence relation generated by: let' 2 mor ([n]; [m]) ,o 2 Km ,x 2 F (o ) ,t 2 [n] (['](t); x) ~ (t; F (')(x)) 3.10. Definition.ZThe pointed homotopy colimit is the following functor: : {pointed diagrams over K} ! Spaces? Z K i W j . F = oe2K [dim(oe)] x F (oe) = [dim(oe)] x {?} ~ K where ~ is an equivalence relation generated by: let' 2 mor ([n]; [m]) ,o 2 Km ,x 2 F (o ) ,t 2 [n] (['](t); x) ~ (t; F (')(x)) 3.11. Example. If F : K ! Spaces? is a constant diagram F (oe) = X and F (') = idX , then: I Z F = K x X ; F = (K x X)=(K x {?}) = K n X K K I In case X = ?, we get that ? = K, where ? : K ! Spaces denotes the K constant diagram whose value is ?. Let F : K ! Spaces be a diagram. There is a natural transformation F ! ? between F and the constant diagram ?. This natural transformation induces the following map: I I F ! ? = K K K Let F : K ! Spaces? be a pointed diagram. This means that there is a natural transformation ? ! F betweenIthe constantIdiagram ? and F . This transformation induces the map K = ? ! F , which is a section of the I K K map F ! K. K CLOSED CLASSES 7 As a consequence we get that the homotopy colimit can be seen as a functor with values in Spaces=K: I : {diagrams over K} ! Spaces=K K I (F : K ! Spaces ) 7! F ! K K which carries out pointed diagrams into maps with a section. If F : K ! Spaces? is a pointed diagram, then the following is a cofib* *ra- tion sequence: I Z K ! F ! F K K As a consequence we get that if F : K ! Spaces? is a pointed diagram over weakly contractible simplicial set, then the unpointed and the pointed homo- topy colimits of F are weakly equivalent. 3.12. Example. Let f : X ! K be a map. Out of f we have constructed a diagram df : sdK ! Spaces (see example 3.7). The main property of df is that in the following commutative diagram, all the horizontal arrows are weak equivalences: I I df -id df -! colimsdK df = X sdK? sdK? ? ? y y ?y ?yf I sdK - d(id) -! colimsdK d(id) = K sdK It impliesIthat every map f : X ! K is weakly equivalent to a map of the form F ! L for some bounded diagram F : L ! Spaces . Since every map L is weakly equivalent to a fibration, we can assume that F : L ! Spaces is a good diagram whose values are weakly equivalent to the homotopy fiber of f (see example 3.7). 3.13. Example. Let _ [n] ! L be a map, where _ [n] is the boundary of [n]. We are going to consider diagrams over L [ _[n][n]. o Let F : L [ _[n][n] ! Spaces be a diagram. One can show that: I I I I F = colim F F ,! F L[ _[n][n] L _[n] [n] 8 WOJCIECH CHACHOLSKI o Let F : L [ _[n][n] ! Spaces be a bounded diagram. If o 2 ([n])n is the only non degenerate simplex and F (o ) = X, one can conclude that in this case: I I F = colim F _[n] x X ,! [n] x X L[ _[n][n] L 3.14. Example. A functor F : I ! Spaces over a small category I defines a bounded diagram Fsd : N (I) ! Spaces over the nerve of I. Let: oe = (a0 '0!a1 '1!::: 'n-1!an) 2 N (I)n Fsd is defined as follows: Fsd(oe) = F (a0) ( Fsd(di : oe ! dioe) = F ('0) : F (a0)! F (a1) ifi = 0 id: F (a0)! F (a0) ifi > 0 It can be shown that in this case: I Fsd = hocolimIF N(I) where hocolimIF denotes the homotopy colimit of F in the sense of Bousfield and Kan [3]. In case of a pointed functor F : I ! Spaces? there is a similar equalit* *y: Z Fsd = phocolimIF N(I) where phocolimIF denotes the pointed homotopy colimit of F in the sense of Bousfield and Kan [3]. 4. Closed Classes In this section we state the definition and give some examples and basic properties of closed classes. The notion of a closed class was introduced by E. Dror Farjoun [6], [7]. The definition presented in this papers is slightly diff* *erent from the one given by E. Dror Farjoun [6, definition 2.1]. We think about a closed class as a class of unpointed and connected simplicial sets. A good example, to keep in mind, is the class of acyclic spaces with respect to some homology theory. 4.1. Definition. (E. Dror Farjoun [6]). A non empty class C of connected sim- plicial sets is said to be a closed class if it is closed under weak equivalenc* *es and taking pointed homotopy colimits. If F : K ! Spaces? Z is a pointed diagram such that for every simplex oe 2 K, F (oe) 2 C, then F 2 C. K Observe that a closed class is assumed to be non-empty. Notice also that since the empty space is not connected, it does not belong to any closed class. CLOSED CLASSES 9 4.2. Notation. o Throughout this article C always denotes a closed class. o By F : K ! C we denote a diagram such that for every simplex oe 2 K, F (oe) belongs to C. o Let f : X ! Y be a map. We say that the homotopy fiber F ib(X !f Y ) belongs to a closed class C, if the homotopy fibers of f over every component of Y belong to C. In particular, the homotopy fibers of f , over various components, are connected and f induces an isomorphism on ss0. If F ib(X !f Y ) belongs to C, then we will write F ib(X !f Y * *) 2 C. 4.3. Remark. The definition of the pointed homotopy colimit 3.10 implies that a class C is closed if and only if: o C is non-empty. o Let X and Y be weakly equivalent simplicial sets. If X 2 C, then Y 2 C. o Let (Xi)i2I be a familyWof simplicial sets. If Xi 2 C, then for any ch* *oice of basepoints in Xi, i2IXi 2 C. o Let X? be a simplicial space. If for every n 0, Xn 2 C, then the realization |X?| 2 C. It follows that a closed class can be characterized as a class of connect* *ed simplicial sets, such that: o C is non-empty. o C is closed under weak equivalences. o C is closed under taking arbitrary wedges. o Let X1 X2 ! X3 be a diagram. If Xi 2 C, then the following simplicial set belongs to C: hocolim(X1 X2 ! X3) o Let ( X0 ! X1 ! X2 ! . . .) be a diagram. If Xi 2 C, then the following simplicial set belongs to C: hocolim(X0 ! X1 ! X2 ! . .). 4.4. Examples. Here is a list of some examples of closed classes: o Let A be a connected simplicial set. The smallest closed class C(A) such that A 2 C(A). This class is called the class of A-cellular space* *s. This class was introduced by E. Dror Farjoun, see [4],[6] and [7]. o The class of acyclic spaces with respect to some homology theory. o The class C(?) of weakly contractible spaces. o The class C(Sn+1 ) of n-connected spaces. o {X 2 cSpaces | H"i(X; G) is trivial fori n}. 10 WOJCIECH CHACHOLSKI o Let A be a pointed and connected Kan simplicial set. {X 2 cSpaces | for any choice of basepoints in X, map?(X; A) ' ?}. o Let A be a connected simplicial set. {X 2 cSpaces | ifY is Kan and the basepoint evaluation map map(A; Y ) ! Y is a weak equivalence, then map(X; Y ) ! Y is also a weak equivalence }. This class is called the class A-acyclic spaces, see [4, definition 16* *.1]. The following two propositions give some examples of elements of a closed class. 4.5. Proposition. (E. Dror Farjoun [6, section 2.3]). If C is a closed class, then ? 2 C. Proof. Let X 2 C and X !? X be the constant map. Notice that the following space is contractible and belongs to C: hocolim(X !? X !? X !? X . .). It implies that ? 2 C. __|_ | 4.6. Proposition. (E. Dror Farjoun [6, theorem 2.8]). Let K and X be sim- plicial sets. If X 2 C, then for any choice of basepoint in X, K n X 2 C (see example 3.11). Proof. Lets consider the constant diagramZX : K ! C , X(oe) = X and X(') = idX . Since C is a closed class, K n X = X 2 C. __|_ | K 5. Closed classes and unpointed homotopy colimits Closed classes are not usually closed under unpointed homotopy colimits (see [6, section 2.3]). This section contains the first approach to this questi* *on, to what extent a closed class is closed under unpointed homotopy colimits. The motivation for the following theorem can be found in the next section. This theorem is going to be applied to prove various properties of closed class* *es with respect to fibrations. Surprisingly all those properties are just particul* *ar cases of this one statement. 5.1. Theorem. Let F : K ! C , GI: K ! C be diagrams and : F ! G be a natural transformation. If h : F ! Y is a map such that Y 2 C, then: K i I I h j hocolim G F ! Y 2 C K K CLOSED CLASSES 11 I 5.2. Lemma. For every diagram F : [n] ! C , F belongs to C. [n] Proof. Let o 2 ([n])n be the non-degenerate simplex. Observe that by choos- ing a vertex in F (o ), we can think about F : [n] ! C as a pointed diagram (this chosen vertex determines a basepoint in F (oe), for all oe 2 [n]). Since * *[n] is contractible, pointed and unpointed homotopy colimits are weakly equiva- lent. It proves the lemma. __|_ | Proof of the theorem. the proof will be by the induction on the dimension of K. It is obvious that the theorem is true for K such that dim(K) = 0. Lets assume that the theorem is true for K such that dim(K) < n. Let dim(L) < n and _[n] ! L be a map. We prove that the theorem holds for K = L[ _[n][n]. Lets consider the following commutative diagram: Y - - id-- Y --id--! Y x? x x ? h ??h ??h I I I F - - - - F --- - ! F L? _[n]? [n]? ?y ?y ?y I I I G - - - - G --- - ! G L _[n] [n] By the inductive assumption: i I I h j hocolim G F ! Y 2 C L L i I I h j hocolim G F ! Y 2 C _[n] _[n] I I According to lemma 5.2, F and G belong to C. Because C is closed [n] [n] under homotopy push-outs we get: i I I h j hocolim G F ! Y 2 C __ K K |_ | 5.3. Corollary. Let F : K ! CI , G : K ! C Ibe diagrams and : F ! G be a natural transformation. If F 2 C, then G 2 C. K K I Proof. Apply theorem 5.1 to the case when Y = F and h = id. __|_ | K 12 WOJCIECH CHACHOLSKI 5.4. ICorollary. Let K be a simplicial set. If F : K ! C is a diagram such that F 2 C, then K 2 C. K Proof. Since there is a natural transformation F ! ? between F and the constantIdiagram ? : K ! Spaces , whose value is ?, corollary 5.3 implies that K = ? belongs to C. __|_ | K 6. Closed classes and fibrations The behavior of a closed class with respect to fibrations has been studied by E. Dror Farjoun [6],[7]. This section contains generalizations of his result* *s. 6.1. Definition. We say that a closed class C is closed under extensions by fibrations if for every fibration sequence Z ! E ! B such that Z and B belong to C, E belongs to C. 2mm Closed classes are not usually closed under extensions by fibrations [7]. 6.2. Examples. Here is a list of some examples of closed classes that are closed under extensions by fibrations: o The class of acyclic spaces with respect to some homology theory. o The class C(?) of weakly contractible spaces. o The class C(Sn+1 ) of n-connected spaces. o Let A be a connected simplicial set. {X 2 cSpaces | ifY is Kan and the basepoint evaluation map map(A; Y ) ! Y is a weak equivalence, then map(X; Y ) ! Y is also a weak equivalence }. The next theorem is a geometric interpretation of theorem 5.1. 6.3. Theorem. Let p1 : E1 ! B , p2 : E2 ! B and s : E1 ! E2 be maps such that p1 = p2Os and the homotopy fibers F ib(E1 p1!B), F ib(E2 p2!B) belong to C. For any map h : E1 ! Y , where Y 2 C: hocolim(E2 s E1 !h Y ) 2 C Proof. Example 3.12 implies that p1 and p2 are weakly equivalent, respectively to maps of the form: I I F ! sdB ; G ! sdB sdB sdB where F has values weakly equivalent to the homotopy fibers of p1 and G has values weakly equivalent to the homotopy fibers of p2. Since the definitions we* *re CLOSED CLASSES 13 natural, s : E1 ! E2 induces a natural transformation : F ! G . Theorem 5.1 implies then: I I hocolim(E2 E1 ! Y ) ' hocolim( G F ! Y ) 2 C sdB sdB __|_ | The following corollaries are particular cases of theorem 6.3. 6.4. Corollary. Let p1 : E1 ! B , p2 : E2 ! B and s : E1 ! E2 be maps such that p1 = p2Os. If the homotopy fibers F ib(E1 p1!B), F ib(E2 p2!B) and E1 belong to C, then so does E2. Proof. Apply theorem 6.3 to the case when Y = E1 and h = id. __|_ | 6.5. Corollary. (E. Dror Farjoun [7]). Let p : E ! B be a map such that the homotopy fiber F ib(E !p B) belongs to C. If E 2 C, then B 2 C. Proof. Apply corollary 6.4 to the case when p1 = p, p2 = idB and s = p. __|* *_ | 6.6. Corollary. (E. Dror Farjoun [7]). Let F ! E !p B be a fibration se- quence. If B and F belong to C, then so does E. Proof. Since B ! F ! E is a fibration sequence such that B and F belong to C, corollary 6.5 implies that E belongs to C. __|_ | 6.7. Corollary. Let F ! E !p B be a fibration sequence and B !s E be a section of p. If h : B ! Y is a map such that Y 2 C, then: colim(E s B !h Y ) 2 C Proof. Apply theorem 6.3 to the case when p1 = idB , p2 = p. __|_ | 6.8. Corollary. Closed classes are closed under split extensions. Let F ! E !p B be a fibration sequence such that p has a section. If F 2 C and B 2 C, then E 2 C. Proof. Apply corollary 6.7 to the case when Y = B and h = idB and s is a section of p. __|_ | 6.9. Corollary. (W. Dwyer [6]). Closed classes are closed under products. If X 2 C and Y 2 C, then X x Y 2 C . Proof. Notice that X ! X x Y ! Y is a fibration sequence with a section. According to corollary 6.8, X x Y belongs to C. __|_ | 14 WOJCIECH CHACHOLSKI 7. Homotopy properties of shapes of diagrams In this section the behavior of a closed class under unpointed homotopy colimits is going to be investigated further (see section 5). A diagram F : K ! Spaces consist of bunch of spaces which are related to each other by various maps. Those relations are coming from the geometry of K. We will try to understand how the geometry of K effects the homotopy colimit functor of diagrams over K. I 7.1. Definition. D(C) = {K | ifF : K ! C is a diagram, then F 2 C} K Class D(C) consists of those simplicial sets that carry enough information so by gluing elements of class C, according to K, we get back a space in C. 7.2. Proposition. D(C) C I Proof. Let K 2 D(C). Since ? 2 C, according to the definition, K = ? K belongs to C. Notice that corollary 5.4 is stronger thanIthis proposition. It says that if there exist a diagram F : K ! C such that F 2 C, then automatically __ K K 2 C. |_ | Observe that lemma 5.2 implies: 7.3. Proposition. For every n, [n] 2 D(C). It turns out that D(C) has nice homotopic properties. The next theorem suggests that to some extant, not geometry but the homotopy type of a simpli- cial set K plays a crucial role toward the homotopic properties of the homotopy colimit functor of diagrams over K. 7.4. Theorem. Class D(C) is closed under weak equivalences. 7.5. Lemma. o Let K L ,! M be a push-out diagram such that L ,! M is a cofibration. If K, L and M belong to D(C), then so does: K [L M = colim(K L ,! M ) o Let be the category associated with an ordinal number (see [8, page 11]). Let G : ! Spaces be a functor such that for every mor- phism ' 2 , G(') is a cofibration. If for every 2 , G() belongs to D(C), then so does colim G . Proof. We will prove only the first part of the lemma. The second part can be proven in the same way. Let F : K [L M ! C be a diagram. According to example 3.13: CLOSED CLASSES 15 I i I I I j F = colim F F ,! F K[LM K L M I I I By the assumption F , F and F belong to C. Since any closed class is K L M closed under taking homotopy push-outs: i I I I j hocolim F F ,! F 2 C K L M Notice that the cofibration assumption implies that the following map is a weak equivalence: i I I I j i I I I j hocolim F F ,! F ! colim F F ,! F K L M K L M I It implies that F belongs to C. __|_ | K[LM 7.6. Lemma. Let 0 k n. [n; k] belongs to D(C), where if o 2 [n]n is the non degenerate simplex, [n; k] is the simplicial subset of [n], generated by simplices {dio }i6=k. Proof. We are going to present [n; k] as a sequence of push-outs of standard simplices. In order to do this we have to introduce some notation: o Let i 2 {0; 1; . .;.n}. i denotes the simplicial subset of [n] generat* *ed by the simplex dio . o {i} denotes the simplicial subset of [n] generated by the vertex {i}. o Let {i; j} 2 {0; 1; . .;.n}. i;jdenotes the simplicial subset of [n] generated by the simplex di-1djo if i > j, or by dj-1 dio if i < j. There are obvious inclusions i;j! i ! [n]. Let X be the colimit of the following diagram: k+2x -id! k+2x id! nx 1x !id k-1x ? ? ? ? ? k+1;k+2? k+2;k+3? . . . n;0? 0;1? . . . k-2;k-1? y y y y y id k+1 k+3 id! 0 -id! 0 ! k-2 Out of the construction of X, we have two natural inclusions k-1 ! X, k+1 ! X. Notice that there is a cofibration map k-1;k+1 _{k}k-1;k+1 ! X which is the wedge of the following maps: k-1;k+1 ! k-1 ! X {k} ! k-1 ! X k-1;k+1 ! k+1 ! X By laborious but straightforward calculation one can show that: 16 WOJCIECH CHACHOLSKI id_id [n; k] = colim k-1;k+1 - k-1;k+1 _{k} k-1;k+1 ! X Since [n; k] is built from standard simplices by push-out process, where the maps involved are cofibrations, according to the lemma 7.5, [n; k] belongs to D(C). __|_ | Proof of the Theorem. The proof will be divided into several steps. Step 1. Let E '! B be a fibration and a weak equivalence. If E 2 D(C), then B 2 D(C). I Proof. Let F : B ! C be a diagram. We have to show that F 2 C. Accord- B ing to section A.1, the following is a pull-back square: I I F Op --- - ! F E? B? ?y ?y E ---p- ! B I I Since p is a fibration and a weak equivalence, F Op ! F is a weak equiva- I E B I lence as well. Because E 2 D(C), F Opbelongs to C. It implies that F 2 C. E B Step 2. Let f : X ! Y be a weak equivalence. If X 2 D(C), then Y 2 D(C). Proof. We are going to construct by the induction a sequence of spaces and inclusions: (X0 ! X1 ! X2 ! . .). together with a sequence of maps: {il: X ! Xl }l0 ; {pl : Xl ! Y }l0 such that Xl 2 D(C), il is a cofibration and a weak equivalence, ilOpl= f and the map colimit(pl) : colimit(Xl) ! Y is a fibration. We denote colimit(pl) by p : X ! Y , in particular X = colimit(Xl). Let X0 = X, i0 = idX and p0 = f . Lets assume that the construction has been carried out for i < l. Let J be the set of all commutative diagrams of the form: [n; k] --- - ! Xl-1 ?? ? y ?ypl-1 [n] --- - ! Y CLOSED CLASSES 17 where [n; k] ! [n] is the canonical inclusion. Xl is defined to be the simpli- cial set that fits into the following push-out square: F l-1 J[n;?k] -- - - ! X? ?y ?y F l J[n] -- - - ! X il is defined to be the following composition: X il-1!Xl-1 ! Xl pl is defined to be the push-out of the following maps: F F Xl-1 pl-1!Y ; J [n; k] ! Y ; J [n] ! Y By the inductive assumption Xl-1 2 D(C). Since Xl is built by gluing lots of [n] along [n; k] to Xl-1 , according to lemma 7.5, Xl 2 D(C). Observe that the natural map i : X = X0 ! X is a weak equivalence. Notice also that pOi = f . By Quillen's small object argument (see [10 ]), p is a fibration. Sin* *ce f and i are weak equivalences, so is p. Lemma 7.5 implies that X 2 D(C). Since p : X ! Y is a fibration and a weak equivalence, according to step 1, Y 2 D(C). Step 3. If X is contractible, then X 2 D(C). Proof. Since ? 2 D(C) and ? ! X is a weak equivalence, step 2 implies that X 2 D(C). I Step 4. Let F : K ! C be a diagram. The homotopy fiber F ib( F ! K) K belongs to C. Proof. Lets choose a connected component of K and a fibration P K ! K such that P K is contractible and the image of P KIis in the chosen component. According to corollary A.2, the homotopy fiber of F ! K over the chosen I K component is weakly equivalent to F. Since P K is contractible, according I P K to step 3, F belongs to C. P K Step 5. Let Z ! E ! B be a fibration sequence. If B 2 D(C) and Z 2 C, then E 2 C. 18 WOJCIECH CHACHOLSKI Proof. We can assume thatIp is a fibration. Example 3.12 implies that E ! B is weakly equivalent to dp ! sdB, where dp : sdB ! Spaces is a good sdB diagram whose values are weakly equivalent to Z. Proposition A.5 gives the following weak equivalence: I I I (dp)Oloe! dp B N(B=oe) sdB I Since N (B=oe) is contractible (dp)Oloe2 C. The assumption B 2 D(C) N(B=oe) implies: I I E ' (dp)Oloe2 C B N(B=oe) Step 6. Let f : X ! Y be a weak equivalence. If Y 2 D(C), then X 2 D(C). Proof. Let FI: X ! C be a diagram. Notice that the homotopy fiber of the composition F ! X f! Y is weakly equivalent to the homotopy fiber I X F ib( F ! X). According to step 4, it belongs to C. Since Y 2 D(C), Step 5 X I implies that F 2 C. This proves that X 2 D(C). __ X |_ | Theorem 7.4 implies an interesting characterization of a closed class (see also[1]): 7.7. Corollary. Non empty class C of connected simplicial sets is closed if it* * is closed under weak equivalences and for every, not necessarilyIpointed diagram, F : K ! C over a contractible simplicial set K, F 2 C. K The definition of a closed class says that it is closed under pointed ho- motopy colimits. It means that for any pointed diagram F : K ! Spaces? the homotopy cofiber: I Z Cof K ! F = F K K belongs to C. The next corollary implies that the dual statement is also true (see [6] and [7] for discussion of similar statements). I 7.8. Corollary. Let F : K ! C be a diagram. F ib( F ! K) 2 C. K The following corollary says that if the the pre-images of simplices have certain properties (belong to a closed class), then so does the homotopy fiber (see also [7]). CLOSED CLASSES 19 7.9. Corollary. Let f : X ! K be a map. If for every simplex oe 2 K: f pullback X ! K [dim(oe)] 2 C then F ib(X !f Y ) 2 C. Proof. AccordingIto example 3.12, f : X ! K is weakly equivalent to a map of the form df ! sdK, where for v = (oe0 ! . . .! oen ) 2 sdK, df is a sdK diagram such that: f df (v) = pullback X ! K [dim(oen )] I __ Corollary 7.8 implies F ib df ! sdK 2 C. |_ | sdK 8. Class D(C) In this section we present other characterizations of the class D(C). We will restrict the class of diagrams on which a simplicial set should be tested * *in order to find out if it belongs to D(C). We will show also that the class D(C) is a closed class and is closed under extensions fibrations. 8.1. Proposition. I D(C) = {K | ifF : K ! C is a bounded diagram, then F 2 C} K Proof. Let: I D0 = {K | ifF : K ! C is a bounded diagram, then F 2 C} K Inclusion D(C) D is obvious. By the same arguments as in theorem 7.4, we can show that the class D0 is closed under weak equivalences.ILet K 2 D0 and F : KI! C be a diagram. According to remark A.6, F is weakly equivalent to Fsd . Since sdK is K sdK weakly equivalentIto K, it belongs to D0.INotice that Fsd is a bounded diagram, therefore Fsd 2 C. It implies that F 2 C and K 2 D(C). __|_ | sdK K 8.2. Proposition. D(C) = {B | ifZ ! E ! B is a fibration sequence and Z 2 C; then E 2 C} Proof. Let: D0 = {B | ifZ ! E ! B is a fibration sequence and Z 2 C; then E 2 C} 20 WOJCIECH CHACHOLSKI I I Let B 2 D and F : B ! C be a diagram. Since F ! F ! B is a I P BI B fibration sequence (see A.2) and F 2 C, we get F 2 C. It implies the P B B inclusion D0 D(C). Let K 2 D(C) and Z ! E !p K be a fibration sequence. AccordingIto example 3.12, E ! K is weakly equivalent to a map of the form F ! L, L where the values of F are weakly equivalent to Z. SinceIL is weakly equivalent to K, it belongs to D(C). As a consequence we get E ' F 2 C. It proves __ L that K 2 D0 and D D0. |_ | 8.3. Corollary. A closed class C is closed under extensions by fibrations if and only if C = D(C). The next corollary says that class D(C) is usually quite big. 8.4. Corollary. If B is such that B 2 C, then B 2 D(C). Proof. See corollary 6.6. __|_ | 8.5. Proposition. I D(C) = {K | ifF : K ! C is a good diagram, then F 2 C} K Proof. Let: I D0 = {K | ifF : K ! C is a good diagram, then F 2 C} K Inclusion D(C) D0 is obvious. By the same arguments as in the theorem 7.4, we can show that the class D0 is closed under weak equivalences. Let B 2 D0 and p : E ! B be a fibration such that the fiber of p belongs to C. AccordingIto example 3.12, p : E ! B is weakly equivalent to a map of the form dp ! sdB, where dp is a good sdB diagramIwhose values are weakly equivalent to the fiber of p. It implies that E ' dp 2 C. This proves the proposition. __|_ | sdB 8.6. Theorem.ILet G : K ! D(C) be a diagram. If K belongs to D(C), then so does G . K I I Proof. According to remark A.6, G is weakly equivalent to Gsd . Theo- I K I sdK rem 7.4 implies that G belongs to D(C) if and only if Gsd does. Since K sdK CLOSED CLASSES 21 Gsd is a bounded diagram, without loss of generality, it is enough to prove the theorem for a bounded diagram G : K ! D(C) . I Let G : D(C) ! b e a bounded diagram and F : G ! C be a diagram. I K I I Theorem A.9 implies that H F is weakly equivalent to F . Since K G I sdK G(v) G has values in D(C), then so does G, therefore F belongs to C. IG(v)I Because K 2 D(C), sdK 2 D(C) and it follows that F belongs to I sdK G(v) C. This proves H F 2 C. __|_ | K G 8.7. Corollary. D(C) is a closed class and D(D(C)) = D(C), therefore D(C) is closed under extensions by fibrations. 9. Theorem of E. Dror Farjoun 9.1. Theorem. Let : E ! B be a natural transformation between diagrams E : K ! Spaces and B : K ! Spaces . If for every simplex oe 2 K the homo- topy fiber F ib(E(oe) -oe!B(oe)) belongs to C, then: I I F ib E -! B 2 C K K 9.2. Lemma. Lets consider the following commutative diagram: E1 - - - - E2 --- - ! E3 ?? ? ? y p1 ?yp2 ?yp3 B1 - - f-- B2 ---g- ! B3 where the maps E2 ! E3, B2 !g B3 are cofibrations. If F ib(p1), F ib(p2) and F ib(p3) belong to C, then: F ib(E1 [E2 E3 ! B1 [B2 B3) 2 C Proof. Without loss of generality we can assume that p1, p2 and p3 are fibra- tions. Let p = colim(p1 p2 ! p3). According to corollary 7.9, it is enough to prove that for every simplex, oe 2 B1 [B2 B3: F (oe) = pullback(E1 [E2 E3 !p B1 [B2 B3 [dim(oe)]) 2 C Let oe 2 B1 [B2 B3. Either oe lies in the image of B1 or B2. Lets assume that it belongs to the image of B1. Let K = pullback([dim(oe)] ! B1 f B2). There 22 WOJCIECH CHACHOLSKI is a natural map K ! B2. Let X1, X2 and X3 be simplicial sets that fit into the following pull-back squares: X1 --- - ! E1 X2 --- - ! E2 X3 --- - ! E3 ?? ? ? ? ? ? y ?yp1 ?y ?yp2 ?y ?yp3 [dim(oe)] --- - ! B1 K --- - ! B2 K --- - ! B3 Observe that the definition gives natural maps X2 ! X1 and X2 ! X3. By straightforward combinatorial calculation one can show: F (oe) = colim(X3 X2 ! X1) Notice that the maps X3 ! K, X2 ! K and X2 ! X1 satisfy the assump- tions of theorem 6.3, therefore hocolim(X3 X2 ! X1) 2 C. Cofibration assumption of the lemma implies: hocolim(X3 X2 ! X1) ' colim(X3 X2 ! X1) It proves the lemma. __|_ | Proof of the theorem. Instead of E : K ! Spaces , B : K ! Spaces we can co* *n- sider bounded diagramsIEsd : sdKI ! Spaces ,IBsd : sdK I! Spaces . Since the homotopy fibers F ib E ! B and F ib Esd ! Bsd are weakly K K sdK sdK equivalent, it is enough to prove the theorem for bounded diagrams. The proof will be by the induction on the dimension of K. If dim(K) = 0, the theorem is obvious. Lets assume that the theorem is true for K such that dim(K) < n. Let L be a simplicial of dimension less than n and _[n] ! L be a map. We prove that the theorem holds for K = L [ _[n][n]. Let o 2 ([n])n be the only non degenerate simplex. Lets consider the following diagram: I I E = colim E _ [n] x E(o ) ,! [n] x E(o ) K? L? ? ? y y ?yidxo ?yidxo I I B = colim B _ [n] x B(o ) ,! [n] x B(o ) K L I I By the inductive assumption F ib E ! B belongs to C. Since the homo- L L topy fiber F ib(E(o ) o! B(o ))also belongs to C, according to lemma 9.2: I I F ib E ! B 2 C __ K K |_ | As a corollary we get the theorem of E. Dror Farjoun CLOSED CLASSES 23 9.3. Corollary. (E. Dror Farjoun [7, theorem I]). Let E : K ! Spaces? and B : K ! Spaces? be pointed diagrams and : E ! B be a natural transfor- mation. If for every simplex oe 2 K, F ib(E(oe) -oe!B(oe)) 2 C , then: Z Z F ib E -! B 2 C K K Proof. Lets consider the following diagram: Z I E ' hocolim ? K ! E K? ? ? K? y ?y ?yid y Z I B ' hocolim ? K ! B K K I I According to theorem 9.1, F ib E ! B belongs to C. Since the ho- K K motopy fiber F ib(K id!K) belongs to C, applying once again theorem 9.1 we get: Z Z F ib E ! B 2 C __ K K |_ | 9.4. Theorem. Let the following be a homotopy push-out square: A ---f- ! B ?? ? yi ?y X ---g- ! Y If the homotopy fiber F ib(A !f B) belongs to C, then so does the homotopy fiber F ib(X !g Y ). Proof. Lets consider the following diagram: i id X? ' hocolim X? A? ! A? ?yg ?yid ?yid ?yf i f Y ' hocolim X A ! B Since F ib(A !f B) belongs to C, theorem 9.1 implies that F ib(X !g Y ) 2 C . * * __|_ | 24 WOJCIECH CHACHOLSKI 9.5. Corollary. Let f : X ! Y be a map. If X belongs to C, then so does the homotopy fiber F ib Y ! Cof (X !f Y ) . Proof. Apply theorem 9.4 to the following homotopy push-out square: X? __________! ?? ?y ?y Y - ! Cof (X !f Y ) __|_ | Appendix A. The homotopy colimit The reference for the proofs of the statements, listed in the appendix, is [5]. A.1. Pulling-back of diagrams. Let F : K ! Spaces be a diagram over K and f : L ! K be a map. We can pull-back F into a diagram F Of over L (F Of will be often denoted simply by F ). Let o and oe be simplices in L and ' : o !* * oe be a morphism in the category associated with L. F Of : L ! Spaces is defined as follows: F Of(oe) = F (f (oe)) F Of (') = F (') The basic property of the pull-back diagram F Of is that the following is a pull-back square: I I F Of --- - ! F L? K? ?y ?y L ---f- ! K As corollary of the this property we get: A.2. Corollary. Let F : K ! Spaces be a diagram and P K ! K be a fibra- tion such that P K is contractible. The following is a fibration sequence: I I F ! F ! K P K K CLOSED CLASSES 25 A.3. Diagrams over colimIG. Let G : I ! Space , F : colimIG ! Spaces be diagrams. There is a family of maps {li: G(i) ! colimIG }i2I which satisfies the universal property of the colimit of the diagram G : I ! Space (see [8, 3x* *3]). Out of this data we can construct a functor: I - ! SpacesI i 7- ! F Oli G(i) I H F OlbI (a '! b) 7- ! ( F Ola G(')-! F Olb) G(a) G(b) This functor has the followingIproperty: I F = colimG F Oli colimIG G(i) A.4. Diagrams over sdK. Let oe be a simplex in K. By K=oe we denote the over category of K (see section 2). There is a functor: K=oe ! K (o ! oe) 7! o ; (o0 ! o1 ! oe) 7! (o0 ! o1) This functor induces a map between simplicial sets: loe: N (K=oe) ! N (K) = sdK (o0 ! . .!.on ! oe) loe7!(o0 ! . .!.on ) One can verify that the family of maps {loe: N (K=oe) ! sdK }oe2K satisfies the universal property of the colimit of the functor: K ! Spaces ; oe 7! N (K=oe) It implies: sdK = colimK N (K=oe) Let F : sdKI ! SpacesI be a diagram. AccordingIto subsection A.3: F = F = colimK F Oloe sdK colimK N(K=oe) N(K=oe) A.5. Proposition.ITheInatural map: I I F Oloe! colimK F Oloe= F K N(K=oe) N(K=oe) sdK is a weak equivalence. A.6. Remark. Let F : K ! Spaces be a diagram. This means that F is a functor over the category associated to K. According to example 3.14,Iit de- fines a bounded diagram Fsd : sdK ! Spaces . It turns out that F is weakly I I K equivalent to Fsd and Fsd = hocolimK F , where hocolimK F is the sdK sdK homotopy colimit of F : K ! Spaces in the sense of Bousfield-Kan. 26 WOJCIECH CHACHOLSKI I A.7. Diagrams over G . Let G : K ! Spaces be a diagram. Out of G we K can construct a new diagram G : sdK ! Spaces . Let: u = (o0 !0 o1 !1 ::: m-1!om) 2 (sdK)m v = (oe0 '0!oe1 '1!::: 'n-1!oen) 2 (sdK)n j : u ! v be a morphism in sdK By definition 3.1, j is a morphism in such that j : [n]! [m] and sdK(j)(u) = v. G : sdK ! Spaces is a diagram defined as follows: G(v) = [dim(oen )] x G(oe0) 8 >>>id x id ifj(0) = 0 ; j(n) = m < id x G( O . .O. ) ifj(0) > 0 ; j(n) = m G(j) = > j(0)-1 0 >>:[ m-1 O . .O. j(n)] x id ifj(0) = 0 ; j(n) < m [ m-1 O . .O. j(n)] x G( j(0)-1O . .O. 0) ifj(0) > 0 ; j(n) < m Observe that G has the values weakly equivalent to the values of G. A.8. Proposition. I G = colimsdK G K I A.9. Theorem. Let G : K ! Spaces and F : G ! Spaces be diagrams. If K G is a bounded diagram,IthenIthe following natural map is a weak equivalence: I I F ! colimsdK F = H F sdK G(v) G(v) K G References 1. A.Amit, Direct limits over diagrams with contractible nerve, Master thesis,* * Hebrrew Univ. (1993). 2. A.K.Bousfield, Localization and periodicity in unstable homotopy theory, pr* *eprint. 3. A.K.Bousfield and D.M.Kan, Homotopy Limits, Completions and Localizations, * *Lect. Notes in Math. 304, Springer (1972) 4. W. Chacholski, Functors CWA and PA , Ph.D. thesis, Univ. of Notre Dame (199* *5). 5. W.Chacholski, Homotopy properties of shapes of diagrams, report No. 6, 1993* */94, Insti- tut Mittag-Leffler. 6. E.Dror Farjoun, Cellular spaces, preprint. 7. E.Dror Farjoun, Cellular inequalities, Proc. to the conf. in Alg. Top. Nort* *heastern Univ. June 1993, Springer Verlag. 8. S. MacLane, Categories for working mathematician, Grad. Texts in Math. 5, S* *pringer (1971). 9. J.Peter May, Simplicial objects in algebraic topology, Van Nostrand Math. S* *tudies 11, (1987). 10. D. Quillen, Homotopical algebra, Lect. Notes in Math. 43, Springer (1967) CLOSED CLASSES 27 11. G.Segal, Classifying spaces and spectral sequences, Inst. haut. Etul. sci.,* * Publ. math. 34, 105-112 (1968). Wojciech Chacholski, Department of Mathematics, University of Notre Dame, Mail Distribution Center, Notre Dame, Indiana 46556-5683 E-mail address: wchacho1@kenna.math.nd.edu