Diagonal fibrations are pointwise fibrations

Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007, pp.1–12 DIA GONAL FIBRA TIONS AR E POINTWISE FIBRA TIONS DEDICA TE D TO THE MEMOR Y OF SA UND ERS MAC LANE(1909-2005) A.M. CEGA RRA and J. REMEDIOS ( c ommunic ate d by Name of Editor ) Abstr a ct On the categ ory of bisimplicial se ts there ar e differe nt Quillen closed model struc tur es asso ciated to v arious defini- tions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bis implicial s et a s a simpli- cial ob ject in the categ o ry of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a K an fibr a tion, that is, the p oint wise fibrations. In another of them, int ro duced by Mo er dijk, a bisimplicial map is a fibration if it induces a Ka n fibr ation of asso cia ted dia gonal simplicial sets, that is, the diagona l fibrations. In this note, we prov e that every diago nal fibration is a p oint wise fibra tion. 1. In tro duction and summary There are several (Quillen) close d model structures on the catego ry of bisimplical sets, see [ 3 , IV, § 3]. This pap er conc e r ns t wo of them, namely , the so-ca lled Bousfield- Kan and Mo er dijk s tr uctures, tha t we briefly reca ll b elow: On the one hand, in the close d mo del s tructure by Bo usfield-Kan, bisimplicial sets are r egarded as diagrams of simplicial sets a nd then fibrations ar e the p oint wise Kan fibrations and weak equiv a lences are the p o in twise w eak homotopy equiv alences. T o be more p rec ise, a bisimplicial set X : ∆ op × ∆ op → Set , ([ p ] , [ q ]) 7→ X p,q , is seen as a “hor izontal” simplicial ob ject in the categor y of “vertical” simplicial sets, X : ∆ op → S , [ p ] 7→ X p, ∗ and then, a bisimplicial map f : X → Y is a fibration (resp. a weak equiv alence) if all simplicial ma ps f p, ∗ : X p, ∗ → Y p, ∗ , p > 0, are Kan fibrations (resp. weak homotopy equiv alences). On the other hand, the Mo erdijk closed mo del str ucture o n the bisimplicial set category is transferred from the ordina ry mo del structure on the simplicial s et category through the diagonal functor, X 7→ diag X : [ n ] 7→ X n,n . Th us, in this closed mo del structure, a bisimplicial map f : X → Y is a fibra tion (resp. a weak Supported by DGI of Spain: ( MTM 2004-01060 and MTM2006-06317), FEDER, and Junt a de Andaluc ´ ıa: P06-FQM-1889. Receiv ed Day Mont h Y ear, revised Da y M on th Y ea r; published on Day Mon th Y ear. 2000 Mathematics Sub ject Cl assification: 55U10, 5P05, 55U35 Key words and phr ases: bisimplicial set, closed mo del structure, fibration, weak homotop y equiv- alence c  2007, A.M. Cegarra and J. Remedios. Permission to cop y f or priv ate use gran ted. Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007 2 equiv alence) if the induced diag onal simplicial map diag f : diag X → diag Y is a Kan fibration (resp. a weak ho motopy equiv alence). Several useful relations hips b et ween these tw o different homotopy theories of bisimplicial sets hav e b een established a nd, p erhaps, the b est known o f them is the following: Theorem. (Bousfield-Kan) L et f : X → Y b e a bi simplicial map such that f p, ∗ : X p, ∗ → Y p, ∗ is a we ak homotopy e qu ivalenc e for e ach p > 0 . Then diag f : diag X → diag Y is a we ak homotopy e quivalenc e . The purp ose o f this br ief note is to state and pr ov e a suitable coun terpart to Bousfield-Ka n’s theorem for fibrations, namely: Theorem 1 . L et f : X → Y b e a bisimplicial map such that diag f : diag X → diag Y is a Kan fi br ation. Then f p, ∗ : X p, ∗ → Y p, ∗ is a Kan fibr ation for e ach p > 0 . Note that the converse of Theorem 1 is not true in g eneral. A co un terexa mple is given in the last s ection of the pap er. Ac kno wledgem en ts. The authors are muc h indebted to the referees , whos e useful observ ations gr eatly improved our exp osition. The se cond author is gr ateful to the Algebra Department in the Universit y of Gr a nada for the excellent atmosphere and ho spitalit y . 2. Some prelimin aries W e use the standard c o n ven tions and terminology which can be fo und in texts on simplicial homotopy theory , e. g. [ 3 ] or [ 6 ]. F o r definiteness o r emphasis we state the following. W e denote by ∆ the ca tegory of finite or de r ed sets of integers [ n ] = { 0 , 1 , . . . , n } , n > 0, with w eakly order-preser ving ma ps betw een them. The category of simplicial sets is the category of functors X : ∆ op → Set, where Set is the category o f se ts . If X is a s implicia l set and α : [ m ] → [ n ] is a ma p in ∆, then we write X n = X [ n ] and α ∗ = X ( α ) : X n → X m . Recall that all maps in ∆ are gener ated by the injections δ i : [ n − 1] → [ n ] (cofaces), 0 6 i 6 n , which miss out the i th elemen t and t he surjections σ i : [ n + 1] → [ n ] (co degenera cies), 0 6 i 6 n , which rep eat the i th element (see [ 5 , VI I, § 5, Pro po sition 2]). Thus, in o r der to define a s implicial set, it suffices to give the sets o f n -simplic es X n , n > 0, together with maps d i = δ ∗ i : X n → X n − 1 , 0 6 i 6 n (the face maps) , s i = σ ∗ i : X n → X n +1 , 0 6 i 6 n (the degener acy maps) , satisfying the well-known basic s implici al identities such as d i d j = d j − 1 d i if i < j , etc. (see [ 5 , p. 175 ]). In addition, we shall write down a list o f other identities betw een some itera ted comp ositio ns of face and degeneracy maps, whic h will be used latter. The pro of o f these equalities is straightforward and left to the reader. Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007 3 Lemma 1. On any simplicial set , the fol lowing e qualities hold: d i d m j = d m j d i + m if i > j (1) d m i = d m − 1 i d j if i 6 j < i + m (2) d i s m j = s m j d i − m if i > j + m (3) d i s m j = s m − 1 j if j 6 i 6 j + m. (4) Let f : X → Y b e a simplicia l map. A collectio n of simplices x i ∈ X n − 1 , i ∈ I , y ∈ Y n , where I ⊆ [ n ] is any subset, is said to b e f -c omp atibl e whenever the following equalities hold: d i x j = d j − 1 x i for all i, j ∈ I , i < j, d i y = f x i for all i ∈ I . The map f is said to b e a Kan fibr ation whenever for every given collec tion of f -compatible simplices x 0 , . . . , x k − 1 , x k +1 , . . . , x n ∈ X n − 1 , y ∈ Y n , there is a simplex x ∈ X n such that d i x = x i for all i 6 = k and f x = y . The next lemma (cf. [ 6 , Lemma 7.4]) w ill be very use ful in our developmen t. F or I any finite set, | I | deno tes its n umber of element s. Lemma 2 . L et f : X → Y b e a Kan fibr ation. Supp o se t hat ther e ar e given a subset I ⊆ [ n ] such that 1 6 | I | 6 n and an f -c omp atible family of simplic es x i ∈ X n − 1 , i ∈ I , y ∈ Y n . Then, t her e exists x ∈ X n such that d i x = x i for al l i ∈ I and f x = y . Pr o of . Supp ose | I | = r . If r = n , the statement is true since f is a Ka n fibra tion. Hence the statement holds for n = 1. W e now pro ceed by induction: Ass ume n > 1 and the result holds for n ′ < n and assume r < n and the result ho lds for r ′ > r . T aking k = max { i | i ∈ [ n ] , i / ∈ I } , we wish to find a simplex x k ∈ X n − 1 , such that the collection of simplices x i ∈ X n − 1 , i ∈ I ∪ { k } , y ∈ Y n , be f -compatible, since then an application of t he induction hypo thesis on r giv es the claim. T o find such an x k , let I ′ ⊆ [ n − 1 ] b e the subset I ′ = { i | i ∈ I , i < k } ∪ { i − 1 | i ∈ I , i > k } , Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007 4 and let x ′ i ∈ X n − 2 , i ∈ I ′ , y ′ ∈ Y n − 1 , be the family of simplices defined by x ′ i = d k − 1 x i for i ∈ I , i < k , x ′ i − 1 = d k x i for i ∈ I , i > k , and y ′ = d k y . This family is f -co mpatible, whence the induction hypothesis on n − 1 g iv es the re quired x k , that is, a ( n − 1 )-simplex o f X s atisfying d i x k = x ′ i for i ∈ I ′ and f x k = y ′ . The category of bi simplicial sets is the categor y of functors X : ∆ op × ∆ op → Set. It is often c o n venien t to see a bisimplicial set X as a (horizontal) simplicial o b j ect in the categor y o f (vertical) simplicial sets. If α : [ p ] → [ p ′ ] a nd β : [ q ] → [ q ′ ] a re any t wo maps in ∆, then we will wr ite α ∗ h : X p ′ ,q → X p,q and β ∗ v : X p,q ′ → X p,q for the images X ( α, id ) and X ( id, β ) r esp e ctiv ely . In particular, the horizo n tal and vertical fa ce and degeneracy maps are d h i = ( δ i ) ∗ h , d v i = ( δ i ) ∗ v , s h i = ( σ i ) ∗ h and s v i = ( σ i ) ∗ v . By comp osing with the diago nal functor ∆ → ∆ × ∆, [ n ] 7→ ([ n ] , [ n ]), we g e t the diagonal functor from bisimplicial sets to simplicial sets, which pro vides a simplicial set dia g X : [ n ] 7→ X n,n , asso ciated to eac h bisimplicial set X , whose f ace and degeneracy o per ators ar e g iven in terms of thos e of X b y the form ulas d i = d h i d v i and s i = s h i s v i , resp ectively . 3. Pro of of Theorem 1 Let p > 0 be any fixed in teger. Then, in order to prov e th at the simplicial set map f p, ∗ : X p, ∗ → Y p, ∗ is a Kan fibration, suppos e that, for some in tegers q > 1 and 0 6 ℓ 6 q , there is given a collection of bisimplices x 0 , . . . , x ℓ − 1 , x ℓ +1 , . . . , x q ∈ X p,q − 1 , y ∈ Y p,q , which is f p, ∗ -compatible, that is, such that d v i x j = d v j − 1 x i for 0 6 i < j 6 q , i 6 = ℓ 6 = j , and d v i y = f x i for 0 6 i 6 q , i 6 = ℓ . W e must therefore find a bisimplex x ∈ X p,q such tha t d v i x = x i for 0 6 i 6 q , i 6 = ℓ, f x = y . (5) T o do tha t, we sta r t by co nsidering the subset I ⊆ [ p + q ] defined b y I = { i | 0 6 i < ℓ } ∪ { p + i | ℓ < i 6 q } , Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007 5 and the family o f dia gonal bisimplices ¯ x i ∈ ( diag X ) p + q − 1 , i ∈ I , (6) ¯ y ∈ ( diag Y ) p + q , where ¯ x i = ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ ( s v ℓ − 1 ) p x i for 0 6 i < ℓ, (7) ¯ x p + i = ( s h 0 ) ℓ ( s h p ) q − ℓ − 1 ( s v ℓ ) p x i for ℓ < i 6 q , (8) ¯ y = ( s h 0 ) ℓ ( s h p ) q − ℓ ( s v ℓ ) p y . (9) The next verifications show that this family (6) o f diagona l bisimplices is actually diag f -compatible: - for 0 6 i < j < ℓ , d i ¯ x j = d h i d v i ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ ( s v ℓ − 1 ) p x j = d h i ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ d v i ( s v ℓ − 1 ) p x j = ( s h 0 ) ℓ − 2 ( s h p ) q − ℓ ( s v ℓ − 2 ) p d v i x j by (4) = ( s h 0 ) ℓ − 2 ( s h p ) q − ℓ ( s v ℓ − 2 ) p d v j − 1 x i = ( s h 0 ) ℓ − 2 ( s h p ) q − ℓ d v j − 1 ( s v ℓ − 1 ) p x i = d h j − 1 ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ d v j − 1 ( s v ℓ − 1 ) p x i by (4) = d h j − 1 d v j − 1 ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ ( s v ℓ − 1 ) p x i = d j − 1 ¯ x i . - for 0 6 i < ℓ < j 6 q , d i ¯ x p + j = d h i d v i ( s h 0 ) ℓ ( s h p ) q − ℓ − 1 ( s v ℓ ) p x j = d h i ( s h 0 ) ℓ ( s h p ) q − ℓ − 1 d v i ( s v ℓ ) p x j = ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ − 1 ( s v ℓ − 1 ) p d v i x j by (4) = ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ − 1 ( s v ℓ − 1 ) p d v j − 1 x i = ( s h 0 ) ℓ − 1 d h p + j − ℓ ( s h p ) q − ℓ ( s v ℓ − 1 ) p d v j − 1 x i by (4) = d h p + j − 1 ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ d v p + j − 1 ( s v ℓ − 1 ) p x i by (3) = d h p + j − 1 d v p + j − 1 ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ ( s v ℓ − 1 ) p x i = d p + j − 1 ¯ x i . Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007 6 - for ℓ < i < j 6 q , d p + i ¯ x p + j = d h p + i ( s h 0 ) ℓ ( s h p ) q − ℓ − 1 d v p + i ( s v ℓ ) p x j = ( s h 0 ) ℓ d h p + i − ℓ ( s h p ) q − ℓ − 1 ( s v ℓ ) p d v i x j by (3) = ( s h 0 ) ℓ ( s h p ) q − ℓ − 2 ( s v ℓ ) p d v j − 1 x i by (4) = ( s h 0 ) ℓ d h p + j − ℓ − 1 ( s h p ) q − ℓ − 1 ( s v ℓ ) p d v j − 1 x i by (4) = d h p + j − 1 ( s h 0 ) ℓ ( s h p ) q − ℓ − 1 d v p + j − 1 ( s v ℓ ) p x i by (3) = d h p + j − 1 d v p + j − 1 ( s h 0 ) ℓ ( s h p ) q − ℓ − 1 ( s v ℓ ) p x i = d p + j − 1 ¯ x p + i . - for 0 6 i < ℓ , d i ¯ y = d h i d v i ( s h 0 ) ℓ ( s h p ) q − ℓ ( s v ℓ ) p y = d h i ( s h 0 ) ℓ ( s h p ) q − ℓ d v i ( s v ℓ ) p y = ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ ( s v ℓ − 1 ) p d v i y by (4) = ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ ( s v ℓ − 1 ) p f x i = f ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ ( s v ℓ − 1 ) p x i = f ¯ x i . and, fina lly , - for ℓ < i 6 q , d p + i ¯ y = d h p + i d v p + i ( s h 0 ) ℓ ( s h p ) q − ℓ ( s v ℓ ) p y = d h p + i ( s h 0 ) ℓ ( s h p ) q − ℓ d v p + i ( s v ℓ ) p y = ( s h 0 ) ℓ d h p + i − ℓ ( s h p ) q − ℓ ( s v ℓ ) p d v i y by (3) = ( s h 0 ) ℓ ( s h p ) q − ℓ − 1 ( s v ℓ ) p d v i y by (4) = ( s h 0 ) ℓ ( s h p ) q − ℓ − 1 ( s v ℓ ) p f x i = f ( s h 0 ) ℓ ( s h p ) q − ℓ − 1 ( s v ℓ ) p x i = f ¯ x p + i . Then, since by h yp othesis diag f : diag X → diag Y is a Kan fibra tion, from Lemma 2 we g et a diagona l bisimplex ¯ x ∈ ( diag X ) p + q such tha t d h i d v i ¯ x = ¯ x i for 0 6 i < ℓ , d h p + i d v p + i ¯ x = ¯ x p + i for ℓ < i 6 q and f ¯ x = ¯ y . Now, us ing the bisimplex ¯ x w e construct the bisimplex Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007 7 x = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ ( d v ℓ ) p ¯ x ∈ X p,q , which, we claim, s atisfies Relations (5 ). Actually : - for 0 < i < ℓ , d v i x = d v i ( d h p +1 ) q − ℓ ( d h 0 ) ℓ ( d v ℓ ) p ¯ x = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ d v i ( d v ℓ ) p ¯ x = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ ( d v ℓ − 1 ) p d v i ¯ x = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ − 1 d h i ( d v ℓ − 1 ) p d v i ¯ x by (2) = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ − 1 ( d v ℓ − 1 ) p d h i d v i ¯ x = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ − 1 ( d v ℓ − 1 ) p ¯ x i = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ − 1 ( d v ℓ − 1 ) p ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ ( s v ℓ − 1 ) p x i by (7) = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ − 1 ( s h 0 ) ℓ − 1 ( s h p ) q − ℓ ( d v ℓ − 1 ) p ( s v ℓ − 1 ) p x i = x i . - for ℓ < i 6 q , d v i x = d v i ( d h p +1 ) q − ℓ ( d h 0 ) ℓ ( d v ℓ ) p ¯ x = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ d v i ( d v ℓ ) p ¯ x = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ ( d v ℓ ) p d v p + i ¯ x by (1) = ( d h p +1 ) q − ℓ − 1 d h p + i − ℓ ( d h 0 ) ℓ ( d v ℓ ) p d v p + i ¯ x by (2) = ( d h p +1 ) q − ℓ − 1 ( d h 0 ) ℓ d h p + i ( d v ℓ ) p d v p + i ¯ x by (1) = ( d h p +1 ) q − ℓ − 1 ( d h 0 ) ℓ ( d v ℓ ) p d h p + i d v p + i ¯ x = ( d h p +1 ) q − ℓ − 1 ( d h 0 ) ℓ ( d v ℓ ) p ( s h 0 ) ℓ ( s h p ) q − ℓ − 1 ( s v ℓ ) p x i by (8) = ( d h p +1 ) q − ℓ − 1 ( d h 0 ) ℓ ( s h 0 ) ℓ ( s h p ) q − ℓ − 1 ( d v ℓ ) p ( s v ℓ ) p x i = x i . and, fina lly , Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007 8 f x = f ( d h p +1 ) q − ℓ ( d h 0 ) ℓ ( d v ℓ ) p ¯ x = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ ( d v ℓ ) p f ¯ x = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ ( d v ℓ ) p ¯ y = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ ( d v ℓ ) p ( s h 0 ) ℓ ( s h p ) q − ℓ ( s v ℓ ) p y by (9) = ( d h p +1 ) q − ℓ ( d h 0 ) ℓ ( s h 0 ) ℓ ( s h p ) q − ℓ ( d v ℓ ) p ( s v ℓ ) p y = y . Hence Theo r em 1 is proved. Let us stress that Theore m 1 not only says that, when diag f : diag X → diag Y is a Kan fibration, all simplicial maps f p, ∗ : X p, ∗ → Y p, ∗ are Kan fibrations, but that the simplicial maps f ∗ ,p : X ∗ ,p → Y ∗ ,p are Kan fibrations as well. This is a consequence of the symmetry of the hypothesis: “ diag f is a K an fibration” , that is, the fact follows by exchanging the vertical and hor izontal direc tions. 4. The con v erse of T heorem 1 is false It is p ossible that, for a bisimplicial map f : X → Y , a ll simplicial ma ps f p, ∗ and f ∗ ,p , p > 0, be K an fibrations and, howev er, diag f b e not a Kan fibratio n: T a k e X to b e the double nerve of an s uita ble double group oid (i.e. a group oid ob ject in the category o f gr oup o ids); then, all simplicia l sets X p, ∗ and X ∗ ,p are K an complexes (since they a re nerves of gro up oids) but, a s we show b elow, the diago na l simplicial set diag X is not necess a rily a Kan complex. W e briefly rec a ll some standard termino logy ab out double gro up oids; see for example [ 2 ] or [ 4 ]. A double group oid consists of ob jects a , b , ..., horizontal and vertical mo rphisms b etw een them a f → b, b ↑ g a , ... , and sq uares σ , τ , etc., of the form c b f o o σ d g ′ O O a f ′ o o g O O . These satisfy axioms such that the ob jects together with the horizo n tal morphisms as well as the ob jects together with the vertical mo r phisms form group oids. F urther - more, the s quares for m a gr oup oid under both hor izontal and vertical juxtap osition, Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007 9 and in the situatio n · · o o · o o σ τ · O O · o o O O · O O o o γ δ · O O · o o O O · o o O O horizontal and vertical comp osition commute in the sens e that ( σ · h τ ) · v ( γ · h δ ) = ( σ · v γ ) · h ( τ · v δ ) . In additio n to the vertical iden tity morphisms, there are horizo n tal identit y s quares for each vertical morphism. Its hor izontal edges are the iden tity a r rows o f the ho r- izontal group oid of arrows. Similarly , there ar e vertical identit y sq uares for each horizontal mor phism with vertical identit y a rrows. These ident ity squar es are com- patible in the sense that vertical and horizontal identit y squares for the vertical and horizontal identit y mo rphisms a re the same. Given a double groupo id G , o ne can construct its bisimplicial nerv e (or double nerve) NN G . A typical ( p, q ) simplex of NN G is a sub divisio n of a square of G a s matrix of p × q horizont ally and vertically co mpo sable s quares o f the form · · o o · o o · · · · · o o σ 11 σ 21 σ p 1 · O O · O O o o · O O o o · · · · O O · O O o o σ 12 σ 22 σ p 2 · O O · O O o o · O O o o · · · · O O · O O o o . . . . . . . . . . . . . . . · · o o · o o · · · · · o o σ 1 q σ 2 q σ pq · O O · O O o o · O O o o · · · · O O · O O o o The bisimplicial face o per ators are induced by horizontal and vertical comp osition of squares, and degenerac y op erator s by a ppropriated identit y squares. W e picture NN G so that the set of ( p, q )-simplices is the set in the p -th row and q -th co lumn. Thu s, the p -th co lumn of NN G , NN G p, ∗ , is the nerve of the “vertical” groupo id whose ob jects are str ing s of p comp osa ble hor izontal morphisms ( a 0 f 1 ← a 1 f 2 ← · · · f p ← a p ) and whos e ar rows are depicted as · · o o · o o · · · · · o o σ 1 σ 2 σ p · O O · O O o o · O O o o · · · · O O · O O o o And, similarly , the q - th row, NN G ∗ ,q , is the “hor izontal” gr oupo id whos e ob jects are the length q seq uences of compo sable vertical morphisms in C and whose arrows are sequences of q vertically compos able sq uares. In particular, NN G 0 , ∗ and NN G ∗ , 0 are, resp ectively , the nerves of the gr oupo ids of vertical and ho rizontal mor phisms of G . Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007 10 Example. Let A, B be subgro ups of a finite g roup G , s uc h that AB 6 = B A , where AB = { ab | a ∈ A, b ∈ B } and B A = { b a | a ∈ A, b ∈ B } ( for inst anc e, we c an take G = S 3 , A = { id, (1 , 2) } and B = { id, (1 , 3) } ). Then, let C = C ( A, B ) denote the double gro upoid with only o ne ob ject, say “ · ”, with horizontal mor phisms · a ← · the elements a ∈ A , with vertical morphis ms · ↑ b · are the elements b of B , both with co mpo s ition giv en b y multiplication in the group, and who se squa res · · a o o · b ′ O O · a ′ o o b O O are lists ( a, b, a ′ , b ′ ) of elements a, a ′ ∈ A and b, b ′ ∈ B such tha t ab = b ′ a ′ . Comp o- sitions of squares are defined, in the natural wa y , by ( a, b, a ′ , b ′ ) · h ( a 1 , b 1 , a ′ 1 , b ) = ( aa 1 , b 1 , a ′ a ′ 1 , b ′ ) , ( a, b, a ′ , b ′ ) · v ( a ′ , b 1 , a ′ 1 , b ′ 1 ) = ( a, bb 1 , a ′ 1 , b ′ b ′ 1 ) , and identities b y id v ( a ) = ( a, e, a, e ) , id h ( b ) = ( e, b, e, b ) , where e is the neutra l element o f the group. So defined, w e claim that the asso ciated simplicial set diag NN C is not a Kan complex. T o prove that, no te that the hypothesis AB 6 = B A im plies the existence of elements a ∈ A and b ∈ B s uc h that ab cannot b e expressed in the form b ′ a ′ , for any a ′ ∈ A and b ′ ∈ B . Now, the identit y s quares ι b = id h ( b ) a nd ι a = i d v ( a ) ca n be pla ced jointly a s in the picture: · · a o o ι a · · a o o · ι b · b O O · b O O This means that, r egarded ι a and ι b as 1- simplices of diago na l simplicial set diag NN C , they are co mpatible, in the s ense that d h 0 d v 0 ι a = · = d h 1 d v 1 ι b . Therefore, if we assume that diag NN C is a Kan complex, we sho uld find a dia gonal bisimplex, say x ∈ NN C 2 , 2 , such that d h 0 d v 0 x = ι b and d h 2 d v 2 x = ι a . This (2 , 2)-simplex Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007 11 x o f the ner ve of the double group oid, should be ther e fore o f the form · · a o o · o o ι a · · a o o · O O ι b · a ′ O O · b ′ o o b O O a b O O what is clea rly impo ssible. Hence, the diago nal of the bisimplicia l nerve of ( A, B ) is not a Ka n co mplex , in spite o f the fact that this bisimplicial set is a p oint wise Ka n complex in both vertical and ho r izontal direc tions. F or a last commen t, w e shall p oint out a fact suggested by a refer e e : the pro per t y prov ed in Theo rem 1 for diagonal fibra tions is not true for diagonal trivial fibr a tions. F or instance: Let E G b e the universal cov er of a non- trivial discre te group G , that is, the s implicial se t with E G n = G n +1 and fa c es g iv en by d i ( x 0 , ..., x n ) = ( x 0 , ..., x i − 1 , x i +1 , ..., x n ), 0 6 i 6 n , and let X = E G ⊗ E G , the bisimplicial set defined b y X p,q = E G p × E G q . Then, diag X is con tra c tible to a o ne point Ka n complex and, howev er, each simplicial set X ∗ ,q = E G × G q is ho mo topy equiv alent to the consta n t simplicial se t G q , a nd therefor e it is not contractible. In co nclusion, X → pt is a diagona l trivial fibration but it is not a p oint wise tr ivial fibr ation. References [1] A. K. Bousfield and D. M. Kan , Homotopy limits, c ompletio ns and lo c al- izations , Lecture Notes in Math., 3 04, Springer, 1 9 72. [2] R. Brow n and C. B. Spencer , Double gr o up oids and cr osse d mo dules , Cahiers T op olog ie G´ eom. Diff´ erentielle, XVI I (1976), pp. 3 43–362 . [3] P. Goerss and J. Jardine , Simplicial Homotopy The ory , P rogres s in Ma th- ematics 174 , Birkh¨ auser, 19 99. [4] G. M. Kell y and R. S. Street , Revi ew of the e lements of 2-c ate gories , in Category Seminar (Pro c. Sem., Sy dney , 1972 /1973 ), Lecture Notes in Math., 420, Springer, 1974, pp. 7 5–103 . [5] S. Mac Lane , Cate gories for the working mathematician , Gradua te T ex ts in Mathematics, V ol. 5, Springer -V erlag , 19 71. [6] J. P. Ma y , Simplicial obje cts in Algebr a ic T o p olo gy , V an Nostr and Mathemat- ical Studies, No. 11, D. V an Nostra nd Co., 1967 . [7] I. Moerdijk , Bi simplicial sets and the gr oup-c ompl etion the or em , in Algebraic K-Theory : connections with geometry and top ology , NA TO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 2 79, K lu wer Academic Publishers, 1989 , pp. 225–2 40. ∆. Journal of Homotopy and R elate d Structu r es , vol. 1(2), 2007 12 A.M. Cegarra acegar ra@ugr .es Departamento de ´ Algebra F acultad de Ciencias Univ er s idad de Granada 18071 Gr anada Spain J. Remedio s jremed @ull.e s Departamento de Matem´ atica F undamental Univ er s idad de La Laguna 38271 La Lag una Spain