Double groupoids and homotopy 2-types

DOUBLE GR OUPOIDS AND HOMOTOPY 2-TYPES A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS Abstract. This work con tributes to clarifying several relationships betw een certain higher categorical structures and the ho motop y ty pes of their classifying spa ces. Double categories (Ehresmann, 1963) hav e we ll-understo od geometric realizations, and here we deal wi th ho- motop y t ypes represen ted by double group oids satisfying a natural ‘ ﬁlling condition’. Any suc h double groupoid c haracteristically ha s associated to it ‘homotop y groups’, whi c h are de- ﬁned using only its algebraic structure. Th us ari s es the notion of ‘weak equiv a lence’ b et w een suc h double groupoids, and a corresponding ‘homotopy category’ is deﬁned. Our main resul t in the paper states that the geometric realization functor i nduces an equiv alence b etw een the homotopy category of double group oids with ﬁll ing condition and the category of homo- top y 2-types (that is , the homotopy catego ry of al l topological s paces wi th the property that the n th homotop y gr oup at any base p oint v anishes for n ≥ 3). A quasi-inv erse functor is explicitly give n by means of a new ‘ homotop y double groupoid’ construction f or topological spaces. Mathematic al Subje ct Classiﬁc ation: 18D05, 2 0L05, 5 5 Q05, 55U40 . 1. Introduction and summar y. Higher-dimensiona l catego ries provide a suitable setting for the treatment of an extensive list of sub jects of recognized mathematical interest. The constructio n of nerves a nd class ifying spaces of higher c a tegorica l structures disco v ers w ays to transp ort ca teg orical co herence to homotopic co he r ence, and it has shown its relev ance as a to ol in algebraic top olo gy , algebra ic geometry , alge braic K -theory , string ﬁeld theo ry , c o nformal ﬁeld theory , and in the study of geometric str uctures o n low-dimensional manifolds. Double gr oup oids , that is, gr o upo id o b j ects in the catego ry of group oids, were intro duced by Ehresmann [15, 1 6] in the late ﬁfties and la ter studied by several peo ple b ecause of their connection with several are a s of mathematics. Roughly , a do uble group oid consists of obj e cts , horizontal and vertic al morphisms , and squ ar es . Each squar e, say α , ha s ob jects as vertices and morphisms as edges, as in · · o o α · O O · o o O O , together with tw o group oid comp os itio ns- the vertic al and horizontal c omp ositions - of squa res, and co mpatible gr oup oid comp ositions of the edges, obey ing several c o nditions (see Section 3 for details). An y double group oid G has a ge ometric re aliza tion |G | , which is the top olo gical space deﬁned by ﬁr st taking the double nerve N N G , which is a bisimplicial set, and then realizing Key wor ds and phr ases. Double groupoid, classifying space, bisimplicial set, Kan complex, geometric real- ization, homotop y t ype. The ﬁrst author ac kno wledge supp ort from the DGI of Spain (Pro jec t: MTM2007-65431); Consejer ´ ıa de Inno v aci´ on de J . de Andaluc ´ ıa ( P06-FQM-1889) ; MEC de Espa˜ na, ‘Ingenio Mathematica(i-Math)’ No. CSD2006- 00032 (consolider-Ingenio 2010). The second author thanks supp ort to Unive rsity of Granada (Beca Pl an Propio 2009). The third author ac kno wledges supp ort f rom DGI of Spain (Pro ject: MTM 2009-12081 ) and tanks the Univ ersity of Granada for i ts support and kind hospitality . 1 2 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS the diago nal to obtain a space: |G | = | diagN N G | . In this pap er, we addres s the homo to p y types obtained in this wa y from double gro upo ids satisfying a natura l ﬁl ling c ondition : An y ﬁlling problem · · o o ∃ ? O O · o o O O ﬁnds a so lution in the double g roup oid. This ﬁlling condition on double gro upo ids is often assumed in the case of double group oids arising in diﬀer en t areas of mathematics, such as in diﬀerential geometry or in weak Hopf algebr a theor y (see the papers by Ma ckenzie [25] and Andruskiewitsch a nd Natale [1], fo r example), and it is satisﬁed for those double gr oup oids that hav e emerged with an interest in algebr aic top olog y , ma inly thanks to the work of Br own, Higgings, Sp encer, et al. , where the co nnection of do uble group oids with cross e d mo dules and a higher Seifert-v an Kamp en Theory has b een established (s e e , for ins tance, the survey pap er [5] and references therein. Thus, the ﬁlling condition is easily prov en for edge symmetric double group oids (also called sp ecial double group oids) with connections (see, for example [7, 10] or [6, 9 , 8], for more recent instances), for double gro upoid ob jects in the categ ory o f g roups (also termed cat 2 -groups, [23, 11, 28]), or, for example, for 2-group oids (regar ded a s double group oids where o ne of the s ide g roup oids of morphisms is dis crete [2 7],[19]). When a double group oid G has the ﬁlling conditio n, then there are character is tically asso c i- ated to it ‘homotopy gro ups’, π i ( G , a ), which we deﬁne using only the a lgebraic structure o f G , and which a re trivial for in tegers i ≥ 3. A ﬁrst ma jor result s tates that: If G is a double gr oup oid with ﬁl ling c ondition, then, for e ach obj e ct a , ther e ar e natur al isomorphisms π i ( G , a ) ∼ = π i ( |G | , | a | ) , i ≥ 0 . The pro o f of this r esult r equires a prio r recog nition of the sig niﬁca nce of the ﬁlling c o ndition on double group oids in the homotopy theor y of simplicial sets; namely , we pr ov e that A double c ate gory C is a double gr oup oid with ﬁl ling c ondition if and only if the simplicia l set diagN N C is a Kan c omplex. This fact can b e seen as a higher version of the well-known fact tha t the nerve of a catego ry is a K an complex if and only if the category is a group oid (see [21], for example). Once we hav e deﬁned the homo topy ca tegory o f double group oids satisfying the ﬁlling con- dition Ho( DG fc ), to b e the lo ca lization of the categ o ry of these double group oids, with re- sp ect to the class of we ak e quivalenc es or do uble functors F : G → G ′ inducing isomorphisms π i F : π i ( G , a ) ∼ = π i ( G ′ , F a ) on the homotopy g roups, w e then obtain an induced functor | | : Ho( DG fc ) → Ho( T op ) , G 7→ |G | , where Ho( T op ) is the lo calizatio n of the catego ry of to p olo gical spaces with res pect to the class of weak equiv a lences. F urther more, we show a new functorial constr uction of a homotopy double gr oup oid Π Π X , for any top olo g ical space X , tha t induces a functor Ho( T op ) → Ho ( DG fc ) , X 7→ Π Π X. A ma in goa l in this pap er is to prov e the following r esult, whose pro of is somewhat indirect since it is given throug h an explicit descriptio n of a le ft a djoint functor, P P ⊣ N N, to the do uble nerve functor G 7→ N N G : Both induc e d functors on the homotopy c ate gories G 7→ |G | and X 7→ Π Π X r estrict by giving mutual ly quasi-inverse e quivalenc e of c ate gories Ho( DG fc ) ≃ Ho ( 2 - t yp es ) , DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 3 where Ho ( 2 - t yp es ) is the full sub catego r y of the homotopy category of to po logical spa ces g iven by those s pa ces X with π i ( X, a ) = 0 for any integer i > 2 and any base point a . F ro m the po in t of view of this fact, the use of double gr oup oids a nd their cla ssifying spaces in homo topy theory go es back to Whitehead [32] and Mac Lane-Whitehead [24] s ince double group oids where one o f the side gro upo ids of morphisms is discrete with only one ob ject (= s trict 2- groups, in the terminology of Baez [3]) a re the same as cros sed mo dules (this obs erv ation is attributed to V erdier in [10]). In this context, w e should mention the w ork by Br own-Higgins [7] and Mo erdijk-Svensson [27] since cros sed modules ov er g roup oids are essentially the same thing as 2- group oids and double group oids where one o f the side gro upo ids of mo rphisms is discrete. Along the sa me line, our result is also a natural 2-dimensional version of the well- known eq uiv alence b et ween the homotopy catego r y of gr oup oids and the homo topy ca tegory of 1-types (for a useful survey of gr o upo ids in topo logy , s ee [4]). The plan of this paper is, brieﬂy , as follows. Af ter this introductor y Section 1 , the pap er is or ganized in six sections. Section 2 aims to make this paper as self-co n tained as p os sible; hence, at the sa me time as ﬁxing no tations and terminolo g y , we also r e v iew necessary asp ects and r esults from the background of (bi)simplicial sets a nd their geometric realiza tions that will b e used throughout the pap er. How ev er, the material in Sectio n 2 is quite standard, so the exp ert reader may s kip most of it. The most original part is in Subsection 2.2, related to the extension condition on bisimplicial sets. In Section 3, a fter recalling the no tion of a double gr oup oid and ﬁxing notations, we mainly intro duce the ho motopy g roups π i ( G , a ), at any o b ject a o f a double group oid with ﬁlling condition G . Section 4 is dedica ted to showing in detail the constructio n of the homotopy double g roup oid Π Π X , characteristically as so ciated to any to p olo gical spa ce X . Here, we prov e tha t a con tin uous map X → Y is a weak homotopy 2-equiv a lence (i.e., it induces bijections on the ho motopy gr o ups π i for i ≤ 2) if a nd only if the induced double functor Π Π X → Π Π Y is a weak eq uiv alence. Next, in Sectio n 5, w e ﬁrs t address the issue of to ha ve a manageable description for the bisimplices in N N G , the double nerve o f a double gro upo id, a nd then w e determine the homotopy t ype of the ge o metric realization |G | of a double group oid with ﬁlling condition. Spec iﬁcally , w e prov e that the homotopy groups of |G | are the sa me as those of G . Our goa l in Section 6 is to prove that the double nerve functor, G 7→ N N G , embeds, a s a r eﬂexive sub category , the category of double group oids satisfying the ﬁlling condition in to a cer tain ca tegory o f bisimplicial sets. The reﬂector functor K 7→ P P K works as a bisimplicial version of Brown’s constr uction in [6, Theorem 2.1]. F urthermor e, as we will pr ov e, the resulting double gr oup oid P P K alw ays r epresents the homotopy 2-type of the input bisimplicial set K , in the sense that there is a natural w eak 2-equiv alence | K | → | P P K | . This result bec o mes crucial in the ﬁnal Sec tion 7 where, bringing int o play all the previous work, the equiv a lence of categ ories Ho( DG fc ) ≃ Ho ( 2 - t yp es ) is achieved. 2. Some preliminaries on bisimplicial sets. This sectio n aims to make this pap er as self-contained as p ossible; Therefor e, while ﬁxing notations a nd terminology , we als o review necessa ry asp ects and r esults from the background of (bi)simplicial sets and their geometric realiza tions used throug hout the pap er. How ever, the material in this se c tion is quite standar d and, in gener al, w e employ the sta nda rd s y m bo lism and nomencla ture to be found in texts on simplicial homotopy theory , mainly in [17] and [26], so the expert reader may skip mo st of it. The most orig inal part is in Subsectio n 2.2 , related to the extension condition and the bihomoto py rela tion on bisimplicial sets. 2.1. Kan complexes: F undamental group oids and homotop y groups. W e s tart by ﬁxing so me notations. In the simplicial category ∆, the genera ting coface and co degeneracy maps are denoted b y d i : [ n − 1 ] → [ n ] and s i : [ n + 1 ] → [ n ] resp ectively . Ho wev er , 4 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS for L : ∆ o → Set a n y simplicia l set, we write d i = L ( d i ) : L n → L n − 1 and s i = L ( s i ) : L n → L n +1 for its corr e spo nding face a nd degeneracy maps . The standar d n -simplex is ∆[ n ] = ∆( − , [ n ]) a nd, as is usual, we identify any simplicial map x : ∆[ n ] → L with the simplex x ( ι n ) ∈ L n , the image by x of the basic s implex ι n = id : [ n ] → [ n ] of ∆[ n ]. Th us, for example, the i th -fac e of ∆[ n ] is d i = ∆( − , d i ) : ∆[ n − 1] → ∆[ n ], the simplicial map with d i ( ι n − 1 ) = d i ( ι n ). Similarly , s i = ∆( − , s i ) : ∆[ n + 1] → ∆[ n ] is the simplicial map that we identify with the degenerated s implex s i ( ι n ) o f ∆[ n ]. The b oundary ∂ ∆[ n ] ⊂ ∆[ n ] is the smallest simplicia l subset containing all the faces d i : ∆[ n − 1] → ∆[ n ], 0 ≤ i ≤ n , of ∆[ n ]. Similarly , for an y given k with 0 ≤ k ≤ n , the k th -horn , Λ k [ n ] ⊂ ∆[ n ], is the smallest simplicial subset co n taining a ll the faces d i : ∆[ n − 1] → ∆[ n ] for 0 ≤ i ≤ n and i 6 = k . F or a more geometric (and useful) description of these simplicial s ets, recall that there are co equalizers G 0 ≤ i n a nd a ny base po in t a . There ar e v a rious constructions o n (bi)simplicial sets that traditionally aid in the algebraic study of homotopy n -types. Below is a brief review of the constructions used in this work. Segal’s ge ometric r e alization functor [31], for simplicial spaces K : ∆ o → T op , is deno ted by K 7→ | K | . Recall that it is deﬁned as the left adjoint to the functor that as so ciates to a space X the simplicial space [ n ] 7→ X ∆ n , where ∆ n = { ( t 0 , . . . , t n ) ∈ R n +1 | X t i = 1 , 0 ≤ t i ≤ 1 } denotes the aﬃne simplex having [ n ] as its set of vertices and X ∆ n is the the function space of contin uous maps from ∆ n to X , given the compact-op en top olo gy . The underlying simplicia l set is the singular c omplex of X , denoted by S X . F or instance, b y reg arding a set a s a discrete spa c e, the (Milnor’s ) geo metr ic realiza tion of a simplicial set L : ∆ o → Set is | L | , whic h is a CW-co mplex whose n -cells are in one-to- one corres p ondence with the n - simplices of L which ar e nondegenera te. The following six facts are well-kno wn: F acts 2.7. (1) F or any sp ac e X , S X is a Kan c omplex. (2) F or any Kan c omplex L , ther e ar e natur al isomorp hisms π i ( L, a ) ∼ = π i ( | L | , | a | ) , for al l b ase vertic es a : ∆[0] → L and n ≥ 0 . (3) A simplici al map b etwe en Kan c omplexes L → L ′ is a homotopy e quivalenc e if and only if the induc e d map on r e alizations | L | → | L ′ | is a homotopy e quivalenc e. (4) F or any Kan c omplex L , the unit of the adjunction L → S | L | is a homotopy e quivalenc e. (5) A c ontinuous m ap X → Y is a we ak homo topy e quivalenc e if and only if the induc e d S X → S Y is a homotopy e quivalenc e. (6) F or any sp ac e X , the c ounit | S X | → X is a we ak ho motopy e quivalenc e. When a bisimplicia l set K : ∆ o × ∆ o → Set is rega r ded as a simplicial ob ject in the simplicial set categ ory and one takes geometric r ealizations, then one obta ins a s implicial spac e ∆ o → T op , [ p ] 7→ | K p, ∗ | , whose Segal r ealization is taken to be | K | , the geometric realization o f K . As there a re natur al homeomor phisms [30, Lemma in pa g e 86] | [ p ] 7→ | K p, ∗ || ∼ = | diag K | ∼ = | [ q ] 7→ | K ∗ ,q || , where diag K is the simplicial set obtained by comp osing K with the diagona l functor ∆ → ∆ × ∆, [ n ] 7→ ([ n ] , [ n ]), one usually takes | K | = | diag K | . Comp osing with the o r dinal sum functor or : ∆ × ∆ → ∆, ([ p ] , [ q ]) 7→ [ p +1 + q ], gives Illusie’s total Dec functor, L 7→ Dec L , from simplicial to bisimplicia l sets [2 1, VI, 1.5]. More s p eciﬁca lly , for any s implicia l set L , Dec L is the bisimplicia l set who s e bisimplices of bidegr ee ( p, q ) ar e the ( p + 1 + q )-s implices of L , x : ∆[ p + 1 + q ] → L , and whose simplicial o pe r ators are given by xd i h = xd i , xs i h = xs i , for 0 ≤ i ≤ p , and xd j v = d p +1+ j , xs j v = xs p +1+ j , for 0 ≤ j ≤ q . The functor Dec has a r ight adjo int [14] (2.2) Dec ⊣ W , DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 9 often called the c o diagonal functor, whose des cription is as follows [2, I II]: for any bisimplicial set K , an n -simplex o f W K is a bisimplicial map n F p =0 ∆[ p, n − p ] ( x 0 ,...,x n ) / / K such that x p d 0 v = x p + 1 d p +1 h , for 0 ≤ p < n , whose faces and degeneracies a re given by ( x 0 , . . . , x n ) d i = ( x 0 d i v , . . . , x i − 1 d 1 v , x i +1 d i h , . . . , x n d i h ) , ( x 0 , . . . , x n ) s i = ( x 0 s i v , . . . , x i s 0 v , x i s i h , . . . , x n s i h ) . The unit and the counit of the adjunction, u : L → W Dec L and v : DecW K → K , are resp ectively deﬁned by u( y ) = ( y s 0 , . . . , y s n ) ( y : ∆[ n ] → L ) v( x 0 , . . . , x p +1+ q ) = x p +1 d 0 h (( x 0 , . . . , x p +1+ q ) : ∆[ p, q ] → Dec W X ) . The following facts are used in our developmen t b elow: F acts 2.8. (1) F or e ach n ≥ 0 , ther e is a natur al Alexander-Whitney typ e diagonal ap- pr oximation φ : Dec∆[ n ] → ∆[ n, n ] , (∆[ p + 1 + q ] x → ∆[ n ]) 7→ (∆[ p ] x ( d p +1 ) q − → ∆[ n ] , ∆[ q ] x ( d 0 ) p +1 − → ∆[ n ]) such that, for any bisimplici al set K , t he induc e d simplicia l map φ ∗ : diag K → W K determines a homotopy e quivalenc e | diag K | ≃ | W K | on the c orr esp onding ge ometric r e alizations [12, Theorem 1 .1] . (2) F or any simplicial map f : L → L ′ , t he induc e d | f | : | L | → | L ′ | is a homotopy e quivalenc e if and only if the induc e d | Dec f | : | Dec L | → | Dec L ′ | is a homotopy e quivalenc e [1 2, Corollar y 7 .2] . (3) F or any simplicial set L and any bisimplicial set K , b oth induc e d maps | u | : | L | → | W Dec L | and | v | : | Dec W K | → | K | ar e homotopy e qu ivalenc es [12 , Pr op o sition 7.1 and discussion below] . (4) If K is any bisimplicial set satisfying the ex tension c ondition, t hen W K is a Kan c omplex [13, Pro po sition 2] . (5) If L is a Kan c omplex, then Dec L s atisﬁ es the extension c onditio n (the pr o of is a straightforward application of [26 , Lemma 7.4]) or [13 , Lemma 1]) . 3. Double groupoids sa tisfying the filling condition: Homo topy gr oups. A (small) double g roup oid [15, 16, 10, 2 2] is a group oid ob ject in the categor y of small group oids. In gene r al, we employ the standard nomenclature co ncerning double categor ie s but, for the sake of clarity , we s ha ll ﬁx some terminology and no tations b elow. A (small) catego ry c a n b e desc rib e d as a system ( M , O , s , t , I , ◦ ), where M is the set of morphisms, O is the s et of ob jects, s , t : M → O ar e the source and target maps, resp ectively , I : O → M is the identit ies map, and ◦ : M s × t M → M is the compo sition map, sub ject to the usual asso ciativity a nd identit y axioms . Therefo re, a double c ate gory provides us with the following data: a set O of obje ct s , a s e t H of horizontal m orphisms , a set V of vertic al morphi sms , and a set C of squar es , together with four ca tegory structures , namely , the c ate gory of horizontal mor- phisms ( H, O , s h , t h , I h , ◦ h ), the c ate gory of vertic al morphisms ( V , O , s v , t v , I v , ◦ v ), the horizontal 10 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS c ate gory of squar es ( C, V , s h , t h , I h , ◦ h ), and the vertic al c ate gory of squar es ( C, H , s v , t v , I v , ◦ v ). These a re sub ject to the following three ax ioms: Axiom 1:    (i) s h s v = s v s h , t h t v = t v t h , s h t v = t v s h , s v t h = t h s v , (ii) s h I v = I v s h , t h I v = I v t h , s v I h = I h s v , t v I h = I h t v , (iii) I h I v = I v I h . Equalities in Axio m 1 allow a square α ∈ C to be depicted in the form (3.1) d b g o o α c w O O a f o o u O O where s h α = u, t h α = w, s v α = f a nd t v α = g , and the four vertices of the square repr esenting α ar e s h s v α = a, t h t v α = d, s h t v α = b a nd s v t h α = c . Moreov er, if we represent iden tit y morphisms by the s ym b o l , then, for any horizontal morphism f , any vertical morphism u , and a ny ob ject a , the ass o ciated identit y squares I v f , I h u a nd I a := I h I v a = I v I h a are resp ectively given in the form · · f o o · I v f · f o o · · · u I h u O O · u O O · · · I a · The equa lities in Axiom 2 b elow show the squares ar e compatible with the bo undaries, whereas Axiom 3 establishes the necessary coherence b etw een the tw o vertical and horizontal comp ositions of squar es. Axiom 2:    (i) s v ( α ◦ h β ) = s v α ◦ h s v β , t v ( α ◦ h β ) = t v α ◦ h t v β , (ii) s h ( α ◦ v β ) = s h α ◦ v s h β , t h ( α ◦ v β ) = t h α ◦ v t h β , (iii) I v ( f ◦ h f ′ ) = I v f ◦ h I v f ′ , I h ( u ◦ v u ′ ) = I h u ◦ v I h u ′ . Axiom 3: In the situation · · o o · o o α β · O O · o o O O · O O o o γ δ · O O · o o O O · o o O O the inter change law holds, that is, ( α ◦ h β ) ◦ v ( γ ◦ h δ ) = ( α ◦ v γ ) ◦ h ( β ◦ v δ ) . A double gr oup oid is a do uble c a tegory such tha t all the four compo nent categorie s are group oids. W e shall use the following notatio n for in v erses in a double g roup oid: f - 1 h denotes the inv er se o f a horizontal morphism f , a nd u - 1 v denotes the inv erse of a vertical mor phism u . F or any squar e α as in (3.1), the ﬁrst one of b d g - 1 h o o α - 1 h a u O O c, f - 1 h o o w O O c a f o o α - 1 v d w - 1 v O O b, g o o u - 1 v O O a c f - 1 h o o α - 1 b u - 1 v O O d, g - 1 h o o w - 1 v O O is the in verse of α in the horizontal gr o upo id of sq uares, the second one denotes the inv erse of α in the vertical group oid of squares, and the third one is the squar e ( α -1 h ) -1 v = ( α -1 v ) -1 h , which is denoted simply by α -1 . The double gr oupo ids we are in terested in s atisfy the condition below. DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 11 Filling condition: Any ﬁ l ling pr oblem · · g o o ∃ ? · O O · o o u O O has a solution; that is, for any horizontal morphism g and any vert ic al morphism u such that s h g = t v u , ther e is a squ ar e α with s h α = u and t v α = g . As w e reca lled in the in tro duction, this ﬁlling c ondition on double gr o upo ids is o ften s a tisﬁed for those double gro upo ids ar ising in algebraic top ology . F urther b elow, in Sections 4 and 6, we p ost tw o new homotopical do uble gr oup o id constructions that relev an t to our delib era tions: one, Π Π X , for topo logical spaces X , and the other, P P K , for bisimplicial sets K , b oth yielding double g r oup oids satisfying the ﬁlling condition. The rema inder of this section is dev oted to deﬁning homotopy gr oups , π i ( G , a ), for double group oids G sa tisfying the ﬁlling condition. The useful observ a tion below is a dir ect c o nsequence of [1, Lemma 1.12]. Lemma 3.1. A double gr oup oid G satisﬁes the ﬁ l ling c ondition if and only if any ﬁl ling pr oblem such as the one b elow has a solution. · · o o ∃ ? · w O O · f o o , O O · · o o ∃ ? · O O · f o o u , O O · · g o o ∃ ? · w O O · o o , O O Hereafter, w e as s ume G is a double g roup oid s atisfying the ﬁlling condition. 3.1. The p ointed sets π 0 ( G , a ) . W e state that tw o o b jects a, b of G are c onne cte d whenever ther e is a pair o f mo rphisms ( g , u ) in G o f the form b · g o o a u , O O that is, where g is a ho rizontal morphism and u a vertical morphism such that s h g = t v u , t h g = b , and s v u = a . Because of the ﬁlling condition, this is eq uiv alent to saying that there is a sq uare in G of the for m b · g o o α · w O O a f o o u , O O and it is also e q uiv alent to saying that there is a pair of matc hing mor phisms ( w , f ) as b · w O O a. f o o If a and b are r ecognized as being connected by means of the pair o f morphisms ( g , u ) as ab ov e , then the pair ( u -1 v , g -1 h ) shows that b is co nnected to a . Hence, being co nnected is a symmetric relatio n on the set of ob jects of G . This relation is clearly reﬂexive thanks to the ident it y morphisms (I h a, I v a ), and it is also transitive. Supp o se a is connected with b , which 12 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS itself is co nnected with another ob ject c . Then, w e hav e mor phisms u , f , v , g as in the diagra m c · g o o · g ′ o o b v O O β · f o o u ′ O O a u O O where β is any s quare with t h β = v and s v β = f , and the dotted g ′ and u ′ are the other sides of β . Consequently , on co nsidering the pair of c ompo sites ( g ◦ h g ′ , u ′ ◦ v u ), we see that a and c are connec ted. Therefore, being connected establishes a n equiv alence relation on the ob jects of the do uble group oid and, ass o ciated to G , w e tak e (3.2) π 0 G = the set of c onne cte d classes of obje cts of G , and we write π 0 ( G , a ) for the set π 0 G p ointed with the cla ss [ a ] of an o b ject a of G . 3.2. The groups π 1 ( G , a ) . Let a b e any g iven o b ject o f G , and let (3.3) G ( a ) =    a x g o o a u O O    be the set o f all pairs of morphisms ( g , u ), where g is a horizontal morphism and u a v ertical morphism in G suc h that t h g = a = s v u and s h g = t v u . Deﬁne a relation ∼ on G ( a ) b y the rule ( g , u ) ∼ ( g ′ , u ′ ) if and only if ther e are tw o s quares α and α ′ in G o f the form a · g o o · w O O α a f o o u O O a · g ′ o o · w O O α ′ a f o o u ′ O O that is, such that t h α = t h α ′ , s v α = s v α ′ , s h α = u, s h α ′ = u ′ , t v α = g , and t v α ′ = g ′ . Lemma 3.2. The r elation ∼ is an e quivalenc e. Pr o of. Since G sa tis ﬁes the ﬁlling conditio n, the re la tion is clear ly reﬂexive, and it is obviously symmetric. T o prov e transitivity , supp ose ( g , u ) ∼ ( g ′ , u ′ ) ∼ ( g ′′ , u ′′ ), so that there a r e square s α, α ′ , β and β ′ as b elow. a · g o o · w O O α a f o o u O O a · g ′ o o · w O O α ′ a f o o u ′ O O a · g ′ o o · w ′ O O β a f ′ o o u ′ O O a · g ′′ o o · w ′ O O β ′ a f ′ o o u ′′ O O Then, we have the horizontally comp osable s q uares a · g ′ o o a g ′ -1 h o o · g o o · w ′ O O β a O O f ′ o o α ′ -1 h · f -1 h o o O O α a f o o u O O whose compo sition β ◦ h α ′ -1 h ◦ h α and β ′ show that ( g , u ) ∼ ( g ′′ , u ′′ ).  DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 13 W e write [ g , u ] for the ∼ -equiv a lence cla s s of ( g , u ) ∈ G ( a ). Now we deﬁne a pr o duct o n (3.4) π 1 ( G , a ) := G ( a )  ∼ as follows: given [ g 1 , u 1 ] , [ g 2 , u 2 ] ∈ π 1 ( G , a ), by the ﬁlling condition o n G , we can choose a square γ with s v γ = g 2 and t h γ = u 1 so tha t we hav e a conﬁgura tion in G o f the form a · g 1 o o · g o o a O O u 1 γ · g 2 o o u O O a u 2 O O where g = t v γ and u = s h γ . Then we deﬁne (3.5) [ g 1 , u 1 ] ◦ [ g 2 , u 2 ] = [ g 1 ◦ h g , u ◦ v u 2 ] Lemma 3.3. The pr o duct is wel l deﬁne d. Pr o of. Let [ g 1 , u 1 ] = [ g ′ 1 , u ′ 1 ] , [ g 2 , u 2 ] = [ g ′ 2 , u ′ 2 ] b e elements of π 1 ( G , a ). Then, there are squa res a · g 1 o o · w 1 O O α a f 1 o o u 1 O O a · g ′ 1 o o · w 1 O O α ′ a f 1 o o u ′ 1 O O a · g 2 o o · w 2 O O β a f 2 o o u 2 O O a · g ′ 2 o o · w 2 O O β ′ a f 2 o o u ′ 2 O O and cho o sing squar es γ a nd γ ′ as in · · g o o a u 1 O O γ · g 2 o o u O O · · g ′ o o a u ′ 1 O O γ ′ · g ′ 2 o o u ′ O O we ha ve [ g 1 , u 1 ] ◦ [ g 2 , u 2 ] = [ g 1 ◦ h g , u ◦ v u 2 ] a nd [ g ′ 1 , u ′ 1 ] ◦ [ g ′ 2 , u ′ 2 ] = [ g ′ 1 ◦ h g ′ , u ′ ◦ v u ′ 2 ]. Now, letting θ be a ny squa re with t v θ = f 1 and s h θ = w 2 , w e hav e squares as in a · g 1 o o · g o o α γ · w 1 O O a o o O O · u O O o o θ β · O O · o o O O a f 2 o o u 2 O O a · g ′ 1 o o · g ′ o o α ′ γ ′ · w 1 O O a o o O O · u ′ O O o o θ β ′ · O O · o o O O a f 2 o o u ′ 2 O O whose cor resp onding comp osites ( α ◦ h γ ) ◦ v ( θ ◦ h β ) and ( α ′ ◦ h γ ′ ) ◦ v ( θ ◦ h β ′ ) s how that [ g 1 ◦ h g , u ◦ v u 2 ] = [ g ′ 1 ◦ h g ′ , u ′ ◦ v u ′ 2 ], as re q uired.  Lemma 3.4. The given mu ltiplic ation t urns π 1 ( G , a ) int o a gr oup. Pr o of. T o see the asso ciativity , let [ g 1 , u 1 ] , [ g 2 , u 2 ] , [ g 3 , u 3 ] ∈ π 1 ( G , a ), a nd choos e γ , γ ′ and γ ′′ any three squar es as in the diag ram (3.6) below. Then we hav e ([ g 1 , u 1 ] ◦ [ g 2 , u 2 ]) ◦ [ g 3 , u 3 ] = [ g 1 ◦ h g ◦ h g ′ , u ◦ v u ′ ◦ v u 3 ] = [ g 1 , u 1 ] ◦ ([ g 2 , u 2 ] ◦ [ g 3 , u 3 ]) . 14 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS (3.6) a · g 1 o o · g o o · g ′ o o a u 1 O O γ · γ ′ g 2 o o O O · o o u O O a u 2 O O γ ′′ · g 3 o o u ′ O O a u 3 O O The iden tit y of π 1 ( G , a ) is [I h a, I v a ]. In eﬀect, if [ g , u ] ∈ π 1 ( G , a ), then the diag rams a x g o o x a O O u I h u a u O O a a a x g o o a I v g x g o o a u O O show that [ g , u ] ◦ [I h a, I v a ] = [ g ◦ h I h x, u ◦ v I v a ] = [ g , u ] = [I h a ◦ h g , I v x ◦ v u ] = [I h a, I v a ] ◦ [ g , u ] . Finally , to see the existence o f inv er ses, let [ g , u ] ∈ π 1 ( G , a ). B y choo sing a ny squa re α with t h α = u -1 v and s v α = g -1 h , that is, of the fo rm a · f o o · u -1 v O O α a g -1 h o o v O O we ﬁnd [ f , v ] := [t v α, s h α ] ∈ π 1 ( G , a ). Since the diagrams a · g o o a g -1 h o o a O O u O O α -1 v · f o o v -1 v O O a v O O a · f o o a f -1 h o o a O O v O O α -1 h · g o o u -1 v O O a u O O show that [ g , u ] ◦ [ f , v ] = [I h a, I v a ] = [ f , v ] ◦ [ g , u ], w e hav e [ g , u ] -1 = [ f , v ].  3.3. The ab elian groups π i ( G , a ) , i ≥ 2 . These are eas ier to deﬁne than the previous ones. F or i = 2, as in [10 , Section 2], we take (3.7) π 2 ( G , a ) = ( a a α a a ) the s et of all squares α ∈ G who se b oundar y edges are s h α = t h α = I v a and s v α = t v α = I h a . By the g eneral E ckman-Hilton argument, it is a co ns equence of the interc hange law that, on π 2 ( G , a ), op erations ◦ h and ◦ v coincide and a re co mm utative. In eﬀect, for α, β ∈ π 2 ( G , a ), α ◦ h β = ( α ◦ v I a ) ◦ h (I a ◦ v β ) = ( α ◦ h I a ) ◦ v (I a ◦ h β ) = α ◦ v β = (I a ◦ h α ) ◦ v ( β ◦ h I a ) = (I a ◦ v β ) ◦ h ( α ◦ v I a ) = β ◦ h α. Therefore, π 2 ( G , a ) is an ab elian group with pro duct (3.8) α ◦ h β = α ◦ v β , DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 15 ident it y I a = I v I h a , and inv erses α -1 h = α -1 v . The higher ho motopy gro ups of the do uble gr o upo id are deﬁned to b e trivial, that is, (3.9) π i ( G , a ) = 0 if i ≥ 3 . 3.4. W eak equiv alences. A double functor F : G → G ′ betw een double catego r ies takes o b jects, horizo n tal and v ertical morphisms, and squares in G to ob jects, horizontal and vertical mor phisms, and squares in G ′ , resp ectively , in s uch a way that all the structur e categ ories ar e preserved. Clearly , each double functor F : G → G ′ , b etw e en double gr o upo ids satisfying the ﬁlling condition, induces maps (group homomorphisms if i > 0) π i F : π i ( G , a ) → π i ( G ′ , F a ) for i ≥ 0 and a any ob ject of G . Call suc h a double functor a we ak e quivalenc e if it induce s isomorphisms π i F for all in tegers i ≥ 0. 4. A homotopy double gr oupoid f or topological sp aces. Our aim here is to provide a new c o nstruction of a double group oid for a topolog ical space that, as w e will see later, captures the ho motopy 2 -type of the space. F or any given spa ce X , the c onstruction of this homotopy double gr oup oid , denoted by Π Π X , is as follows: The ob jects in Π Π X ar e the paths in X , that is, the co n tin uous maps u : I = [0 , 1] → X . The group oid o f horiz o nt al mo rphisms in Π Π X is the category with a unique mor phism be- t ween ea ch pair ( u ′ , u ) of paths in X such that u ′ (1) = u (1), a nd, similarly , the group oid o f vertical morphisms in Π Π X is the categ ory having a unique mor phism b etw een each pair ( v , u ) of paths in X s uc h that v (0 ) = u (0). A s quare in Π Π X , [ α ], with a b ounda r y as in (4.1) v ′ v o o [ α ] u ′ O O u o o O O is the equiv alence c la ss, [ α ], of a map α : I 2 → X whose eﬀect on the b oundar y ∂ ( I 2 ) is such that α ( x, 0) = u ( x ) , α (0 , y ) = v ( y ) , α (1 , 1 − y ) = u ′ ( y ), and α (1 − x, 1) = v ′ ( x ), for x, y ∈ I . W e call such an application a “s quare in X” and dr aw it as (4.2) · · v ′ o o u ′   α · v O O u / / · Two such mappings α, α ′ are equiv alent, a nd then repr esent the sa me square in Π Π X , whe ne ver they ar e r e lated by a ho mo topy r e lative to the sides o f the s quare, that is, if there exists a con- tin uous ma p H : I 2 × I → X such that H ( x, y , 0) = α ( x, y ) , H ( x, y , 1) = α ′ ( x, y ) , H ( x, 0 , t ) = u ( x ) , H (0 , y , t ) = v ( y ) , H ( x, 1 , t ) = v ′ (1 − x ) and H (1 , y, t ) = u ′ (1 − y ), for x, y , t ∈ I . Given the squares in Π Π X w ′ w o o [ β ] v ′′ v ′ o o O O v o o O O [ α ′ ] [ α ] u ′′ O O u ′ o o O O u o o O O the c orresp onding comp osite squares 16 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS v ′′ v o o [ α ′ ] ◦ h [ α ] u ′′ O O u o o O O w ′ w o o [ β ] ◦ v [ α ] u ′ O O u o o O O are deﬁned to b e those repr esented by the square s in X · · v ′′ o o u ′′   · ^ ^ < < < v ′   < < < u ′ · v O O u / / α α ′ · · β α · w ′ o o u ′        v ′ · · w O O u / / @ @    v · obtained, respectively , b y pasting α ′ with α , and β with α , along their common pa ir of sides. That is, (4.3) [ α ′ ] ◦ h [ α ] = [ α ′ ◦ h α ] , [ β ] ◦ v [ α ] = [ β ◦ v α ] where ( α ′ ◦ h α )( x, y ) =        α (2 x, x + y ) if x ≤ y , x + y ≤ 1 , α ( x + y , 2 y ) if x ≥ y , x + y ≤ 1 , α ′ ( x + y − 1 , 2 y − 1) if x ≤ y , x + y ≥ 1 , α ′ (2 x − 1 , x + y − 1) if x ≥ y , x + y ≥ 1 , and ( β ◦ v α )( x, y ) =        α (2 x − 1 , 1 − x + y ) if x ≥ y , x + y ≥ 1 , α ( x − y , 2 y ) if x ≥ y , x + y ≤ 1 , β (1 + x − y , 2 y − 1) if x ≤ y , x + y ≥ 1 , β (2 x, y − x ) if x ≤ y , x + y ≤ 1 . It is no t hard to see that b oth the horiz o nt al a nd vertical comp ositions of s quares in Π Π X are well deﬁned. F or example, to prove that [ α ] = [ α 1 ] and [ α ′ ] = [ α ′ 1 ] imply [ α ′ ◦ h α ] = [ α ′ 1 ◦ h α 1 ], let H , H ′ : I 2 × I → X be homotopies ( r el ∂ ( I 2 )) from α to α 1 and from α ′ to α ′ 1 resp ectively . Then, a homoto p y F : I 2 × I → X is deﬁned by F ( x, y , t ) =        H (2 x, x + y , t ) if x ≤ y , x + y ≤ 1 , H ( x + y , 2 y , t ) if x ≥ y , x + y ≤ 1 , H ′ ( x + y − 1 , 2 y − 1 , t ) if x ≤ y , x + y ≥ 1 , H ′ (2 x − 1 , x + y − 1 , t ) if x ≥ y , x + y ≥ 1 , showing that α ′ ◦ h α and α ′ 1 ◦ h α 1 represent the same square in Π Π X . The horizontal identit y s quare on a vertical morphism ( v , u ) is v v I h ( v , u ) = [ e h ] u O O u O O where e h ( x, y ) =  v ( y − x ) if x ≤ y , u ( x − y ) if x ≥ y , whereas, for a ny horizo n tal mor phism ( u ′ , u ), its corres p onding vertical identit y squa re is u ′ u o o I v ( u ′ , u ) = [ e v ] u ′ u o o where e v ( x, y ) =  u ( x + y ) if x + y ≤ 1 , u ′ (2 − x − y ) if x + y ≥ 1 . Theorem 4.1. Π Π X is a double gr oup oid satisfying the ﬁl ling c ondition. DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 17 Pr o of. The ho rizontal comp os itio n of square s in Π Π X is asso c ia tive since, for any three c ompo s- able squares, say · · o o · o o · o o [ α ′′ ] [ α ′ ] [ α ] · O O · o o O O · o o O O · o o O O , a relative homotopy ( α ′′ ◦ h α ′ ) ◦ h α H → α ′′ ◦ h ( α ′ ◦ h α ) is given by the formula H ( x, y , t ) =                          α ( 4 x 2 − t , (2+ t ) x +(2 − t ) y 2 − t ) if x ≤ y , (2 − t )(1 − y ) ≥ (2+ t ) x, α ( (2 − t ) x +(2+ t ) y 2 − t , 4 y 2 − t ) if x ≥ y , (2 − t )(1 − x ) ≥ (2+ t ) y , α ′ ( t (1+ x − y )+2( x + y − 1) ,x +3 y − 2+ t (1+ x − y ) ) if x ≤ y , (2 − t )(1 − y ) ≤ (2+ t ) x, (1+ t ) x ≤ (3 − t )(1 − y ) , α ′ ( 3 x + y − 2+ t (1 − x + y ) ,t (1 − x + y )+2( x + y − 1) ) if x ≥ y , (2 − t )(1 − x ) ≤ (2+ t ) y , (1+ t ) y ≤ (3 − t )(1 − x ) , α ′′ ( x +3 y − 3+ t (1+ x − y ) 1+ t , t − 3+4 y 1+ t ) if x ≤ y , (1+ t ) x ≥ (1 − y )(3 − t ) , α ′′ ( t − 3+4 x 1+ t , 3 x + y − 3+ t (1 − x + y ) 1+ t ) if x ≥ y , (1+ t ) y ≥ (3 − t )(1 − x ) . And, s imilarly , we prov e the asso cia tivit y for the vertical comp osition o f squar es in Π Π X . F o r ident ities, let [ α ] b e any squa re in Π Π X as in (4.1). Then, a r elative homotopy b etw een α and α ◦ h e h is given by the map H : I 2 × I → X deﬁned by H ( x, y , t ) =              v ( y − x ) if x ≤ y , x ≤ 1 2 (1 − t )(1+ x − y ) , u ( x − y ) if x ≥ y , x ≤ 1 2 (1 − t )(1+ x − y ) , α ( x + y − 1+ t (1+ x − y ) 1+ t , 2 y + t − 1 1+ t ) if 1 2 (1 − t )(1+ x − y ) ≤ x ≤ y , α ( 2 x + t − 1 1+ t , x + y − 1+ t (1 − x + y ) 1+ t ) if 1 2 (1 − t )(1 − x − y ) ≤ y ≤ x . Therefore, [ α ] ◦ h I h ( v , u ) = [ α ]; and similar ly we prov e the remaining needed equalities: [ α ] = I h ◦ h [ α ] = [ α ] ◦ v I v = I v ◦ v [ α ]. Let us now describ e in v erse square s in Π Π X . F or a ny given square [ α ] as in (4.1), its resp ective horizontal and vertical inv erses v v ′ o o [ α ] -1 h u O O u ′ o o O O , u ′ u o o [ α ] -1 v v ′ O O v o o O O are repr esented by the squares in X , α -1 h , α -1 v : I 2 → X , deﬁned resp ectively by the formulas α -1 h ( x, y ) = α (1 − y , 1 − x ) , α -1 v ( x, y ) = α ( y , x ) . The equality [ α -1 h ] ◦ h [ α ] = I h ( v , u ) holds, thank s to the homotopy H : I 2 × I : → X deﬁned by H ( x, y , t ) =            α ( 2 x (1 − t ) , (1 − 2 t ) x + y ) if x ≤ y , x + y ≤ 1 , α ( x +(1 − 2 t ) y , 2 y (1 − t ) ) if x ≥ y , x + y ≤ 1 , α ( 2( ty − t − y +1) , 2( ty − t +1) − x − y ) if x ≤ y , x + y ≥ 1 , α ( (2 x − 2) t +2 − x − y , (2 x − 2) t +2 − 2 x ) if x ≥ y , x + y ≥ 1 . And, s imilarly , one sees the remaining equalities [ α ] ◦ h [ α ] -1 h = I h , [ α ] ◦ v [ α ] -1 v = I v and [ α ] -1 v ◦ v [ α ] = I v . 18 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS By co ns truction of Π Π X , conditions (i) and (ii) in Axiom 1 are c le arly satisﬁed. F or (iii) in Axiom 1 , we need to prove that, for any path u : I → X , the equality I h ( u, u ) = I v ( u, u ) holds. But this follows from the rela tive homotopy H : e h → e v deﬁned b y H ( x, y , t ) =                                  u ( y − x ) if x ≤ y , (1 − t )(1 − y ) ≥ (1+ t ) x , u ( 2 y − 1+ t (1+ x − y ) ) if x ≤ y, x + y ≤ 1 , (1 − t )(1 − y ) ≤ (1+ t ) x , u ( x − y ) if x ≥ y , (1 − t )(1 − x ) ≥ (1+ t ) y , u ( 2 x − 1+ t (1 − x + y ) ) if x ≥ y , x + y ≤ 1 , (1 − t )(1 − x ) ≤ (1+ t ) y , u ( 1 − 2 x + t (1+ x − y ) ) if x ≤ y , x + y ≥ 1 , (1+ t )(1 − y ) ≥ (1 − t ) x , u ( y − x ) if x ≤ y , x + y ≥ 1 , (1+ t )(1 − y ) ≤ (1 − t ) x , u ( 1 − 2 y + t (1 − x + y ) ) if x ≥ y, x + y ≥ 1 , (1+ t )(1 − x ) ≥ (1 − t ) y , u ( x − y ) if x ≥ y , (1+ t )(1 − x ) ≤ (1 − t ) y . The given deﬁnition of how squares in Π Π X comp ose makes the conditio ns (i) and (ii) in Axiom 2 clear, and the remaining condition (iii) ho lds since, for a n y three paths u, v , w : I → X with u (1) = v (1) = w (1), there is a relative homoto p y betw een e v ( w, v ) ◦ h e v ( v , u ) and e v ( w, u ) , deﬁned b y H ( x, y , t ) =                                  u ( y − x + 4 x 1+ t ) if x ≤ y , (1+ t )(1 − y ) ≥ (3 − t ) x , v ( 2 − 3 x − y + t (1+ x − y ) ) if x ≤ y , x + y ≤ 1 , (1+ t )(1 − y ) ≤ (3 − t ) x , u ( x − y + 4 y 1+ t ) if x ≥ y , (1+ t )(1 − x ) ≥ (3 − t ) y , v ( 2 − x − 3 y + t (1 − x + y ) ) if x ≥ y , x + y ≤ 1 , (1+ t )(1 − x ) ≤ (3 − t ) y , v ( x +3 y − 2+ t (1+ x − y ) ) if x ≤ y , x + y ≥ 1 , (3 − t )(1 − y ) ≥ (1+ t ) x , w ( y − x + 4( y − 1) 1+ t ) if x ≤ y , (3 − t )(1 − y ) ≤ (1+ t ) x , v ( 3 x + y − 2+ t (1 − x + y ) ) if x ≥ y , x + y ≥ 1 , (3 − t )(1 − x ) ≥ (1+ t ) y , w ( x − y + 4(1 − x ) 1+ t ) if x ≥ y , (3 − t )(1 − x ) ≤ (1+ t ) y . And, similarly , one proves the e q uality I h ( w, u ) ◦ v I h ( w, u ) = I h ( w, u ) , for any three paths in X , u , v , w : I → X with u (0) = v (0 ) = w (0). Then, it only remains to pr ov e the interchange law in Axiom 3 . T o do s o, let w ′′ w ′ o o w o o [ δ ] [ β ] v ′′ O O v ′ o o O O v o o O O [ γ ] [ α ] u ′′ O O u ′ o o O O u o o O O be square s in Π Π X . Then, the req uired equality follows from the exis tence of the relative homotopy ( δ ◦ h β ) ◦ v ( γ ◦ h α ) → ( δ ◦ v γ ) ◦ h ( β ◦ v α ) deﬁned by the map H : I 2 × I → X such that H ( x, y , t ) = DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 19 1 0 71 62 53 4 α ( x + y − 2 ty , 4 y ) 1 − x +2 ty ≥ 5 y , x − 3 y ≥ 2 ty , 2 0 71 62 53 4 α ( 2( x − y ) 1+ t , 2( x − tx + y +3 ty ) 2+ t − t 2 ) 2+ t − t 2 − 6 x +4 tx +2 y ≥ 8 ty , (3+2 t ) y ≥ x ≥ y , 3 0 71 62 53 4 α ( t 2 − 2 t +2 x − 2 y +4 ty 1 − 2 t +2 t 2 , 3 t 2 − t (1+4 x )+2( x + y ) 2 − 4 t +4 t 2 ) t 2 + t (4 x − 3) ≥ 2( x + y − 1) , t 2 − 2 x +6 y ≥ t (4 x +8 y − 3) , t 2 +6 x + t (8 y − 4 x − 1) ≥ 2(1+ y ) , 4 0 71 62 53 4 α ( t − 2( x + y ) t − 2 , 2 t ( x +3 y − 1) − 8 y t 2 − t − 2 ) 1 ≥ x + y , x − 1 ≥ (2 t − 5) y , 2 x − t 2 − 6 y ≥ t (3 − 4 x − 8 y ) , 5 0 71 62 53 4 v ′ ( 3 − t − 4 x ) x ≥ y , 1 ≥ x + y , 2( x + y − 1) ≥ t 2 + t (4 x − 3) , 6 0 71 62 53 4 γ ( 4 x − 3 , x + y − 1 − 2 t ( x − 1) ) 5 x + y − 5 ≥ 2 t ( x − 1) , 2 t ( x − 1)+3 x ≥ y +2 , 7 0 71 62 53 4 γ ( 6+ t 2 − 8 x + t (6 x +2 y − 7) t 2 − t − 2 , 2( x + y − 1) 2 − t ) x + y ≥ 1 , 5+2 t ( x − 1) ≥ 5 x + y , 9 t +6 x ≥ 4+ t 2 +8 tx +2 y +4 ty , 8 0 71 62 53 4 γ ( t + t 2 − 4 ty +2( x + y − 1) 2 − 4 t +4 t 2 , 1+ t 2 +4 t ( x − 1) − 2 x +2 y 1 − 2 t +2 t 2 ) t + t 2 +2( x + y − 1) ≥ 4 ty , t 2 +2(1+ x − 3 y ) ≥ t (8 x − y − 3) , 4+ t 2 − 6 x +2 y ≥ t (9 − 8 x − 4 y ) , 9 0 71 62 53 4 γ ( t 2 − 2( x + y − 1)+ t (3 − 6 x +2 y ) t 2 − t − 2 , 1+ t − 2 x +2 y 1+ t ) 8 tx +6 y − 4 ty ≥ 2+3 t + t 2 +2 x, x ≥ y , 2+ y ≥ 2 t ( x − 1)+3 x , 10 0 71 62 53 4 v ′ ( 4 y − 1 − t ) x ≥ y , x + y ≥ 1 , 2+4 ty ≥ t + t 2 +2 x +2 y , 11 0 71 62 53 4 β ( 4 x,x + y − 2 tx ) 1+2 tx ≥ y +5 x, y ≥ 3 x +2 tx , 12 0 71 62 53 4 β ( 2 t ( y +3 x − 1) − 8 x t 2 − t − 2 , t − 2( x + y ) t − 2 ) 1 ≥ x + y , y +(5 − 2 t ) x ≥ 1 , 2 y + t (3 − 4 y − 8 x ) − 6 x ≥ t 2 , 13 0 71 62 53 4 β ( 3 t 2 − t (1+4 y )+2( x + y ) 2 − 4 t +4 t 2 , t 2 − 2 t +2 y − 2 x +4 tx 1 − 2 t +2 t 2 ) t 2 + t (4 y − 3) ≥ 2( x + y − 1) , t 2 + t (3 − 4 y − 8 x )+6 x ≥ 2 y , t 2 +6 y − 2(1+ x ) ≥ t (1+4 y − 8 x ) , 14 0 71 62 53 4 β ( 2( y − ty + x +3 tx ) 2+ t − t 2 , 2( y − x ) 1+ t ) 2+ t +4 ty +2 x ≥ t 2 +6 y +8 tx, (3+2 t ) x ≥ y ≥ x , 15 0 71 62 53 4 v ′ ( 3 − t − 4 y ) y ≥ x, 1 ≥ x + y , 2( x + y − 1) ≥ t 2 + t (3 − 4 y ) , 16 0 71 62 53 4 δ ( 2 t (1 − y )+ x + y − 1 , 4 y − 3 ) 5 y + x ≥ 5+2 t ( y − 1) , 2 t ( y − 1)+3 y ≥ x +2 , 17 0 71 62 53 4 δ ( 2( x + y − 1) 2 − t , 6+ t 2 − 8 y + t (6 y +2 x − 7) t 2 − t − 2 ) x + y ≥ 1 , 9 t +6 y − 8 ty − 2 x − 4 tx ≥ 4+ t 2 , 5+2 t ( y − 1) ≥ 5 y + x , 18 0 71 62 53 4 δ ( 1+ t 2 +4 t ( y − 1) − 2 y +2 x 1 − 2 t +2 t 2 , t + t 2 − 4 tx +2( x + y − 1) 2 − 4 t +4 t 2 ) t + t 2 − 4 tx ≥ 2(1 − x − y ) , t 2 +2(1+ y − 3 x ) ≥ t (8 y − 4 x − 3) , 4+ t 2 − 6 y +2 x ≥ t (9 − 8 y − 4 x ) , 19 0 71 62 53 4 δ ( 1+ t − 2 y +2 x 1+ t , t 2 − 2( x + y − 1)+ t (3 − 6 y +2 x ) t 2 − 2 − t ) 8 ty +6 x − 4 tx ≥ 2+3 t + t 2 +2 y , y ≥ x , 2 − 3 y + x ≥ 2 t ( y − 1) , 20 0 71 62 53 4 v ′ ( 4 y − 1 − t ) y ≥ x, x + y ≥ 1 , 2+4 tx ≥ t + t 2 +2 y +2 x , where parts n ( /) .* -+ , in the homotopy H ( x, y , t ) a bove co r resp ond to the are a s with ( x, y ) ∈ I 2 shown in Figure 1 b elow.               ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?                                          j j j j j j j j j j j j j j j j ? ? ? ? ? ? ?        T T T T T T T T T T T T T T T T                        ? ? ? ? ? ? ? * * * * * * * * * * * * * * * * ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?                 * * * * * * * * * * * * * * * *        ? ? ? ? ? ? ? j j j j j j j j j j j j j j j j T T T T T T T T T T T T T T T T        11 '! &" %# $ 13 '! &" %# $ 15 '! &" %# $ 10 '! &" %# $ 8 '! &" %# $ 6 '! &" %# $ 1 '! &" %# $ 3 '! &" %# $ 5 '! &" %# $ 20 '! &" %# $ 18 '! &" %# $ 16 '! &" %# $ 2 '! &" %# $ 14 ' ! &" %# $ 4 ' ! &" %# $ 7 '! &" %# $ 19 ' ! &" %# $ 9 '! &" %# $ 17 '! &" %# $ 12 ' ! &" %# $ Figure 1. 20 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS Finally , w e o bserve that Π Π X satisﬁes the ﬁlling condition. Supp ose a conﬁguration of mor- phisms in Π Π X v ′ v o o u O O is given. This means we have pa ths u, v , v ′ : I → X with u (0) = v (0) and v (1) = v ′ (1). Since the inclusion ∂ I ֒ → I is a coﬁbratio n, the map f : ( { 0 } × I ) ∪ ( I × ∂ I ) → X with f (0 , t ) = v ( t ), f ( t, 0 ) = u ( t ) and f ( t, 1 ) = v ′ (1 − t ) for 0 ≤ t ≤ 1, has a n extensio n to a map α : I × I → X , which prec isely r e presents a square in Π Π X of the form v ′ v o o [ α ] u ′ O O u o o O O where u ′ : I → X is the path u ′ ( t ) = α (1 , 1 − t ). Hence, Π Π X v eriﬁes the ﬁlling condition.  In the prev ious Section 3 w e introduce d homotopy gro ups for double gr o upo ids sa tisfying the ﬁlling condition. The next prop osition pr ovides gr eater sp eciﬁcs on the relationship b etw een the homoto p y gro ups of the asso c ia ted homotopy double g roup oid Π Π X to a top olog ical space X and the corresp onding for X . Theorem 4 .2. F or any sp ac e X , any p ath u : I → X , and 0 ≤ i ≤ 2 , ther e is an isomorphism π i (Π Π X , u ) ∼ = π i ( X, u (0)) . Pr o of. F or any tw o points x, y ∈ X , the consta n t paths c x and c y are in the same connected comp onent of Π Π X if and only if there is a pair of mo rphisms in Π Π X o f the form c y u o o c x O O or, equiv alently , if and o nly if there is a path u : I → X in X such that u (1 ) = y and u (0) = x . Then, we have an injectiv e map π 0 X → π 0 Π Π X , [ x ] 7→ [ c x ] , which is also surjective since, for any path u in X , we hav e a vertical mo r phism u ← c u (0) in Π Π X ; whence the a nnounced bijection π 0 X ∼ = π 0 Π Π X . Next, we prov e that there is an is o morphism π 1 (Π Π X , u ) ∼ = π 1 ( X, u (0)) for an y given path u : I → X . T o do so, we shall use the fundamen tal group oid Π X o f the space X ; that is, the g roup oid whose ob jects are the p oints of X and whose mor phisms are the (relativ e to ∂ I ) homotopy classes [ v ] o f pa ths v : I → X . Simply by c hecking the co nstruction, we see that an element [( u, v ) , ( v , u )] ∈ π 1 (Π Π X , u ) is determined by a path v : I → X , with v (0) = u (0 ) and v (1) = u (1). Mo reov er, for any other such v ′ : I → X , it holds that [( u , v ) , ( v , u )] = [( u, v ′ ) , ( v ′ , u )] in π 1 (Π Π X , u ) if and only if ther e are squar es in Π Π X of the form u v o o [ α ] w O O u o o O O u v ′ o o [ α ′ ] w O O u o o O O or, equiv alen tly , if a nd only if there a re sq uares in X , α, α ′ : I 2 → X with b oundaries as in · · u o o w   α · v O O u / / · · · u o o w   α ′ · v ′ O O u / / · DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 21 Since this last c ondition simply means that, in the fundamental group oid Π X , the e quality [ v ] = [ v ′ ] holds, we conclude with bijections π 1 (Π Π X , u ) ∼ = Hom Π X ( u (0) , v (1)) ∼ = π 1 ( X, u (0)) [( u, v ) , ( v , u )]  / / [ v ]  / / [ u ] -1 ◦ [ v ] T o see that the comp os ite bijection φ : [( u, v ) , ( v , u )] 7→ [ u ] -1 ◦ [ v ] is actually an iso mor- phism, let v 1 , v 2 : I → X be paths in X , b oth from u (0 ) to u (1). Then, [( u, v 1 ) , ( v 1 , u )] ◦ [( u, v 2 ) , ( v 2 , u )] = [( u, v ) , ( v , u )], where v o ccur s in a co nﬁguration such as u v 1 o o v o o u O O [ γ ] v 2 O O o o u O O for s ome (any) squar e γ : I 2 → X in X with b oundary as below. · · v 1 o o u   · v O O v 2 / / γ · It fo llows tha t, in Π X , [ v ] = [ v 1 ] ◦ [ u ] -1 ◦ [ v 2 ] and therefor e φ [( u, v 1 ) , ( v 1 , u )] ◦ φ [( u, v 2 ) , ( v 2 , u )] = [ u ] -1 ◦ [ v 1 ] ◦ [ u ] -1 ◦ [ v 2 ] = [ u ] -1 ◦ [ v ] = φ ([( u, v 1 ) , ( v 1 , u )] ◦ [( u , v 2 ) , ( v 2 , u )]) . Finally , we consider the case i = 2 . Let u : I → X b e any pa th with u (0) = x . Then, the mapping [ α ] 7→ I h ( c x , u ) ◦ v [ α ] ◦ v I h ( u, c x ), which c arries a sq uare [ α ] ∈ π 2 (Π Π X , u ), to the comp osite o f c x c x [ e h ] u O O u O O [ α ] u u [ e h ] c x O O c x O O establishes an isomorphism π 2 (Π Π X , u ) ∼ = π 2 (Π Π X , c x ). Now, it is clear that bo th π 2 (Π Π X , c x ) a nd π 2 ( X, x ) ar e the s a me ab elian group of r e lative to ∂ I 2 homotopy cla s ses of maps I 2 → X whic h are cons tant x along the fo ur sides of the square.  The construction of the double group oid Π Π X from a space X is easily seen to be functoria l and, moreov e r , the isomorphisms in Theorem 4.2 above b eco me natural. Then, we hav e the next cor ollary . Corollary 4.3. A c ontinuous map f : X → Y is a we ak homotopy 2 -e quivalenc e if and only if the induc e d double functor Π Π f : Π Π X → Π Π Y is a we ak e qu ivalenc e. 5. The geometric realiza tion of a do uble gr oupoid. Hereafter, we shall r egard each order ed set [ n ] a s the categ ory with exactly one arrow j → i when 0 ≤ i ≤ j ≤ n . Then, a non-dec reasing ma p [ n ] → [ m ] is the same as a functor. 22 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS The geometric realization, or classifying space, of a catego ry C , [3 0], is |C | := | N C | , the geometric r ealization of its nerve [1 8] N C : ∆ o → Set , [ n ] 7→ F unc ([ n ] , C ) , that is, the simplicial s e t who se n -simplices a re the functor s F : [ n ] → C , o r tuples of arrows in C F =  F i F i,j ← − F j  0 ≤ i ≤ j ≤ n such that F i,j ◦ F j,k = F i,k and F i,i = I F i . If G is a double ca tegory , then its geometric realization, |G | , is deﬁned by ﬁrst taking the double nerve N N G , whic h is a bisimplicial set, and then r ealizing to obtain a s pace |G | := | N N G | . T o have a mana geable description handle descr iptio n for the bisimplices in N N G , we can use the fo llowing construction: If A and B a re ca tegories, let A ⊗ B be the double catego ry whose ob jects ar e pair s ( a, b ), where a is an ob ject of A and b is an ob ject of B ; ho rizontal morphisms are pairs ( f , b ) : ( a, b ) → ( c, b ), with f : a → c a morphism in A ; vertical morphisms ar e pairs ( a, u ) : ( a, b ) → ( a, d ) with u : b → d in B ; and a squar e in A ⊗ B is g iven by e ach mor phism ( f , u ) : ( a, b ) → ( c, d ) in the pro duct categor y A × B , b y stating its bo undary as in ( c, d ) ( f , u ) ( a, d ) ( f , d ) o o ( c, b ) ( c, u ) O O ( a, b ) ( f , b ) o o ( a, u ) O O Comp ositions in A ⊗ B are deﬁned in the evident wa y . Then, the double nerve N N G o f a double category G is the bisimplicial set N N G : ∆ o × ∆ o → Set , ([ p ] , [ q ]) 7→ DF unc ([ p ] ⊗ [ q ] , G ) , whose ( p, q )-bis implice s ar e the double functors F : [ p ] ⊗ [ q ] → G or conﬁgura tions of squares in G o f the form F r i F r,s i,j F r j F r i,j o o 0 ≤ i ≤ j ≤ p 0 ≤ r ≤ s ≤ q , F s i F r,s i O O F s j F s i,j o o F r,s j O O such that F r,s i,j ◦ h F r,s j,k = F r,s i,k , F r,s i,j ◦ v F s,t i,j = F r,t i,j , F r,s i,i = I h F r,s i , and F r,r i,j = I v F r i,j . But note tha t the double category [ p ] ⊗ [ q ] is free o n the bigraph ( j − 1 ,r − 1) ( j,r − 1) o o 0 ≤ i ≤ j ≤ p 0 ≤ r ≤ s ≤ q , ( j − 1 ,r ) O O ( j,r ) o o O O DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 23 and therefore, giving a double functor F : [ p ] ⊗ [ q ] → G as ab ov e is equiv alen t to sp ecifying the p × q conﬁguration o f squar es in G F r − 1 j − 1 F r − 1 , r j − 1 ,j F r − 1 j F r − 1 j − 1 ,j o o 1 ≤ j ≤ p 1 ≤ r ≤ q . F r j − 1 F r − 1 , r j − 1 O O F r j F r j − 1 ,j o o F r − 1 , r j O O Thu s, ea ch vertical simplicial s e t N N G p, ∗ is the nerve of the “vertical” catego ry ha ving as ob jects strings of p -co mpos able horizo n tal mo rphisms a 0 ← a 1 ← · · · ← a p , whose arr ows consist of p horizontally comp osable squares as in b 0 b 1 o o · o o · b p o o a 0 O O a 1 O O o o · O O o o · O O a p O O o o And, similarly , each horizontal simplicial se t N N G ∗ ,q is the nerve of the “horizo n tal” category whose ob jects ar e the length q sequences of comp osable vertical mo rphisms o f G , with length q sequences of vertically comp osable square s as morphisms b etw een them. F or instance, if A and B a re catego ries, then N N( A ⊗ B ) = N A ⊗ N B . In particula r , N N([ p ] ⊗ [ q ]) = ∆[ p ] ⊗ ∆[ q ] = ∆[ p, q ] , is the standar d ( p, q )-bisimplex. It is a well-kno wn fa ct that the nerve N C of a categ ory C satisﬁes the Kan ex tens ion condition if a nd only if C is a gr o upo id, and, in s uch a case , every ( k , n )-horn Λ k [ n ] → N C , for n ≥ 2, has a unique extensio n to an n -simplex o f N C Λ k [ n ] / /  _   N C ∆[ n ] ∃ ! = = (see [2 1, P r op ositions 2 .2.3 and 2 .2.4], fo r ex ample). F or double categories G , we hav e the following: Theorem 5.1. L et G b e a double c ate gory. The fol lowing statements ar e e qu ivalent: (i) G is a double gr oup oid satisfying the ﬁl ling c ondition. (ii) The bisimpl icial set N N G satisﬁes t he extension c ondition. (iii) The simplic ial set diagN N G is a Kan c omplex. Pr o of. (i) ⇒ (ii) Since G is a double group oid, a ll simplicial sets N N G p, ∗ and N N G ∗ ,q are nerves o f group oids. Therefor e, every extension problem of the form ∆[ p ] ⊗ Λ l [ q ] / /  _   N N G ∆[ p, q ] ∃ ! 9 9 or Λ k [ p ] ⊗ ∆[ q ] / /  _   N N G ∆[ p, q ] ∃ ! 9 9 24 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS has a solution and it is unique. Suppo se then a n extension problem o f the form (5.1) Λ k,l [ p, q ] / /  _   N N G ∆[ p, q ] : : If p ≥ 2, then the restricted map Λ k [ p ] ⊗ ∆[ q ] ֒ → Λ k,l [ p, q ] → N N G has a unique extension to a bisimplex ∆[ p, q ] → N N G , which is a s o lution to (5.1) (which in fact has a unique solution if p ≥ 2 or q ≥ 2). Hence, we reduce the pro of to the case in w hich p = 1 = q , with the four p oss ibilities k = 0 , 1 and l = 0 , 1. But any such extensio n problem has a solutio n thanks to Le mma 3.1. F or example, let us discus s the case k = 0 = l : A bisimplicial map Λ 0 , 0 [1 , 1] ( − ,w ; − ,g ) / / N N G consists of tw o bisimplicial maps w : ∆[0 , 1] → N N G and g : ∆[1 , 0] → N N G , such that w d 1 v = g d 1 h . That is, a vertical mo rphism w of G and a ho rizontal mor phism g o f G , such tha t b oth have the same target. By Lemma 3.1, there is a square α in G of the form · α · g o o · w O O · o o O O which deﬁnes a bis implicial map F : ∆[1 , 1] → N N G such that F 0 , 1 0 , 1 = α . Then F d 1 h = w , F d 1 v = g , and the diag r am b elow commutes, a s req uired. Λ 0 , 0 [1 , 1]  _   ( − ,w ; − ,g ) / / N N G ∆[1 , 1] F 6 6 n n n n n n n n n (ii) ⇒ (i) The simplicial sets N N G 0 , ∗ , N N G ∗ , 0 , N N G 1 , ∗ , and N N G ∗ , 1 are resp ectively the nerves of the four comp onent categ ories of the double category G . Since all these simplicial sets satisfy the Ka n extension condition, it follows that the four catego ry structures in volv ed a re gro upo ids; that is, G is a double g roup oid. F ur ther more, for any given ﬁlling problem in G , · · g o o ∃ ? · O O · o o u O O we ca n solve the extension problem Λ 1 , 0 [1 , 1]  _   ( u, − ; − ,g ) / / N N G ∆[1 , 1] F 6 6 and the square F 0 , 1 0 , 1 has u as ho rizontal sourc e and g as v ertical target. Thus G satisﬁes the ﬁlling co nditio n. (i) ⇒ (iii) The higher dimensio nal part o f the pro o f is in the following lemma, tha t we establish fo r future refere nce . DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 25 Lemma 5. 2. If G is any double gr oup oid and n is any inte ger such that n ≥ 3 , then every extension pr oblem Λ k [ n ] / /  _   diagN N G ∆[ n ] 8 8 has a solut ion and it is unique. Pr o of. Let F = ( F r,s i,j ) : [ n ] ⊗ [ n ] → G denote the double functor we a re lo oking for so lving the given extensio n problem. Recall that to give such an F is equiv a le n t to sp ecifying the n × n conﬁguratio n o f squar es F r − 1 j − 1 F r − 1 , r j − 1 ,j F r − 1 j F r − 1 j − 1 ,j o o 1 ≤ j ≤ n 1 ≤ r ≤ n . F r j − 1 F r − 1 , r j − 1 O O F r j F r j − 1 ,j o o F r − 1 , r j O O W e claim that F exists a nd, moreov er, that it is completely showed f rom a ny three o f its (known) faces [ n − 1] ⊗ [ n − 1 ] d m ⊗ d m − → [ n ] ⊗ [ n ] F − → G , m 6 = k ; there fore, b y the input data Λ k [ n ] → diag N N G . In eﬀect, since each m th -face consists of all squares F r,s i,j such that m / ∈ { i, j, r , s } , once we have selected any three in teg ers m, p, q with m < p < q and k / ∈ { m, p, q } , w e know explicitly all squares F r,s i,j except those in which m, p and q appear in the lab els, that is: F m,p q,j , F m,q p,j , F j,p m,q , and so on. In the case where k ≥ 3, if we tak e { m, p, q } = { 0 , 1 , 2 } then we hav e given all squares F r,s i,j , except those with { 0 , 1 , 2 } ⊆ { r , s, i, j } . In pa rticular, we hav e all F r,r +1 i,i +1 , except four of them, namely , F 0 , 1 2 , 3 , F 0 , 1 1 , 2 , F 1 , 2 0 , 1 , and F 2 , 3 0 , 1 , whic h, howev e r, are uniquely determined b y the e quations F 0 , 1 2 , 3 ◦ v F 1 , 2 2 , 3 = F 0 , 2 2 , 3 , F 2 , 3 0 , 1 ◦ h F 2 , 3 1 , 2 = F 2 , 3 0 , 2 , F 0 , 1 1 , 2 ◦ h F 0 , 1 2 , 3 = F 0 , 1 1 , 3 , F 1 , 2 0 , 1 ◦ v F 2 , 3 0 , 1 = F 1 , 3 0 , 1 , that is, F 0 , 1 2 , 3 = F 0 , 2 2 , 3 ◦ v ( F 1 , 2 2 , 3 ) -1 v , and so on. The other p ossibilities for k a re discussed in a similar wa y : If k = 2, then we select { m, p, q } = { 0 , 1 , n } and deter mine F completely b y taking in to account the tw o equations F 0 , 1 n − 1 ,n ◦ v F 1 , 2 n − 1 ,n = F 0 , 2 n − 1 ,n , F n − 1 ,n 0 , 1 ◦ h F n − 1 ,n 1 , 2 = F n − 1 ,n 0 , 2 . If k = 1, then we take { m, p, q } = { 0 , 2 , 3 } and ﬁnd the unknown squa r es F 2 , 3 0 , 1 and F 0 , 1 2 , 3 by the e quations F 1 , 2 0 , 1 ◦ v F 2 , 3 0 , 1 = F 1 , 3 0 , 1 and F 0 , 1 1 , 2 ◦ h F 0 , 1 2 , 3 = F 0 , 1 1 , 3 , resp ectively . Finally , in the case where k = 0, we take { m, p, q } = { n − 2 , n − 1 , n } and we determine the non-given four s quares o f the family ( F r,r +1 i,i +1 ), that is, F n − 2 ,n − 1 n − 1 ,n , F n − 1 ,n n − 2 ,n − 1 , F n − 3 ,n − 2 n − 1 ,n , and F n − 1 ,n n − 3 ,n − 2 by means of the four equatio ns F n − 3 ,n − 2 n − 3 ,n − 1 ◦ h F n − 3 ,n − 2 n − 1 ,n = F n − 3 ,n − 2 n − 3 ,n , F n − 3 ,n − 1 n − 3 ,n − 2 ◦ v F n − 1 ,n n − 3 ,n − 2 = F n − 3 ,n n − 3 ,n − 2 , F n − 3 ,n − 2 n − 1 ,n ◦ v F n − 2 ,n − 1 n − 1 ,n = F n − 3 ,n − 1 n − 1 ,n , and F n − 1 ,n n − 3 ,n − 2 ◦ h F n − 1 ,n n − 2 ,n − 1 = F n − 1 ,n n − 3 ,n − 1 . This completes the pro of of the lemma.  W e now r e turn to the pro of of (i) ⇒ (iii) in Theorem 5.1. After Lemma 5.2 ab ov e, it r emains to prove that every extension problem Λ k [2] / /  _   diagN N C ∆[2] ∃ ? : : 26 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS for k = 0 , 1 , 2 , has a solution. In the ca se wher e k = 0, the data fo r a simplicial ma p ( − , τ , σ ) : Λ 0 [2] → diagN N G consists of a co uple of squar es in G of the form a σ · o o · O O · O O o o a τ · o o · O O · O O o o and an extension solution ∆[2] / / diagN N G a mount s to a diagram of sq uares as in a σ · o o x · o o · O O y · o o O O z · o o O O · O O · o o O O · o o O O such that ( σ ◦ h x ) ◦ v ( y ◦ h z ) = τ . T o se e that such squa res x , y , a nd z exist, we form the conﬁguratio n (a ctually , a 3-s implex of dia gN N G ) a σ · o o σ -1 h · o o α -1 v · o o · O O σ -1 v · O O o o σ -1 · o o O O α · o o O O · O O β -1 h · O O o o β a o o O O τ · o o O O · O O · o o O O · o o O O · O O o o where α and β are any found tha nk s G satisﬁes the ﬁlling condition. Then, we take x = σ -1 h ◦ h α -1 v , y = σ -1 v ◦ v β -1 h , and z = ( σ -1 ◦ h α ) ◦ v ( β ◦ h τ ). The case in which k = 2 is dual of the case k = 0 ab ov e, and the case when k = 1 is easier: A s implicial map ( σ , − , τ ) : Λ 1 [2] → diag N N G amounts to a couple of squar es in G of the form a σ · o o · O O · O O o o · τ · o o · O O a O O o o and an extens ion solution ∆[2] / / diagN N G is given b y any conﬁguration of s quares in G of the fo rm · τ · o o x · o o · O O y a O O o o σ · o o O O · O O · o o O O · O O o o Since G s atisﬁes the ﬁlling co ndition (reca ll Lemma 3 .1), it is clea r that ﬁlling sq ua res x a nd y as a bove exist, and therefore the requir e d extension map exists. (iii) ⇒ (i) By [13, Theorem 8], a ll simplicial sets N N G p, ∗ and N N G ∗ ,q satisfy the Kan extension condition. In par ticular, the nerves of the four co mpone nt ca tegories of the double category G , that is, the simplicial sets N N G 0 , ∗ , N N G ∗ , 0 ,N N G 1 , ∗ , and N N G ∗ , 1 are all Kan c omplexes. By [21, Prop ositions 2.2.3 and 2.2.4 ], it follows that the four ca tegory structure s inv o lved are g r oup oids, and so G is a double g roup oid. T o see that G satisﬁes the ﬁlling condition, supp ose that a ﬁlling pro blem · · g o o ∃ ? · O O · o o u O O DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 27 is given. Since the s implicial map Λ 1 [2] (I h u, − , I v g ) / / diagN N C ha s an extension to a 2 - simplex ∆[2] / / diagN N G , w e conclude the existence of a diag ram of square s in G of the form · I v g · g o o · o o · · o o g I h u · O O · α O O · O O u o o · u O O and then, particular ly , the existence of a s quare α as is r e q uired.  W e now s ta te o ur main result in this section. Theorem 5 .3. L et G b e a double gr oup oid satisfying the ﬁl ling c ondition. Then, for e ach obje ct a of G , t her e ar e natur al isomorph isms (5.2) π i ( G , a ) ∼ = π i ( |G | , | a | ) , i ≥ 0 . Pr o of. By taking into acco un t F act 2.7 (1), we sha ll identif y the ho motopy gro ups of |G | with those o f the K an co mplex (b y Theo rem 5.1) dia gN N G , which a r e deﬁned, as w e noted in the preliminary Sectio n 2 , using only its simplicia l s tructure. T o compare the π 0 sets, obser ve that the 0-simplices a ∈ diagN N G 0 = N N G 0 , 0 are precisely the ob jects o f G . F ur ther more, tw o 0-simplices a, b ar e in the s a me connected comp onent of diagN N G if and only if there is a s q uare (i.e., a 1-simplex) o f the form b ∃ ? · o o · O O a, o o O O that is, since G s a tisﬁes the ﬁlling condition, if and only if a and b are connected in G (see Subsetion 3 .1). Thus, π 0 |G | = π 0 G . W e now compa re the π 1 groups. An elemen t [ α ] ∈ π 1 ( |G | , | a | ) is the equiv alence class of a square α in G of the fo r m a α · g o o · O O a o o u O O and [ α ] = [ α ′ ] if and only if there is a conﬁguration of squar es in G of the form a α · x g o o · g ′ o o · y O O a I a O O u o o a u ′ O O · O O a o o a such that ( α ◦ h x ) ◦ v y = α ′ . By recalling now the deﬁnition o f the ho motopy g roup π 1 ( G , a ), we obs erve that, if [ α ] = [ α ′ ] in π 1 ( |G | , | a | ), then, by the existence of the squares α and α ◦ h x , we hav e [ g , u ] = [ g ◦ h g ′ , u ′ ] in π 1 ( G , a ); tha t is,[t v α, s h α ] = [t v α ′ , s h α ′ ]. It follows tha t there is a well-deﬁned map Φ : π 1 ( |G | , | a | ) − → π 1 ( G , a ) . [ α ] 7− → [ g , u ] = [t v α, s h α ] 28 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS This map is actually a gr oup homomo rphism. T o see that, let a α 1 · g 1 o o · O O a o o u 1 O O a α 2 · g 2 o o · O O a o o u 2 O O be squares represe nting elemen ts [ α 1 ] , [ α 2 ] ∈ π 1 ( |G | , | a | ). Then, its pro duct in the homotopy group π 1 ( |G | , | a | ) is [ α 1 ] ◦ [ α 2 ] = [( α 1 ◦ h β ) ◦ v ( γ ◦ h α 2 )], where β and γ a re an y squares in G deﬁning a conﬁgura tion of the fo r m (i.e., a 2-simplex o f diag N N G ) a α 1 · g 1 o o β · g o o · γ O O a α 2 O O o o · o o u O O · O O · o o O O a o o u 2 O O Hence, Φ([ α 1 ] ◦ [ α 2 ]) = [ g 1 ◦ h g , u ◦ v u 2 ] = [ g 1 , u 1 ] ◦ [ g 2 , u 2 ] = Φ([ α 1 ]) ◦ Φ([ α 2 ]) , and therefo re Φ is a homo morphism. F rom the ﬁlling condition on G , it follows that Φ is a sur jectiv e map. T o pr ove that it is a lso injectiv e, supp ose Φ[ α 1 ] = Φ[ α 2 ], where [ α 1 ] , [ α 2 ] ∈ π 1 ( |G | , a ) are as above. This means that there a re sq ua res in G , sa y x 1 and x 2 , of the form a x 1 · g 1 o o · w O O a f o o u 1 O O a x 2 · g 2 o o · w O O a f o o u 2 O O with which we can form the following three 2-s implice s of diag N N G · x 1 · I h u 1 o o · · x -1 v 1 ◦ v α 1 O O · I a o o O O · O O · O O · o o · , · x 2 · I h u 2 o o · · x -1 v 2 ◦ v α 2 O O · I a o o O O · O O · O O · o o · , · x 1 · x -1 h 1 ◦ h x 2 o o · o o · I v f O O · I a o o O O · O O · · o o · . The ﬁrst one shows that [ x 1 ] = [ α 1 ] in the g roup π 1 ( |G | , a ), the second that [ x 2 ] = [ α 2 ], and the third that [ x 1 ] = [ x 2 ]. Whence [ α 1 ] = [ α 2 ], as re q uired. Finally , we s how the isomorphisms π 1 ( |G | , | a | ) ∼ = π i ( G , a ), for i ≥ 2. F or i ≥ 3, it follows from Lemma 5.2 tha t π i ( |G | , a ) = 0 , a nd the result beco mes obvious. F o r the case i = 2 , it is als o a consequence of the afore-mentioned Lemma 5.2 tha t the homotopy relation b et ween 2-simplices in diag N N G is trivial. Then, the group π 2 ( |G | , | a | ) consists of all 2-simplices in diag N N G of the form · I a · σ · · σ -1 · I a · · · for σ ∈ π 2 ( G , a ), whence the isomor phism b ecomes clear.  Corollary 5 .4. A double functor F : G → G ′ is a we ak e quivalenc e if and only if the induc e d c el lular map on r e alizations | F | : |G | → |G ′ | is a homotopy e quivalenc e. DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 29 6. A left adjoint to the d ouble ner ve functor. Recall from Theorem 5.1 (ii) that the double nerve N N G , of any double gr oup o id satisfying the ﬁlling condition, s atisﬁes the extension condition. Mor e ov er , since b oth simplicial sets N N G ∗ , 0 and N N G 0 , ∗ are nerves o f gro upo ids , all homotopy gr oups π 2 (N N G ∗ , 0 , a ) and π 2 (N N G 0 , ∗ , a ) v anish. Our go al in this section is to prove that the double nerve functor, G 7→ N N G , embeds, as a reﬂexive sub categ ory , the categ ory of double g r oup oids with ﬁlling condition into the categor y of those bisimplicial sets K that sa tisfy the extensio n condition and s uc h that π 2 ( K ∗ , 0 , a ) = 0 = π 2 ( K 0 , ∗ , a ) for all v ertices a ∈ K 0 , 0 . That is , there is a reﬂector functor for suc h bisimplicial sets K 7→ P P K, which works a s a bisimplicial version of Brown’s construction in [6 , Theorem 2.1 ]. F urthermore, as we will prove, the resulting double group oid P P K a lwa y s repr esents the homotop y 2-type of the input bisimplicial set K , in the sense that there is a natura l weak 2 -equiv alence | K | → | P P K | . F or any given bisimplicial set K , under the ass umption that it satisﬁes the extension condition and b oth the Kan complexes K ∗ , 0 and K 0 , ∗ hav e trivial gr oups π 2 , the deﬁnition of the homotopy double gr oup oid P P K is as follows: The ob jects o f P P K are the vertices a : ∆[0 , 0 ] → K o f K . The groupo id of horiz ont al morphisms is the horizontal fundamen ta l gro upo id P K ∗ , 0 , and the g roup oid of vertical mor phisms is the vertical fundamental gro upo id P K 0 , ∗ (see the la st part of Subsec tio n 2.2). Thus, a horiz o nt al mor phism [ f ] h : a → b is the horizontal homotopy class o f a bisimplex f : ∆[1 , 0 ] → K with f d 0 h = a and f d 1 h = b , whereas a vertical morphism in P P K , [ u ] v : a → b , is the vertical homotopy class o f a bisimplex u : ∆[0 , 1] → K with u d 0 v = a and ud 1 v = b . A squar e of P P K is the bihomotopy cla ss [[ x ]] of a bis implex x : ∆[1 , 1] → K , with b oundary · · [ xd 1 v ] h o o · [ xd 1 h ] v O O · [ xd 0 v ] h o o [ xd 0 h ] v O O which is well deﬁned thanks to Lemma 2.6. The horizontal co mpo s ition of squares in P P K is the only one making the corres po ndence  xd 0 h [ x ] h → xd 1 h  [ ] 7− →  [ xd 0 h ] v [[ x ]] → [ xd 1 h ] v  a sur jective ﬁbration of gro up oids from the horizontal fundamental gr oup oid P K ∗ , 1 to the horizontal gr oupo id of squares in P P K . T o deﬁne this comp o sition, we shall need the following: Lemma 6.1. L et x, y : ∆[1 , 1] → K b e bisimplic es such that [ xd 0 h ] v = [ y d 1 h ] v . Then, ther e is a bisimplex x ′ : ∆[1 , 1] → K such that [ x ′ ] v = [ x ] v and x ′ d 0 h = y d 1 h . Pr o of. Once an y vertical homotopy from xd 0 h to y d 1 h is selected, say α : ∆[0 , 2] → K , let β : ∆[1 , 2] → K b e any bis implex s olving the extensio n pro ble m Λ 1 , 2 [1 , 2] ( α, − ; xd 0 v s 0 v , x , − ) / /  _   K. ∆[1 , 2] β 5 5 30 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS Then, we take x ′ = β d 2 v : ∆[1 , 1 ] → K . Since β becomes a vertical homotopy from x to x ′ , we hav e [ x ] v = [ x ′ ] v . Mor eov er, x ′ d 0 h = β d 2 v d 0 h = β d 0 h d 2 v = αd 2 v = y d 1 h , as required.  Remark. N ote that, for any s uch bisimplex x ′ as in the lemma, we have [[ x ′ ]] = [[ x ]] and x ′ d i v = xd i v for i = 0 , 1. Now deﬁne the horizontal comp osition of squares in P P K by (6.1) [[ x ]] ◦ h [[ y ]] = [[ x ′ ] h ◦ h [ y ] h ] if [ x ] v = [ x ′ ] v and x ′ d 0 h = y d 1 h , where [ x ′ ] h ◦ h [ y ] h is the comp osite in the fundamental g r oup oid P K ∗ , 1 , that is, (6.2) [[ x ]] ◦ h [[ y ]] = [[ γ d 1 h ]] for γ : ∆[2 , 1] → K any bisimplex with γ d 2 h = x ′ and γ d 0 h = y . In view of Lemma 6.1, our pr o duct is g iven for all squares [[ x ]] and [[ y ]] with s h [[ x ]] = t h [[ y ]]. W e also have the lemma below. Lemma 6.2. The horizontal c omp osition of squar es in P P K is wel l deﬁne d. Pr o of. W e ﬁrst pr ov e that the square in (6.1) do es no t dep end on the choice of x ′ . T o do s o, suppo se x ′′ : ∆[1 , 1] → K is ano ther bisimplex such that [ x ] v = [ x ′′ ] v and x ′′ d 0 h = y d 1 h , and let β , β ′ : ∆[1 , 2] → K b e vertical homotopies from x to x ′ and fro m x to x ′′ resp ectively . Then, bo th bisimplice s β d 0 h : ∆[0 , 2] → K and β ′ d 0 h : ∆[0 , 2] → K hav e the same vertical faces . Since the 2 nd homotopy groups o f the Kan co mplex K 0 , ∗ v anish, it follows that β d 0 h and β ′ d 0 h are vertically homotopic (F act 2.2). Cho ose ω : ∆[0 , 3 ] → K any vertical homotopy from β d 0 h to β ′ d 0 h , and then let Γ : ∆[1 , 3] → K be a solution to the extensio n problem Λ 1 , 3 [1 , 3] ( ω , − ; xd 0 v s 0 v s 1 v , x s 1 v , β , − ) / /  _   K . ∆[1 , 3] Γ 3 3 Then, the bisimplex e β = Γ d 3 v : ∆[1 , 2 ] → K has vertical faces e β d 0 v = Γ d 3 v d 0 v = Γ d 0 v d 2 v = xd 0 v s 0 v s 1 v d 2 v = xd 0 v s 0 v , e β d 1 v = Γ d 3 v d 1 v = Γ d 1 v d 2 v = xs 1 v d 2 v = x, e β d 2 v = Γ d 3 v d 2 v = Γ d 2 v d 2 v = β d 2 v = x ′ , so tha t e β is another v ertical homo to p y from x to x ′ , and moreover e β d 0 h = Γ d 3 v d 0 h = Γ d 0 h d 3 v = ω d 3 v = β ′ d 0 h , that is, e β and β ′ hav e b o th the same hor izontal 0-face, say α . Now let Φ : ∆[1 , 3] → K and θ : ∆[2 , 2] → K b e solutions to the following extension pro blems Λ 1 , 3 [1 , 3] ( αs 1 v , − ; xd 0 v s 0 v s 1 v , e β , β ′ , − ) / /  _   K ∆[1 , 3] Φ 3 3 Λ 1 , 2 [2 , 2] ( y s 1 v , − , Φ d 3 v ; γ d 0 v s 0 v , γ , − ) / /  _   K ∆[2 , 2] θ 3 3 where γ : ∆[2 , 1] → K is any bisimplex suc h that γ d 2 h = x ′ and γ d 0 h = y . Then, θ is actually a vertical ho mo topy from γ to γ ′ = θd 2 v , and this bisimplex γ ′ satisﬁes that γ ′ d 2 h = θ d 2 v d 2 h = θ d 2 h d 2 v = Φ d 3 v d 2 v = Φ d 2 v d 2 v = β ′ d 2 v = x ′′ , γ ′ d 0 h = θ d 2 v d 0 h = θ d 0 h d 2 v = y s 1 v d 2 v = y . DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 31 Hence, [ x ′ ] h ◦ h [ y ] h = [ γ d 1 h ] h whereas [ x ′′ ] h ◦ h [ y ] h = [ γ ′ d 1 h ] h . Since the bisimplex θ d 1 h : ∆[1 , 2 ] → K is a vertical homo topy from γ d 1 h to γ ′ d 1 h , we conclude that [[ γ d 1 h ]] = [[ γ ′ d 1 h ]], that is, [[ x ′ ] h ◦ h [ y ] h ] = [[ x ′′ ] h ◦ h [ y ] h ], as required. Suppo se now x 0 , x 1 , y : ∆[1 , 1] → K bisimplices with [[ x 0 ]] = [[ x 1 ]] and [ x 0 d 0 h ] v = [ y d 1 h ] v . Then, for some x : ∆[1 , 1] → K , w e hav e [ x 0 ] v = [ x ] v and [ x ] h = [ x 1 ] h . Let x ′ 0 : ∆[1 , 1 ] → K b e any bisimplex with [ x ′ 0 ] v = [ x ] v and x ′ 0 d 0 h = y d 1 h . Since [ x ′ 0 ] v = [ x 0 ] v , w e hav e (6.3) [[ x 0 ]] ◦ h [[ y ]] = [[ x ′ 0 ] h ◦ [ y ] h ] . Letting β : ∆[1 , 2] → K b e a n y vertical homoto p y from x to x ′ 0 and δ : ∆[2 , 1] → K b e any horizontal homotopy from x 1 to x , w e can choo se θ : ∆[2 , 2 ] → K , a bisimplex making commutativ e the diag ram Λ 1 , 2 [2 , 2] ( β d 0 h s 0 h , − , β ; δ d 0 v s 0 v , δ, − ) / /  _   K ∆[2 , 2] θ 4 4 i i i i i i i i i i i i i i i i i i i Then, β 1 = θ d 1 h : ∆[1 , 2 ] → K is a vertical ho motopy fr om x 1 to x ′ 1 := β 1 d 2 v , and since x ′ 1 d 0 h = β 1 d 2 v d 0 h = β 1 d 0 h d 2 v = θd 1 h d 0 h d 2 v = θd 0 h d 0 h d 2 v = β d 0 h d 2 v = β d 2 v d 0 h = x ′ 0 d 0 h = y d 1 h , we have (6.4) [[ x 1 ]] ◦ h [[ y ]] = [[ x ′ 1 ] h ◦ h [ y ] h ] . As θ d 2 v : ∆[2 , 1 ] → K is a horizo n tal homotopy from x ′ 1 to x ′ 0 , we hav e [ x ′ 0 ] h = [ x ′ 1 ] h . Therefore , comparing (6 .3) with (6.4), we obtain the desire d conclusion, that is, [[ x 0 ]] ◦ h [[ y ]] = [[ x 1 ]] ◦ h [[ y ]] . Finally , suppo se x, y 0 , y 1 : ∆[1 , 1 ] → K with [[ y 0 ]] = [[ y 1 ]] and [ xd 0 h ] v = [ y 0 d 1 h ] v . Then, [ y 0 ] v = [ y ] v , [ y ] h = [ y 1 ] h , for so me y : ∆[1 , 1] → K . Le t x ′ : ∆[1 , 1] → K be such that [ x ] v = [ x ′ ] v and x ′ d 0 h = y d 1 h . Since x ′ d 0 h = y 1 d 1 h , w e hav e (6.5) [[ x ]] ◦ h [[ y 1 ]] = [[ x ′ ] h ◦ h [ y 1 ] h ] = [[ x ′ ] h ◦ h [ y ] h ] = [[ γ d 1 h ]] , for γ : ∆[2 , 1] → K a ny bisimplex with γ d 2 h = x ′ and γ d 0 h = y . No w, as [ y 0 ] v = [ y ] v , we ca n select a vertical homotopy δ : ∆[1 , 2 ] → K from y to y 0 , and then a bisimplex β 0 : ∆[1 , 2 ] → K making commutativ e the diagr a m Λ 1 , 2 [1 , 2] ( δd 1 h , − ; x ′ d 0 v s 0 v , x ′ , − ) / /  _   K. ∆[1 , 2] β 0 4 4 i i i i i i i i i i i i i i i i i This bisimplex β 0 bec omes a vertical homotopy from x ′ to x ′ 0 := β 0 d 2 v , and this x ′ 0 veriﬁes that x ′ 0 d 0 h = y 0 d 1 h . Hence, (6.6) [[ x ]] ◦ h [[ y 0 ]] = [[ x ′ 0 ] h ◦ h [ y 0 ] h ] . But, b y taking θ : ∆[2 , 2 ] → K an y bisimplex solving the ex tens ion pro blem Λ 1 , 2 [2 , 2] ( δ, − , β 0 ; γ d 0 v s 0 v , γ , − ) / /  _   K, ∆[2 , 2] θ 4 4 i i i i i i i i i i i i i i i i i 32 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS we obtain a bisimplex γ 0 := θ d 2 v : ∆[2 , 1 ] → K satis fying that γ 0 d 0 h = y 0 and γ 0 d 2 h = x ′ 0 , whence [[ x ]] ◦ h [[ y 0 ]] = [[ γ 0 d 1 h ]] . As the bisimplex θd 1 h : ∆[1 , 2] → K is ea s ily recognized to b e a vertical homotopy from γ d 1 h to γ 0 d 1 h , w e conclude [[ γ d 1 h ]] = [[ γ 0 d 1 h ]]. Cons equent ly , the re quired eq ua lit y [[ x ]] ◦ h [[ y 0 ]] = [[ x ]] ◦ h [[ y 1 ]] follows by co mparing (6.5) with (6.6).  Simply by exchanging the horizo n tal and vertical directions in the for egoing discussio n, we also hav e a w ell-deﬁned vertical comp osition of sq uares [[ x ]] and [[ y ]] in P P K , whenever [ xd 0 v ] h = [ y d 1 v ] h , which is given by (6.7) [[ x ]] ◦ v [[ y ]] = [[ x ′ ] v ◦ v [ y ] v ] if [ x ] h = [ x ′ ] h and x ′ d 0 v = y d 1 v , where [ x ′ ] v ◦ v [ y ] v is the co mpo site in the fundamen tal gr oup oid P K 1 , ∗ , that is, (6.8) [[ x ]] ◦ v [[ y ]] = [[ γ d 1 v ]] for γ : ∆[1 , 2] → K any bisimplex with γ d 2 v = x ′ and γ d 0 v = y . Theorem 6.3. P P K is a double gr oup oid satisfyi ng the ﬁl ling c onditio n. Pr o of. W e ﬁrst obs erve tha t, with b oth deﬁned ho rizontal and vertical comp ositions, the squares in P P K form group oids. The asso ciativity fo r the horizo n tal composition o f squares in P P K follows from the asso c iativity of the comp osition of mor phisms in the fundamental group oid P K ∗ , 1 . In eﬀect, let [[ x ]] , [[ y ]] and [[ z ]] be three hor izontally comp osable squares in P P K . By changing representatives if neces sary , w e can ass ume that xd 0 h = y d 1 h and y d 0 h = z d 1 h . Then, [[ x ]] ◦ h ([[ y ]] ◦ h [[ z ]]) = [[ x ]] ◦ h [[ y ] h ◦ h [ z ] h ] = [[ x ] h ◦ h ([ y ] h ◦ h [ z ] h )] = [([ x ] h ◦ h [ y ] h ) ◦ h [ z ] h ] = [[ x ] h ◦ h [ y ] h ] ◦ h [[ z ]] = ([[ x ]] ◦ h [[ y ]]) ◦ h [[ z ]] . The horizo n tal identit y square on the vertical mo rphism represented by a bis implex u : ∆[0 , 1] → K is (6.9) I h [ u ] v = [[ us 0 h ]] (recall Lemma 2.6), as can b e eas ily deduced from the fact that [ us 0 h ] h is the identit y morphism on u in the g roup oid P K ∗ , 1 . Thus, for example, for an y x : ∆[1 , 1] → K , [[ x ]] ◦ h I h [ xd 0 h ] v = [[ x ] h ◦ h [ xd 0 h s 0 h ] h ] = [[ x ] h ] = [[ x ]] . The horizontal in v erse in P P K of a squa re [[ x ]] is [[ x ]] -1 h = [[ x ] -1 h ], where [ x ] -1 h is the inv erse of [ x ] h in P K ∗ , 1 , as is easy to verify: [[ x ]] ◦ h [[ x ] -1 h ] = [[ x ] h ◦ h [ x ] -1 h ] = [[ xd 1 h s 0 h ]] = I h [ xd 1 h ] v . Similarly , we see that the as so ciativity fo r the vertical comp os itio n of squar e s in P P K follo ws from the asso ciativity of the comp osition in the fundamental group oid P K 1 , ∗ ), that the vertical ident it y square on the horiz ont al morphis m repr esented b y a bisimplex f : ∆[1 , 0] → K is I v ([ f ] h ) = [[ f s 0 v ]], and that the vertical in verse in P P K of a square [[ x ]] is [[ x ] − 1 v ], whe r e [ x ] -1 v denotes the inv erse of [ x ] v in P K 1 , ∗ . W e are now rea dy to prove that P P K is actually a double gro upo id. Axiom 1 is eas ily veriﬁed. Th us, for example, given a ny x : ∆[1 , 1] → K , s h s v [[ x ]] = s h [ xd 0 v ] h = xd 0 v d 0 h = xd 0 h d 0 v = s v [ xd 0 h ] v = s v s h [[ x ]] , DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 33 or, given any f : ∆[1 , 0] → K , s h I v [ f ] h = s h [[ f s 0 v ]] = [ f s 0 v d 0 h ] v = [ f d 0 h s 0 v ] v = I v f d 0 h = I v s h [ f ] h , and so on. Also, for any a : ∆[0 , 0] → K , I h I v a = I h [ as 0 v ] v = [[ as 0 v s 0 h ]] = [[ as 0 h s 0 v ]] = I v [ as 0 h ] h = I v I h a. F or Axiom 2 (i), let [[ x ]] and [[ y ]] b e t wo horizo nt ally comp osa ble square s in P P K . W e can assume that xd 0 h = y d 1 h , and then [[ x ]] ◦ h [[ y ]] = [[ γ d 1 h ]], for any γ : ∆[2 , 1 ] → K with γ d 2 h = x and γ d 0 h = y . Hence, s v ([[ x ]] ◦ h [[ y ]]) = [ γ d 1 h d 0 v ] h = [ γ d 0 v d 1 h ] = [ γ d 0 v d 2 h ] h ◦ h [ γ d 0 v d 0 h ] h = [ γ d 2 h d 0 v ] h ◦ h [ γ d 0 h d 0 v ] = [ xd 0 v ] h ◦ h [ y d 0 v ] h = s v [[ x ]] ◦ h s v [[ y ]] , t v ([[ x ]] ◦ h [[ y ]]) = [ γ d 1 h d 1 v ] h = [ γ d 1 v d 1 h ] h = [ γ d 1 v d 2 h ] h ◦ h [ γ d 1 v d 0 h ] h = [ γ d 2 h d 1 v ] h ◦ h [ γ d 0 h d 1 v ] h = [ xd 1 v ] h ◦ h [ y d 1 v ] h = t v [[ x ]] ◦ h t v [[ y ]] . Axiom 2 (ii) is prov ed analogo usly , and fo r (iii), let f , f ′ : ∆[1 , 0 ] → K b e maps with f d 0 h = f ′ d 1 h . Then, [ f ] h ◦ h [ f ′ ] h = [ γ d 1 h ] h , for γ : ∆[2 , 0] → K any bisimplex with γ d 2 h = f and γ d 0 h = f ′ , and we hav e the equalities: I v ([ f ] h ◦ h [ f ′ ] h ) = I v [ γ d 1 h ] h = [[ γ d 1 h s 0 v ]] = [[ γ s 0 v d 1 h ]] = [[ γ s 0 v d 2 h ]] ◦ h [[ γ s 0 v d 0 h ]] = I v [ f ] h ◦ h I v [ f ′ ] h . And similarly one sees that I h ([ u ] v ◦ v [ u ′ ] v ) = I h [ u ] v ◦ v I h [ u ′ ] v for any u, u ′ : ∆[0 , 1 ] → K with ud 0 v = u ′ d 1 v . T o v erify Axio m 3 , that is, to prov e that the interc hange law holds in P P K , let · [[ x ]] · o o [[ x ′ ]] · o o · O O [[ y ]] · o o O O [[ y ′ ]] · O O o o · O O · o o O O · o o O O be squares in P P K . By an itera ted use of Lemma 6.1 (a nd its co rresp onding version for vertical direction), we can assume that xd 0 h = x ′ d 1 h , xd 0 v = y d 1 v , x ′ d 0 v = y ′ d 1 v and y d 0 h = y ′ d 1 h . Let α : ∆[2 , 1] → K and β : ∆[1 , 2] → K b e bisimplicia l maps such that αd 2 h = y , αd 0 h = y ′ , β d 2 v = x ′ and β d 0 v = y ′ ; therefore, [[ y ]] ◦ h [[ y ′ ]] = [[ αd 1 h ]] and [[ x ′ ]] ◦ v [[ y ′ ]] = [[ β d 1 v ]]. Now we select bisimplices γ : ∆[1 , 2] → K and δ : ∆[2 , 1] → K as r esp ective s o lutions to the following extension problems: Λ 1 , 1 [1 , 2]  _   ( β d 1 h , − ; y , − ,x ) / / K ∆[1 , 2] γ 6 6 Λ 1 , 1 [2 , 1]  _   ( x ′ , − ,x ; αd 1 v , − ) / / K ∆[2 , 1] δ 6 6 Then [[ x ]] ◦ v [[ y ]] = [[ γ d 1 v ]] , [[ x ]] ◦ h [[ x ′ ]] = [[ δ d 1 h ]] and, moreov er, we can ﬁnd a bisimplex θ : ∆[2 , 2] → K ma king the triangle below co mm utative. Λ 1 , 1 [2 , 2]  _   ( β , − ,γ ; α, − , δ ) / / K ∆[2 , 2] θ 6 6 34 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS Letting φ = θ d 1 h : ∆[1 , 2 ] → K and ψ = θ d 1 v : ∆[2 , 1 ] → K , we have the equalities: φd 2 v = θ d 2 v d 1 h = δ d 1 h , φd 0 v = θ d 0 v d 1 h = αd 1 h , ψ d 2 h = θd 2 h d 1 v = γ d 1 v , ψ d 0 h = θd 0 h d 1 v = β d 1 v , whence, ([[ x ]] ◦ h [[ x ′ ]]) ◦ v ([[ y ]] ◦ h [[ y ′ ]]) = [[ δ d 1 h ]] ◦ v [[ αd 1 h ]] = [[ φd 1 v ]] , ([[ x ]] ◦ v [[ y ]]) ◦ h ([[ x ′ ]] ◦ h [[ y ′ ]]) = [[ γ d 1 v ]] ◦ h [[ β d 1 v ]] = [[ ψ d 1 h ]] . Since φd 1 v = θd 1 h d 1 v = θd 1 v d 1 h = ψ d 1 h , the interchange law follows. Thu s, P P K is a double group o id and, moreov er, it satisﬁes the ﬁlling c ondition: given mor- phisms · · [ g ] h o o · [ u ] v O O represented b y bisimplices u : ∆[0 , 1] → K and g : ∆[1 , 0] → K with g d 0 h = ud 1 v , if x : ∆[1 , 1 ] → K is any solution to the extensio n problem Λ 0 , 1 [1 , 1]  _   ( − , g ; u, − ) / / K ∆[1 , 1] x 5 5 then the bihomoto p y class of x is a s quare in P P K , · [[ x ]] · [ g ] h o o · O O · o o [ u ] v O O , as re q uired.  The construction of the double gr oup oid P P K is clearly functorial on K , and we hav e the following: Theorem 6.4. The double nerve c onstruction, G 7→ N N G , emb e ds, as a r eﬂexive su b c ate gory, the c ate gory of double gr oup oids satisfying the ﬁl ling c ondition into the c ate gory of those bisimplicia l sets K that satisfy t he extension c ondition and su ch that π 2 ( K ∗ , 0 , a ) = 0 = π 2 ( K 0 , ∗ , a ) for al l vertic es a ∈ K 0 , 0 . The r eﬂe ctor functor for su ch bisimplicial sets is given by the ab ove describ e d homotopy double gr oup oid c onstruction K 7→ P P K . Thus, P P N N G = G , and ther e ar e natu r al bisimp licial maps (6.10) ǫ ( K ) : K → N NP P K, such that P P ǫ = id and ǫ N N = id . Pr o of. F rom Theorem 5 .1(ii), if G is any double group oid satisfying the ﬁlling co ndition, then its double nerve N N G satisﬁes the ex tens ion conditio n and, since b oth simplicial sets N N G ∗ , 0 and N N G 0 , ∗ are nerves o f gro upo ids , all homotopy gr oups π 2 (N N G ∗ , 0 , a ) and π 2 (N N G 0 , ∗ , a ) v anish. Moreov er, since the bihomotopy r elation is trivial on the bisimplices ∆[ p, q ] → N N G , for p ≥ 1 or q ≥ 1, it is e a sy to see that P PN N G = G . F or any bisimplicial set K in the hypo thesis o f the theorem, there is a natur al bisimplicia l map ǫ = ǫ ( K ) : K → N NP P K , DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 35 that takes a bisimplex x : ∆[ p, q ] → K , o f K , to the bisimplex ǫx : [ p ] ⊗ [ q ] → P P K , o f N NP P K , deﬁned b y the p × q conﬁguratio n of squa res in P P K ǫ r i x ǫ r,s i,j x ǫ r j ǫ r i,j x o o 0 ≤ i ≤ j ≤ p 0 ≤ r ≤ s ≤ q , ǫ s i x ǫ r,s i x O O ǫ s j x ǫ s i,j x o o ǫ r,s j x O O where ǫ r,s i,j x = [[ xd p h · · · d j +1 h d j − 1 h · · · d i +1 h d i − 1 h · · · d 0 h d q v · · · d s +1 v d s − 1 v · · · d r +1 v d r − 1 v · · · d 0 v ]] , ǫ r,s j x = [ xd p h · · · d j +1 h d j − 1 h · · · d 0 h d q v · · · d s +1 v d s − 1 v · · · d r +1 v d r − 1 v · · · d 0 v ] v , ǫ r i,j x = [ xd p h · · · d j +1 h d j − 1 h · · · d i +1 h d i − 1 h · · · d 0 h d q v · · · d r +1 v d r − 1 v · · · d 0 v ] h , ǫ r i x = xd p h · · · d i +1 h d i − 1 h · · · d 0 h d q v · · · d r +1 v d r − 1 v · · · d 0 v . Since a straightforw ard veriﬁcation sho ws that P P ǫ ( K ) is the iden tit y map on P P K , for an y K , and ǫ (N N G ) is the identit y map on N N G , for any double gr oupo id G , it follows that N N is right a djoin t to P P, with ǫ and the iden tit y b eing the unit a nd the counit of the adjunction resp ectively .  With the next theor em we show that the do uble group oid P P K r epresents the same homo to p y 2-type as the bisimplicial set K . Theorem 6. 5. L et K b e any bisimplicial set satisfying the extens ion c ondition and such that π 2 ( K 0 , ∗ , a ) = 0 = π 2 ( K ∗ , 0 , a ) for al l b ase vert ic es a . Then, the induc e d map by unit of the adjunction | ǫ | : | K | → | N NP P K | = | P P K | is a we ak homotopy 2 -e quivalenc e. Pr o of. By F acts 2.8 (1) a nd (3 ) a nd Theor em 5.1 , the map | ǫ | : | K | → | N NP P K | is, up to na tur al homotopy equiv ale nces, induced by the simplicial ma p W ǫ : W K → W N NP P K , where b oth W K and WN NP P K ar e Ka n-complexes. A t dimension 0, we have the equalities W K 0 = K 0 , 0 = WN NP P K 0 , and the map W ǫ is the ident it y on 0- simplices. At dimensio n 1 , the map W ǫ : ( x 0 , 1 , x 1 , 0 ) 7→ ([ x 0 , 1 ] v , [ x 1 , 0 ] h ) , is clea rly sur jective, whence we c o nclude that the induced π 0 W ǫ : π 0 W K → π 0 WN NP P K ( 5.2 ) ∼ = π 0 P P K is a bijection and also that, for any vertex a ∈ K 0 , 0 , that induced on the π 1 -groups π 1 W ǫ : π 1 (W K , a ) → π 1 (WN N P P K , a ) ( 5.2 ) ∼ = π 1 (P P K , a ) is sur jective. T o see that π 1 W ǫ is actually an iso mo rphism, suppos e that ( x 0 , 1 , x 1 , 0 ) ∈ W K 1 , with x 0 , 1 d 1 v = a = x 1 , 0 d 0 h , re presents an ele ment in the kernel o f π 1 W ǫ . This implies the existence of a bisimplex x : ∆[1 , 1 ] → K whose bihomo topy class is a square in P P K with bo undary as in a [[ x ]] a [ as 0 h ] h · [ x 0 , 1 ] v O O a [ x 1 , 0 ] h o o [ as 0 v ] v 36 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS Using Lemma 6.1 t wic e (o ne in eac h direction), we can ﬁnd a bisimplex x 1 , 1 : ∆[1 , 1] → K , such that [[ x 1 , 1 ]] = [[ x ]], x 1 , 1 d 1 v = as 0 h , a nd x 1 , 1 d 0 h = a s 0 v . Moreover, s ince [ x 1 , 1 d 0 v ] h = [ x 1 , 0 ] h and [ x 1 , 1 d 1 h ] v = [ x 0 , 1 ] v , there are bisimplices x 2 , 0 : ∆[2 , 0 ] → K and x 0 , 2 : ∆[0 , 2 ] → K , with faces a s in the pictur e a a as 0 v o o x 1 , 1 a as 0 h o o · x 0 , 2 b b E E E E E E E E x 0 , 1 O O a o o as 0 v O O a b b E E E E E E E E x 2 , 0 x 1 , 0 as 0 h O O This amounts to saying that the tr iplet ( x 0 , 2 , x 1 , 1 , x 2 , 0 ) is a 2-simplex o f W K which is a ho- motopy from ( x 0 , 1 , x 1 , 0 ) to ( as 0 v , as 0 v ). Then, ( x 0 , 1 , x 1 , 0 ) represents the identit y element of the group π 1 (W K, a ). This proves that π 1 W ǫ is a n isomorphism. Let us now analyze the homomorphism π 2 W ǫ : π 2 (W K , a ) → π 2 (WN NP P K , a ) ( 5.2 ) ∼ = π 2 (P P K , a ) . An ele ment of π 2 (P P K , a ) is a square in P P K of the form a [[ x ]] a [ as 0 h ] h o o a [ as 0 v ] v O O a [ as 0 h ] h o o [ as 0 v ] v O O and the homomor phism π 2 W ǫ is induce d by the mapping a a as 0 v o o x 1 , 1 a as 0 h o o a x 0 , 2 b b E E E E E E E as 0 v O O a o o as 0 v O O a b b E E E E E E E x 2 , 0 as 0 h as 0 h O O 7− → [[ x 1 , 1 ]] . That π 2 W ǫ is surjective is prov en using a par allel arg umen t to that given previously for proving that π 1 W ǫ is injective (given [[ x ]], using Lemma 6.1 twice, we can ﬁnd x 1 , 1 : ∆[1 , 1] → K , e tc.). T o prov e that π 2 W ǫ is also injective, supp ose ( x 0 , 2 , x 1 , 1 , x 2 , 0 ) as ab ov e, r epresenting an element of π 2 (W K, a ) in to the k ernel of π 2 W ǫ , that is, suc h that [[ x 1 , 1 ]] = [[ as 0 h s 0 v ]]. Then, there is a bisimplex y : ∆[1 , 1] → K such that [ x 1 , 1 ] v = [ y ] v and [ y ] h = [ as 0 h s 0 v ] h , whence we can ﬁnd bisimplices α ′ : ∆[1 , 2 ] → K and β ′ : ∆[2 , 1 ] → K suc h that α ′ d 0 v = y d 0 v s 0 v , α ′ d 1 v = y , α ′ d 2 v = x 1 , 1 , β ′ d 0 h = as 0 h s 0 v , β ′ d 1 h = as 0 h s 0 v , β ′ d 2 h = y . Let us now choose θ : ∆[2 , 2] → K and θ ′ : ∆[1 , 3 ] → K as resp ective solutions to the fo llowing extension problems Λ 2 , 0 [2 , 2]  _   ( as 0 h s 0 v s 0 v , as 0 h s 0 v s 0 v , − ; − ,β ′ d 1 v s 0 v , β ′ ) / / K ∆[2 , 2] θ 2 2 ∆[1] ⊗ Λ 2 [3]  _   ( θ d 2 h d 0 v s 0 v , θ d 2 h , − ,α ′ ) / / K ∆[1 , 3] θ ′ 3 3 DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 37 Then, for α = θ ′ d 2 v : ∆[1 , 2] → K and β = θd 0 v : ∆[2 , 1 ] → K , we have the equalities (6.11) αd 0 v = β d 2 h , αd 1 v = as 0 h s 0 v , αd 2 v = x 1 , 1 , β d 0 h = as 0 h s 0 v , β d 1 h = as 0 h s 0 v . By Lemma 2 .2, as the 2 nd homotopy groups of K 0 , ∗ v anish and both bisimplices αd 0 h and as 0 v s 0 v hav e the sa me vertical faces, there is a vertical homotopy ω : ∆[0 , 3] → K from a s 0 v s 0 v to αd 0 h . And similarly , since β d 1 v and x 2 , 0 hav e the sa me hor izontal faces and the 2 nd homotopy groups of K ∗ , 0 are all trivial, there is a hor izontal homotopy , sa y ω ′ : ∆[3 , 0] → K , from β d 1 v to x 2 , 0 . Now, let Γ : ∆[1 , 3] → K and Γ ′ : ∆[3 , 1] → K be bisimplices solving, resp ectively , the extension problems Λ 1 , 2 [1 , 3]  _   ( ω , − ; αd 0 v s 1 v , as 0 v s 0 v s 0 h , − , α ) / / K ∆[1 , 3] Γ 3 3 Λ 3 , 0 [3 , 1]  _   ( as 0 v s 0 h s 0 h , as 0 v s 0 h s 0 h , β , − ; − , ω ′ ) / / K ∆[3 , 1] Γ ′ 2 2 and take x 1 , 2 = Γ d 2 v : ∆[1 , 2] → K and x 2 , 1 = Γ ′ d 3 h : ∆[2 , 1] → K . Then, the same equalities as in (6.11) hold for x 1 , 2 instead of α and x 2 , 1 instead of β , and moreov e r x 1 , 2 d 0 h = as 0 v s 0 v and x 2 , 1 d 1 v = x 2 , 0 . Finally , by ta king x 0 , 3 : ∆[0 , 3] → K any bisimplex with x 0 , 3 d 0 v = x 1 , 2 d 1 h , x 0 , 3 d 1 v = as 0 v s 0 v , x 0 , 3 d 2 v = as 0 v s 0 v and x 0 , 3 d 3 v = x 0 , 2 , and x 3 , 0 : ∆[3 , 0 ] → K any horizo ntal homo top y fro m as 0 h s 0 h to x 2 , 1 d 0 v (whic h exis t thanks to Lemma 2.2), we hav e the 3- s implex ( x 0 , 3 , x 1 , 2 , x 2 , 1 , x 3 , 0 ) of W K , which is easily recog nized as a homotopy fr o m ( as 0 v s 0 v , as 0 h s 0 v , as 0 h s 0 h ) to ( x 0 , 2 , x 1 , 1 , x 2 , 0 ). Consequently , ( x 0 , 2 , x 1 , 1 , x 2 , 0 ) repr esents the iden tit y of the gro up π 2 (W K , a ). Therefore, π 2 W ǫ is an isomorphis m, and the pro o f is complete  7. The equiv alence of homotopy ca tegories Recall that the ca tegory of weak homotopy t y p es is deﬁned to b e the lo caliza tion of the category of top ologica l spaces with resp ect to the class of weak equiv alences, and the c ate gory of homotop y 2 -typ es , her eafter denoted b y Ho( 2 - t yp es ), is its full subca tegory given by thos e spaces X with π i ( X, a ) = 0 for any integer i > 2 a nd any base p oint a . W e now deﬁne the homotopy c ate gory of double gr oup oids satisfying the ﬁl ling c ondition , denoted by Ho ( DG fc ), to b e the lo calization o f the catego ry DG fc , of these double group oids, with r esp ect to the class o f weak equiv alence s , as deﬁned in Subsec tio n 3 .4. By Corollar ies 5.4 a nd 4.3, both the geometric realiza tio n functor G 7→ |G | and the homotopy double g r oup oid funtor X 7→ Π Π X induce equally denoted functors (7.1) | | : Ho( DG fc ) → Ho( 2 - t ypes ) , (7.2) Π Π : Ho( 2 - t yp es ) → Ho ( DG fc ) . One o f the main goa ls in this section is to pr ov e the following: Theorem 7.1. Both induc e d functors ( 7.1 ) and ( 7.2 ) ar e mutu al ly quasi-inverse, est ablishing an e quivalenc e of c ate gories Ho( DG fc ) ≃ Ho ( 2 - t yp es ) . The pro of of this Theo rem 7.1 is s o mewhat indirec t. Previo usly , we s hall establish the following re sult, wher e KC is the ca tegory of Kan complexes and Ho( L ∈ KC | π i L = 0 , i > 2) is the full sub categor y of the homotopy ca tegory of K an complexes g iven by those L such that π i ( L, a ) = 0 for all i > 2 and base vertex a ∈ L 0 : 38 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS Theorem 7 .2. Th er e ar e adjoint funct ors, WN N : DG fc → K C , the right adjoi nt, and P P D ec : K C → DG fc , t he left adjoint, that induc e an e quivalenc e of c ate gories Ho( DG fc ) ≃ Ho ( L ∈ KC | π i L = 0 , i > 2) . Pr o of. The pair of adjoint functors P P Dec ⊣ WN N is obtaine d by compositio n of the pair of adjoint functor s Dec ⊣ W, reca lle d in (2.2), with the pair of adjoint functors P P ⊣ N N, stated in Theorem 6.4. F or any double gr oup oid G ∈ DG fc , its do uble nerve N N G satisﬁes the extension condition, b y Theo rem 5.1, and ther efore, b y F act 2.8 (4), the simplicial se t WN N G is a Kan complex. Conversely , if L is any Kan co mplex, then the bisimplicial set Dec L sa tisﬁes the extension condition by F act 2.8 (5) and, mo reov er, π 2 ( Dec L ∗ , 0 , a ) = 0 = π 2 ( Dec L 0 , ∗ , a ) for all vertices a , since b oth augmented simplicial sets Dec L ∗ , 0 d 0 → L 0 and Dec L 0 , ∗ d 1 → L 0 hav e simplicial con tractions, given resp ectively by the families of degeneracies ( s p : L p → L p +1 ) p ≥ 0 and ( s 0 : L q → L q +1 ) q ≥ 0 . Therefor e, in acco rdance with Theorem 6.4, the co mpo s ite functor L 7→ P P Dec L is well deﬁned on K a n complexe s . By F act 2.7 (3), the homotopy equiv alences in F act 2.8 (1), and Cor ollary 5.4, it follows that a double functor F : G → G ′ , in DG fc , is a weak equiv alence if and only if the induce d simplicial map WN N F : WN N G → W N N G ′ is a homotopy equiv a lence. By F acts 2 .7 (3) and 2 .8 (2), Theorem 6.5, and Co rollary 5.4, if f : L → L ′ is any simplicial map betw een Kan complexes L, L ′ such that π i ( L, a ) = 0 = π i ( L ′ , a ′ ) for all i ≥ 3 and ba se vertices a ∈ L 0 , a ′ ∈ L ′ 0 , then f is a homoto p y equiv alence if and only if the induced P P Dec f : P P Dec L → P P Dec L ′ is a weak equiv ale nc e o f double group oids. If L is any K an complex such that π i ( L, a ) = 0 for all i ≥ 3 and all ba se v ertices a ∈ L 0 , then the unit of the adjunction L → W N NP P D ec L is a homotopy equiv alence since it is the comp osition of the simplicial maps L u / / W Dec L W ǫ ( Dec L ) / / WN NP P Dec L, where u is a homotopy equiv alence by F act 2.8(3) and F act 2.7 (3), a nd then W ǫ ( Dec L ) is also a ho mo topy equiv a lence by Theo rem 6.5 and F act 2 .7 (3). Finally , the c o unit P Pv(N N G ) : P PDec WN N G → P PN N G = G , at a ny double gr oup oid G , is a weak equiv alence, thanks to F act 2 .8(3), Theore m 6.5, and Coro llary 5.4. This ma kes the pr o of complete.  Since, by F acts 2.7, the adjoint pair of functors | | ⊣ S : T op ⇆ KC induces mutually quasi-inv erse equiv alences of c a tegories Ho( 2 - t yp es ) ≃ Ho( L ∈ KC | π i L = 0 , i > 2) , the fo llowing follows from Theorem 7.2 ab ov e, and F act 2.8(1): Theorem 7.3. The induc e d functor ( 7.1 ) , | | : Ho( DG fc ) → Ho( 2 - t yp es ) , is an e qu ivalenc e of c ate gories with a quasi-inverse t he induc e d by t he functor X 7→ P P Dec S X . Theorem 7.3 gives ha lf of Theorem 7 .1. The remaining pa rt, that is, that the induced functor (7.2) is a quasi-inverse equiv alence of (7 .1), follows from the prop osition b elow. Prop ositio n 7.4 . The two induc e d functors Π Π , P PDec S : Ho( 2 - ty p es ) → Ho( DG fc ) ar e natu - r al ly e quivalent. Pr o of. The pro of cons is ts in displaying a natural double functor η : P PDecS X → Π Π X, which is a weak equiv alence for any top ologica l space X . This is as follows: DOUBLE GROUPOIDS AND HOMOTOPY 2-TYPES 39 On ob jects of P PDecS X , the double functor η carries a contin uo us map u : ∆ 1 → X to the path η u : I → X g iven by η u ( x ) = u (1 − x, x ). On horizontal mo rphisms of P P DecS X , η a cts by ( g d 0 [ g ] h − → g d 1 ) η 7→ ( η gd 0 → η gd 1 ) , the unique ho rizontal mor phism in Π Π X from the path η gd 0 to the pa th η gd 1 , for any con tin uous map g : ∆ 2 → X . This cor resp ondence is well deﬁned since η gd 0 (1) = g d 0 (0 , 1) = g (0 , 0 , 1) = g d 1 (0 , 1) = η gd 1 (1), and, moreover, if [ g ] h = [ g ′ ] h in DecS X , then g d i = g ′ d i for i = 0 , 1. And, similarly , on vertical mor phisms, η is g iven by ( g d 1 [ g ] v − → g d 2 ) η 7→ ( η gd 1 → η gd 2 ) . On squares in P PDecS X , η is deﬁned by · [[ α ]] · [ αd 3 ] h o o η αd 1 d 2 η αd 0 d 2 o o · [ αd 1 ] v O O · [ αd 2 ] h o o [ αd 0 ] v O O η 7→ η αd 1 d 1 O O [ η α ] η αd 0 d 1 o o O O where, fo r any contin uous map α : ∆ 3 → X , the map η α : I × I → X is g iven by the formula η α ( x, y ) = α ( xy , (1 − x )(1 − y ) , (1 − x ) y , x (1 − y )) . T o see that η is well deﬁned o n squares in P PDecS X , s uppo s e [[ α 1 ]] = [[ α 2 ]]. This means that [ α 1 ] h = [ α ] h and [ α 2 ] v = [ α ] v , for some α : ∆ 3 → X , in the bisimplicial set Dec S X . Then, there a re maps β , γ : ∆ 4 → X such that the following equalities hold: β d 0 = α 1 d 0 s 0 , β d 1 = α 1 , β d 2 = α = γ d 3 , γ d 4 = α 2 , γ d 2 = α 2 d 2 s 2 ; whence the equalities of s q uares in Π Π X , [ η α 1 ] = [ η α ] = [ η α 2 ], follow fro m the homotopies F 1 , F 2 : I 2 × I → X , resp ectively deﬁned b y the formulas F 1 ( x, y , t ) = β ( xy , t (1 − x )(1 − y ) , (1 − t )(1 − x )(1 − y ) , (1 − x ) y , x (1 − y )) , F 2 ( x, y , t ) = γ ( xy , (1 − x )(1 − y ) , (1 − x ) y , tx (1 − y ) , (1 − t ) x (1 − y )) . Most of the details to conﬁrm η is actually a double functor are routine and e a sily veriﬁable. W e leave them to the rea der since the only o nes with any diﬃculty are those a) and b) prov en below. a) F o r ω : ∆ 4 → X , [ η ω d 1 ] = [ η ω d 2 ] ◦ h [ η ω d 0 ] and [ η ω d 3 ] = [ η ω d 4 ] ◦ v [ η ω d 2 ]. b) F or g : ∆ 2 → X , [ η gs 2 ] = I v ( η gd 1 , η gd 0 ) a nd [ η gs 0 ] = I h ( η gd 2 , η gd 1 ). How ever, all these equalities in a) and b) hold thanks to the rela tive homotopies H 1 : η ω d 1 → η ω d 2 ◦ h η ω d 0 , H 2 : η ω d 3 → η ω d 4 ◦ v η ω d 2 , H 3 : η gs 2 → e v , H 4 : η gs 0 → e h , which ar e, res pectively , deﬁned by the maps H i : I 2 × I → X such that 40 A. M. CEGARRA, B. A. HEREDIA, AND J. REMEDIOS H 1 ( x, y , t ) =                          ω ((1- t ) xy , 2 tx ( x + y ) , (1- x )(1- y )+ tx (2 x -2+ y ) ,y (1- x )+ tx (1-2 x - y ) ,x (1- y )+ tx (1-2 x - y )) if x + y ≤ 1 , x ≤ y, ω ((1- t ) xy , 2 ty ( x + y ) , (1- x )(1- y )+ ty (2 y -2+ x ) ,y (1- x )+ ty (1- x -2 y ) , x (1- y )+ ty (1- x -2 y )) if x + y ≤ 1 , x ≥ y, ω ( xy + t (1- y )(1- x -2 y ) , 2 t (1- y )(2- x - y ) , (1- t )(1- x )(1- y ) ,y (1- x )- t (1- y ) ( 2- x -2 y ) , (1- y )( x - t (2- x -2 y )) if x + y ≥ 1 , x ≤ y, ω ( xy + t (1- x )(1-2 x - y ) , 2 t (1- x )(2- x - y ) , (1- t )(1- x )(1- y ) , (1- x )( y - t (2-2 x - y )) ,x (1- y )- t (1- x )(2-2 x - y )) if x + y ≥ 1 , x ≥ y, H 2 ( x, y , t ) =                          ω ( t (1- x )(2 x -1- y )+ xy , (1- x )(1- y )+ t (1- x )(2 x -1- y ) , (1- t )(1- x ) y , 2 t (1- x )(1- x + y ) ,x (1- y )+ t (1- x )( y -2 x )) if x + y ≥ 1 , x ≥ y, ω ( xy + ty ( x -2 y ) , (1- x )(1- y )+ ty ( x -2 y ) , (1- t )(1- x ) y , 2 ty (1- x + y ) , x (1- y )+ ty (2 y -1- x )) if x + y ≤ 1 , x ≥ y, ω ( xy + t (1- y )(2 y - x -1) , (1- x )(1- y )+ t (1- y )(2 y -1- x ) , (1- x ) y + t (1- y )( x -2 y ) , 2 t (1- y )(1+ x - y ) , (1- t ) x (1- y )) if x + y ≥ 1 , x ≤ y, ω ( xy + tx ( y -2 x ) , (1- x )(1- y )+ tx ( y -2 x ) ,y (1- x )+ tx (2 x -1- y ) , 2 tx (1+ x - y ) , (1- t ) x (1- y )) if x + y ≤ 1 , x ≤ y, H 3 ( x, y , t ) = ( g ((1- t ) xy , (1- x )(1- y ) − txy ,x + y +2 xy ( t -1)) if x + y ≤ 1 , g ( xy + t ( x + y -1- xy ) , (1- t )(1- x )(1- y ) ,x + y -2 xy +2 t (1- x )(1- y )) if x + y ≥ 1 , H 4 ( x, y , t ) = ( g (1- x - y +2 xy +2 tx (1- y ) , (1- x ) y + tx ( y -1) , (1- t ) x (1- y )) if x ≤ y, g (1- x - y +2 xy +2 ty (1- x ) , (1- t )(1- x ) y ,x (1- y )+ ty ( x -1)) if x ≥ y. This double functor, η : P P Dec S X → Π Π X , which is cle a rly na tural on the top olog ical s pace X , is actually a weak equiv alence since, for any 1-simplex u : ∆ 1 → X and in teger i ≥ 0, the induced map π i η : π i (P P Dec S X , u ) → π i (Π Π X , η u ) o c c urs in this comm utative dia gram π i (P P Dec S X , u ) π i η   π i ( | P P Dec S X | , u ) Th.5.3 ∼ = o o π i ( | Dec S X | , u ) Th.6.5 ∼ = o o F act2.8(1) ∼ =   π i ( | W Dec S X | , u ) F act2.8(3) ∼ =   π i (Π Π X , η u ) Th.4.2 ∼ = / / π i ( X, u (1 , 0)) π i ( | S X | , u (1 , 0)) F act2.7(6) ∼ = o o in which a ll other maps are bijections (g r oup isomorphisms for i ≥ 1) by the references in the lab els.  References [1] N. Andruskiewitsch, S. Natale, T ensor catego ries att ac hed to double groupoids. Adv. Math. 200 (20 06), 539-583. [2] M. Artin, B. Mazur, On the V an Kamp en Theorem, T op ology 5 (1966), 179-189. [3] J.C. Baez, A.D. 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Dep ar t amen to de ´ Algebra, Universidad de G ranada, 18071 Gran a da, S p ain E-mail addr ess : acegar ra@ugr.es E-mail addr ess : fieras h@correo. ugr.es Dep ar t amen to de Ma tem ´ atica Fundament al, Un iv ersidad de La Laguna. 38271 La Laguna, S p ain E-mail addr ess : jremed @ull.es

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