The classifying topos of a topological bicategory
For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB. It is shown that the category of geometric morphisms Hom(S…
Authors: Igor Bakovic, Branislav Jurco
The classif ying top os of a top ological bicategory Igor Bak o vi ´ c Departmen t of Mathematics F acult y of Natural Sciences and Mathematics Univ ersit y of Split T eslina 12/I I I, 21000 Split, Croatia and Branisla v Jurˇ c o Max Planc k Institute for Mathematics Viv atsgasse 7, 531 11 Bonn, German y Abstract F or an y topolog ical bicategory B , the Duskin ner ve N B of B is a s implicial space. W e introduce the cla ssifying topo s B B of B as the Deligne topo s o f sheav es S h ( N B ) on the simplicial space N B . It is shown that the categor y o f geometric morphisms Hom( S h ( X ) , B B ) from the top o s o f sheaves S h ( X ) on a top ologica l space X to the Deligne classifying top os is naturally equiv a lent to the category of principa l B -bundles. As a simple consequence, the geometric rea liz ation | N B | of the nerv e N B of a lo ca lly contractible to p o logical bicategory B is the cla ssifying space of principal B -bundles, giving a v ar iant of the result of Baas, B¨ okstedt and Kro derived in the co nt ext of bicategoric a l K-theor y . W e also define classifying top oi of a top ologica l bicateg ory B using sheav es on other types of nerves of a bicategory given by Lack a nd Paoli, Simpson and T amsamani by means of bisimplicia l spa ces, and we examine their prop er ties . 1 This research wa s in part supp orted by the Croatian Ministry of Science, Education and Sp ort, Pro ject No. 098-0982930-299 0. 2 The first auth or ac knowledges supp ort from Europ ean Commission under the MR TN-CT-2006-035505 HEPTOOLS Marie Curie R esearch T raining Netw ork 3 Authors email addresses: ibak ovic@gmail .com, branislav.jurco@googlemail .com 1 In tro duction In a recen t pap er by Baas, B¨ okstedt and K r o [1 ] it wa s sho wn that the geometric realization | N B | of the Duskin nerve N B [11] of a goo d top ologica l bicatego ry B is the classifying space of charted B -bund les. The bicategory is calle d goo d if its Duskin n erv e N B is a go o d s implicial space, i.e. all d egeneracy maps are closed cofibr ations. Sp ecial cases of top ological 2- groups and Lie 2-g roup s we re discussed in [2] and in [15], resp ectiv ely . The result of [1] generaliz es the we ll known fact that the geometric realiza tion | N C | of th e nerve N C of a lo cally con tractible to p ologic al category C is the classifying space of principal C -bu ndles (on CW complexes). This is very nicely describ ed by Mo erd ijk in [21]. There, also the classifying top os B C of a top ological category C is describ ed as the Deligne top os of shea ves S h ( N C ) on th e nerve N C and it is sho w n that the cate gory of geometric morphisms Ho m ( S h ( X ) , B C ) from the top os of shea ve s S h ( X ) on a top ological space X to the D eligne top os is n aturally equiv alent to the catego ry of pr incipal C -bu ndles. As a simple co nsequen ce, it is sh o wn that the geometric realizatio n | N C | of the nerve N C of a lo cally co ntrac tible top ological categ ory C is the classifying space of principal C -bu ndles. One purp ose of this note is to intro d uce the classifying top os B B of a top ological bicategory B as the top os of shea v es S h ( N B ) on the Duskin nerve N B o f the bicategory B , whic h is a simplicial space. The category of geometric m orphisms Hom( S h ( X ) , B B ) from the top os of sh ea ves S h ( X ) on a top ological sp ace X to the classifying top os is naturally equiv alen t to the catego ry of (suitably defined) principal B -bun dles. As a simple consequence, the geometric r ealizat ion | N B | of th e n erv e N B of a lo cally contract ible top ological bicategory B is the classifying space of principal B -bundles. Hence, we h a v e a v arian t of the r esult of Baas, B¨ okstedt and Kr o. Another p u rp ose of this note is to define classifying top oi of a top ologica l b icatego ry B using shea v es on other t yp es of nerves of the bicategory B , th e nerv es acc ording to Lac k & P aoli [17] (or Simpson [23] and T amsamani [2 5]), whic h can b e view ed as bisimplicial spaces. Again, the category of top os morp hisms f rom the top os of sh ea ves S h ( X ) on a top ological space X to the corresp onding classifying top os is naturally equiv alen t to the resp ectiv e category of (suitably defined) prin cipal B -bund les. As a simple consequence, the geometric realization of an y of these nerv es of a lo cally contrac tible top ological bicategory B is the classifying space of the resp ectiv e principal B -bund les. In Section 2, w e recall some prerequisites from [21] rega rdin g shea ves on a simplicial space and augmente d linear orders ov er to p ologi cal spaces. In Section 3, w e recall, again from [21], th e kn own facts ab out classifying spaces and top oi of top ological categories (and the corresp onding pr in cipal bund les). W e d escrib e a ge neralization to the ca se of bicate- gories, b ased on the Duskin nerve , in Section 4. F urther preliminaries n eeded for the subse- quen t discussion of alternativ e definitions of classifying spaces and top oi of bicatego ries are giv en in S ection 5. Finally , in section 6, we describ e a mo d ification of th e classifying top os of a top ologic al bicate gory (and the corresp onding pr incipal bu ndles) b ased on alternativ e definitions o f the nerves according to L ack & Paoli, S impson and T amsamani. 2 This article is meant to b e the fir s t one in the sequel within a program, initiated by the authors, of classifying top oi of h igher order structures in top ology . It is a v ast generalizatio n of the program initiated b y Mo erdij k in [21] on the relation b et w een classifying sp aces and classifying top oi. Mo erdijk’s lect ure notes arose out of an imp ortan t question: What do es the classifying space of a sm all category classify? In the article titled by the same question [29], W eiss pro v ed the classifying pr op ert y of the classifying sp ace for sligh tly differen t geometric ob j ects then those of Moer d ijk, sh o wing that the answer may not b e un ique. Therefore, th is article ma y be seen as an (one p ossible) answ er to the follo wing ques- tion: What do es the classifying space of a top ological bicategory classify? Bicategories are the w eak est p ossible generalizatio n of ordinary catego ries to the immediate next le ve l of dimension. Lik e categories, bicategories do ha ve a genuine simplicial set asso ciated with them, their Dusk in nerve [11]. Based on un publish ed work of Rob erts on the charact eriza- tion of the nerv e of a strict n -categ ory , Street p ostulated in [24] an equiv alence b et w een the catego ry of strict ω -c ate gories and a cat egory of certain t yp es of simplicial sets whic h are called c omplicial sets . The Str e et-R ob erts c onje ctur e w as pro v ed b y V erit y in [26], and in his subsequent pap ers [27] and [28] he ga v e a charac terization of we ak ω -c ate gories . Under th is c haracterizati on, one should b e able to captur e classifying sp aces and top oi of bicategories and other higher dimensional categ ories, at least in so f ar as th ese concepts ha ve found satisfactory defin itions. F ollo wing suc h r easoning, w e ma y define the classifying space of a w eak ω -category as a geometric realization of the complicial set w h ic h is its nerve, and the classifying top os of a we ak ω -catego ry as a top os of shea v es on that complicial set. It w ould b e in terested to compare this app roac h to classifying spaces of wea k ω - catego ries with classifying sp aces of crossed complexes defin ed b y Brown and Higgi ns in [8], since there is a w ell kno wn equiv alence b et w een strict ω -gr oup oids and cr osse d c omplexes pro v ed in [7] by the same authors. In particular, it wo uld b e in teresting to see wh ether the metho ds w e dev elop ed w ould allo w to define a classifying sp ace of a weak ω -category b y taking a fundamenta l crossed complex of its coheren t sim p licial n erv e. Ho wev er, this article is not so cosmological in its scop e, and its main contribution is to put together some esta blished results on classifying spaces and cla ssifying topoi in a new w a y , with consequences for the theory of bicateg ories. Since w e are follo wing Mo erd ijk’s approac h to cla ssifying sp aces and classifying top oi, we will omit all pro ofs, whic h can b e found in Moerd ijk’s lecture notes. Ac knowledgmen ts W e w ould lik e to thank Ronnie Bro wn for useful comments on the relev ance of this work to general notions of classifying spaces of crossed complexes. W e w ould also lik e to express our thanks to the referee of ”Homol ogy , Homotop y and Applications” whose careful reading resulted in comment s and suggestions whic h ha v e impro v ed the stru cture and the con tent of this article. The first au th or w ould lik e to thank for a hospitalit y of Max Planc k Institute for Physics in Mun ic h, w h ere the p art of this researc h w as made, and to Max Planc k In stitute for Mat hematics in Bonn, where he w as su pp orted by the Croatian-German bilateral DAA D p rogram ”Homolog ical algebra in geometry a nd ph ysics”. 3 2 Simplicial spaces and linear orders o v er top ological spaces In this sectio n, w e recall some prerequisites r egarding sh ea ves on a simplicial space and augmen ted linear ord ers o v er topological space s. Almost all the definitions and theorems are take n v erbatim from [21], where pro ofs of all statemen ts of this section ca n b e found . 2.1 T opological spaces Let us recall that a closed set in a (topological) space X is irr e ducible if it can not b e written as a un ion of tw o smaller closed sets. The space X is sob er if ev ery irreducible set is the closure { x } of the one p oin t set { x } of a unique x ∈ X . Ev ery Hausdorff sp ace is sob er. In this note all spaces will b e sob er by assumption. A s pace X is lo c al ly e quic onne cte d (LEC) if the diagonal map X → X × X is a closed cofibration. F or example, CW-complexes are LEC. A space X is lo c al ly c ontr actible if it has a basis of con tractible sets. Examples of lo - cally contrac tible spaces are lo cally equiconnected spaces and in p articular CW complexes. F or a lo cally con tractible s p ace the ´ etale homotop y groups π n ( S h ( X ) , x 0 ) are naturally isomorphic to the ordin ary homotopy groups π n ( X, x 0 ) f or eac h n . 2.2 Shea ves as ´ etale spaces Throughout this article, w e will consider shea v es as shea v es of cross-sect ions of ´ etale spaces. Recall that a bund le p : E → X ov er X is said to b e ´ etale sp ac e o ve r X if for eac h e ∈ E there exists an op en set V ⊂ E , with e ∈ V , such that p ( V ) ⊂ X is open in X an d the restriction p | V : V → p ( V ) o v er V is a homeomorphism. There is a w ell kno wn equiv alence E tal e ( X ) Γ / / S h ( X ) Λ o o where Γ : E tal e ( X ) → S h ( X ) is a fun ctor w hic h assigns to eac h ´ etale space p : E → X o ver X the sheaf of all cross-sections of E . The functor Λ : S h ( X ) → E tal e ( X ) assigns to eac h sheaf S the ´ etale space of germes of S , where the germ at the p oint x ∈ X is an equiv alence class ger m x s represen ted b y s ∈ S ( U ) under the equiv alence relatio n, whic h relates t wo elemen ts s ∈ S ( U ) and t ∈ S ( V ) if there is some op en set W ⊂ U ∩ V suc h that x ∈ W and s | W = t | W . The stalk of th e sheaf S at th e p oin t x ∈ X is the set S x = { g er m x s : s ∈ S ( U ) , x ∈ U } of all germs at x , wh ic h is formally a filtered colimit S x = l im − → x ∈ U S ( U ) of the restrictio n S ( x ) : O x ( X ) op → S et of the sheaf S to the filtered category O x ( X ) op of op en neighborh o o ds of the p oin t x ∈ X . Then Λ S is an ´ etale space p : ` x ∈ X S x → X whose sheaf of cross sections is canonically isomorphic to S . Therefore, we will use simultaneo usly terms sh ea ves and ´ etale spaces in the rest of this article. 4 2.3 T opoi I n the follo w ing, a top os will alwa ys mean a Grothendiec k top os. S h ( X ) will denote top os of shea v es on a (to p ologi cal) space X . A sob er space X can b e reco ve red from the top os S h ( X ), which is the faithfu l image of the space X in the wo rld of top oi. F urther, Hom( S h ( X ) , S h ( Y )) will denote the category of geometric morphisms from S h ( X ) to S h ( Y ). W e will use the same notation Hom( F , E ) also in the more general case of any t wo top oi F and E . By definition a ge ometric morphism f ∈ Hom( F , E ) is a pair of fu n ctors f ∗ : E → F and f ∗ : F → E , f ∗ b eing left adjoin t to f ∗ and also f ∗ b eing left exact, i.e. preserving fi nite limits. Let us recall that a geometric morph ism f : F → E b et ween lo cally connected top oi is a we ak homo topy e quivalenc e if it induces an isomorphisms on ´ etale homotop y (pr o)groups π 0 ( F ) ∼ = π 0 ( E ) and π n ( F , p ) ∼ = π n ( E , f q ), for n ≥ 1 for an y base p oin t q ∈ F . F or the collection of homotop y classes of geometric morphism from F to E the usu al notation [ F , E ] will b e used. 2.4 The singular functor The follo wing construction of a sin gular functor is tak en from [16], wh ere Kelly d escrib ed it in the con text of enric h ed V -categ ories for an y symmetric monoidal closed catego ry V , whic h is complete and cocomplete. Let F : A → E b e a functor from the small cate gory A . The singular functor of F is the fu nctor E ( F , 1) : E → [ A op , V ] whic h is obtained as the comp osite of the Y oneda embedd ing Y on : E → [ E op , V ] follo wed b y the functor [ F op , V ] : [ E op , V ] → [ A op , V ] giv en by restriction along a functor F . More explicitly , the sin gu lar functor E ( F , 1) sends any ob ject E in E to th e f unctor E ( F ( − ) , E ) : A op → V whic h tak es an ob ject A in A to the hom-ob ject E ( F ( A ) , E ) in V . I f the category E is co complete, th en the sin gular fu n ctor has a left ad j oin t L : [ A op , V ] → E defined for eac h presh eaf P : A op → V as the col imit L ( P ) = l im − → ( R A P π P / / A F / / E ) where R A P is th e so call ed Grothendiec k construction [20] on a presheaf P : A op → V . 5 2.5 Grothendiec k nerve as a singular functor Eac h ordinal [ n ] = { 0 < 1 < . . . < n } can b e seen as a category with ob jects 0 , 1 , . . . n, and a unique arro w i → j for eac h 0 ≤ i ≤ j ≤ n . Also, an y monotone map b et ween t wo ordinals may b e seen as a functor. In this wa y , ∆ b ecomes a full s ub category of Cat 1 with a fu lly faithful inclusion fun ctor J : ∆ → Cat 1 F or any small category B , the comp osite of the Y oneda em b edd in g Y on : B → [ B op , Set] follo wed by the restriction fun ctor [ B op , Set] → [∆ op , Set] along J giv es a singular fun ctor of J . In more d etails, the singular fun ctor of J d efines the Grothendiec k nerve f unctor N : Cat 1 → [∆ op , Set] whic h sends an y category C to the simplicial set N C w h ic h is the nerve of C whose n - simplices a re defined by the set N C n = [ J ([ n ]) , C ] where the righ t side d enotes the set of f unctors f rom an ord inal [ n ] to the category C . The nerv e fun ctor is fully faithful, w hic h means that the simplicial sk eletal category ∆ is an adequate su b category of th e category Cat 1 in the sense of Isb ell [13], [14]. W e also sa y that the corresp ondin g em b edd ing is dense, in the sense of Kelly [16]. 2.6 Simplicial spaces Let ∆ b e the s im p lical mo del cat egory h a ving as ob j ects n onempt y finite sets (o rdin als) [ n ] = { 0 , 1 , . . . , n } , for n ≥ 0, and as arro ws order-preserving functions α : [ n ] → [ m ]. A simplicial sp ac e (set) is a contra v ariant functor from ∆ into the category of spaces (set s). Its v alue at [ n ] is denoted Y n and its action on arro w α : [ n ] → [ m ] as Y ( α ) : Y m → Y n . A simplicial space Y is called lo c al ly c ontr actible if eac h Y n has a basis of con tractible sets. F or a sim p licial space Y th e ge ometric r e alization | Y | will alwa ys mean the th ic k ened (fat) geometric realizat ion. This is defined as a top ologica l space obtained from the disjoint sum P n ≥ 0 X n × ∆ n b y th e th e equ iv alence relations ( α ∗ ( x ) , t ) ∼ ( x, α ( t )) for all injectiv e (order-preserving) arro ws α : [ n ] → [ m ] ∈ ∆, an y x ∈ X m and an y t ∈ ∆ n , where ∆ n is the standard top ologic al n -simp lex. If all degeneracies are closed cofibrations, i.e. the simplicial space is a go o d simplicial space, th is geometric real ization is homotop y equiv alen t to th e geometric realization of the underlyin g simplicial s et of Y , which is defined as ab o ve but allo w ing for all arrows in ∆. In particular, Y is go o d if all spaces Y n are lo cally equiconnected [1]. Geometric realizatio n of a locally contract ible simplicial space is a lo cally con tractible space. 6 Definition 2.1. A sheaf S on a simplicial space Y is define d to b e a system of she aves S n on Y n , for n ≥ 0 , to gether with she af maps S ( α ) : Y ( α ) ∗ S n → S m for e ach α : [ n ] → [ m ] . These maps ar e r e qu ir e d to satisfy the fol lowing functoria lity c onditions: i) (normaliza tion) S (id [n] ) = id S n , and ii) for any α : [ n ] → [ m ] , β : [ m ] → [ k ] the fol lowing diagr am Y ( β ) ∗ Y ( α ) ∗ S n ∼ = Y ( β ) ∗ S ( α ) / / Y ( β ) ∗ S m S ( β ) Y ( β α ) ∗ S n S ( β α ) / / S k is c ommutative. A morp hism f : S → T of she aves on Y c onsists of maps f n : S n → T n of she aves on Y n for e ach n ≥ 0 , which ar e c omp atible with th e structur e ma ps S ( α ) and T ( α ) . This defines the c ate gory S h ( Y ) of she aves on th e simplicial sp ac e Y . Prop osition 2.2. The c ate g ory S h ( Y ) of she aves on a simplicial sp ac e is a top os. Theorem 2.3. F or any simplicial sp ac e Y the top oi S h ( Y ) and S h ( | Y | ) have the same we ak homotop y typ e. Definition 2.4. A linear order ov er a top ological space X is a she af p : L → X on X to gether with a subshe af O ⊆ L × X L suc h that for e ach p oint x ∈ X the stalk L x is nonempty and line arly or der e d by the r elation y ≤ z iff ( y , z ) ∈ O x , for y , z ∈ L x . A mapping L → L ′ b etwe en two line ar or ders over X is a mapping of she aves r estricting for e ach x ∈ X to an or der pr eserving map of stalks L x → L ′ x . This defines a c ate gory of line ar or ders on X . Example 2.5. An op e n or der e d c overing U = { U i } i ∈ I of a top olo gic al sp ac e X , is a c overing indexe d over a p artial ly or der e d set I , which r estricts to a total or dering on every finite subset { i 0 , . . . , i n } of I whenever the finite interse ction U i 0 ,...,i n = U i 0 ∩ . . . ∩ U i n is nonempty. When a she af p : L → X is give n by the pr oje ction p : ` i ∈ I U i → X fr om the disjoint union of op en sets in the op en or der e d c overing U the subshe af p [2] : L × X L → X is give n by the induc e d pr oje ction p [2] : ` i,j ∈ I U ij → X fr om the family { U ij } i,j ∈ I of double interse ctions of op en sets U . The family of i nclusions i ij : U ij ֒ → ` i,j ∈ I U ij , for e ach U ij 6 = ∅ such that i < j , defines a subshe af O = ` i
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