Exit paths and constructible stacks

EXIT P A THS AND CONSTR UCTIBLE ST A CKS. D A VID TREUMANN Abstract. F or a Whitney stratiﬁcation S of a space X (or more generally a top olog- ical stratiﬁcation in the s ense of Goresky and Ma cPherson) we in troduce the notion of an S -constructible stack of categories on X . The motiv ating example is the stack of S - constructible per verse shea ves. W e introduce a 2-categ ory E P ≤ 2 ( X, S ), called the exit-path 2-catego ry , which is a natura l stratiﬁed version of the fundamen tal 2-gro upo id. Our main result is that the 2-category o f S -c o nstructible stac ks o n X is equiv a le nt to the 2-catego ry of 2-functors 2F unct( E P ≤ 2 ( X, S ) , Cat ) from the exit-path 2-ca tegory to the 2-categor y of small categor ies. 1. Introduction This pap er is concerned with a generalization of the following w ell- known and v ery old theorem: Theorem 1.1. L et X b e a c onne cte d, lo c al ly c on tr actible top olo gic al sp ac e. The c ate gory of lo c al ly c on stant she aves of sets on X is e quivalent to the c ate gory of G -sets, wher e G is the fundamental gr oup of X . W e wish to generalize t his theorem in tw o dire ctions. In one direction w e will conside r shea v es whic h a re not necessarily lo cally constant – namely , constructible shea ve s. In the second direction w e will consider shea v es of “higher-categorical” ob jects – these generaliza- tions of shea ves are usually called stacks . Putting these together, w e get the “constructible stac ks” of the title. In this pap er, w e introduce an ob ject – the exit-p ath 2-c ate gory – whic h pla ys f or cons tructible stac ks the same role the fundamen tal group pla ys fo r lo cally constant shea v es. 1.1. Exit paths and constructible shea v es. A sheaf F on a space X is called “con- structible” if the space may b e decomp osed in to suitable pieces with F lo cally constan t on eac h piece. T o get a go o d theory one needs to imp ose some conditions on the decomposition – for our purp oses the notio n o f a top olo gic al str atiﬁc ation , in tro duced in [7 ], is the most con v e- nien t. A top ological stratiﬁcation S of X is a decompo sition of X into top olog ical manifolds, called “strata,” whic h are required to ﬁt together in a nice wa y . (T op olo gical stratiﬁcations are more general than Whitney stratiﬁcations and Thom-Mat her stratiﬁcations [21], [16], and they are less general than Sieb enmann’s CS sets [20] and Quinn’s manifo ld stratiﬁed spaces [19]. A precise deﬁnition is giv en in [7] and in section 3.) A sheaf is called S - c onstructible if it is lo cally constan t along eac h stratum of ( X , S ). Date : August 2 007. 1 2 DA VID TREUMAN N MacPherson observ ed (unpublished) tha t , for a ﬁxed stratiﬁcation S of X , it is p ossible to give a description of the S -constructible shea v es on X in terms of mo no drom y along certain paths. A path γ : [0 , 1] → X has the exit pr op erty with res p ect to S if, for eac h t 1 ≤ t 2 ∈ [0 , 1], the dimension of the stratum containing γ ( t 1 ) is less than or equal to t he dimension of the stratum con taining t 2 . He re is a picture of an exit pat h in the plane, where the plane is stratiﬁed b y the origin, the rays of the a xes, and t he in teriors of the quadrants: The concatenation of t w o exit paths is an exit path, and passing to ho mo t op y classes yields a category w e call E P ≤ 1 ( X , S ). That is, the ob jects of E P ≤ 1 ( X , S ) are p oin ts of X , and the morphisms are homotop y classes of exit pa t hs b et w een p oints . (W e req uire that the homotopies h : [0 , 1] × [0 , 1] → X hav e the pro p ert y that eac h h ( t, − ) is an exit path, and that h do es not inte rsect stra t a in a pathological wa y . W e conjecture that the latt er “tameness” condition can b e remo v ed.) Then w e hav e the follo wing a nalog of theorem 1.1: Theorem 1.2 (MacPherson) . L et ( X , S ) b e a top olo gic al ly str atiﬁe d sp ac e. The c ate gory of S -c o n structible she aves of sets is e quivalen t to the c ate gory F unct  E P ≤ 1 ( X , S ) , Sets  of Sets -value d functors on E P ≤ 1 ( X , S ) . Example 1.3. Let D b e the op en unit disk in the complex plane, and let S b e the stratiﬁ- cation b y the orig in { 0 } and its complemen t D − { 0 } . Then E P ≤ 1 ( D , S ) is equiv a len t to a category with t w o ob jects, one la b eled b y 0 and one la b eled b y some other p oint x ∈ D − { 0 } . The arrow s in this category are generated b y arrows α : 0 → x and β : x → x ; β generates the automor phism group Z and w e ha v e β ◦ α = α . The map α is represen ted by an exit path from 0 to x in D and the path β is represen t ed b y a lo op around 0 based at x . It follo ws that an S -constructible sheaf (of , say , complex v ector spaces) on D is giv en b y t w o v ector spaces V and W , a morphism a : V → W and a morphism b : W → W , with the prop ert y that b is in v ertible and that b ◦ a = a . Example 1.4. Let P 1 = C ∪ ∞ b e the Riemann sphere, and let S b e the stratiﬁcation of P 1 b y one p oin t {∞} and the complemen t C . Then E P ≤ 1 ( P 1 , S ) is equiv alen t t o a category with t w o ob jects, one lab eled by ∞ and the other lab eled b y 0 ∈ C . The only nontrivial a r r ow in this category is repres en ted b y an exit path from ∞ to 0; all suc h paths are homotopic to eac h other. It fo llo ws t ha t a n S - constructible sheaf on P 1 is giv en b y t w o v ector spaces V and W , and a single morphism V → W . 1.2. Perv er se sheav es. Let ( X, S ) b e a top ologically stratiﬁed space. F or each function p : S → Z fro m connected strata of ( X , S ) to inte gers, there is an a b elian category P ( X , S, p ) of “ S -constructible p ervers e shea v es on X of p erv ersit y p ,” intro duced in [1]. It is a full sub- category o f the derive d category of shea v es on X ; its ob jects are complexes of shea v es whose EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 3 cohomology shea v es are S -constructible, and whose deriv ed restriction and corestriction to strata satisfy certain cohomology v anishing conditions dep ending on p . It is diﬃcult to lay hands on the ob jects and esp ecially the morphisms of P ( X , S, p ), although w e g et Sh S ( X ) as a sp ecial case when p is constan t. There is a small industry dev oted to ﬁnding concrete descriptions of the category P ( X, S, p ) in terms of “linear algebra data,” similar to the description of S -constructible shea v es given in the examples a b o v e ([15], [5], [3], [2], [2 5 ]). Here ( X , S ) is usually a complex analytic space, with complex analytic strata, and p is the “middle p erv ersit y” whic h associates to eac h stratum its complex dimension. The ﬁrst example was found b y D eligne: Example 1.5. Let D and S b e as in example 1.3 . The category P ( X , S, p ), where p is the middle p erv ersit y , is equiv a lent to the category of tuples ( V , W , m, n ), where V a nd W a re C -v ector spaces, m : V → W and n : W → V are linear maps, and 1 W − mn is in v ertible. A top ological inte rpretation of this description w as g iven b y MacPherson and Vilonen [15]. If ( L, S ) is a compact top o logically stratiﬁed space then the op en cone C L on L has a natural top ological stratiﬁcation T , in whic h the cone p oin t is a new stratum. MacPherson and Vilonen gav e a description of p erv erse shea v es on ( C L, T ) in terms of p erv erse shea v es on L , generalizing Deligne’s description 1.5. One of the imp o rtan t pro p erties of p ervers e shea v es is t ha t they for m a stack ; it means that a pervers e sheaf on a space X ma y b e described in the c harts of an op en co v er of X . A top ologically stratiﬁed space has an o p en cov er in whic h the c harts are of the form C L × R k . The stac k prop erty together with the MacPherson-Vilonen construction giv e an inductiv e strategy for computing categories of p ervers e shea v es. One of the motiv at io ns for the theory in t his pap er is to analyze this strategy systematically; see [23]. 1.3. Constructible stacks. In this pap er, w e introduce the notion of a c onstructible stack on a top olo gically stratiﬁed space. Our main example is the stac k P o f S -constructible p erv erse shea v es discusse d in section 1.2 . Our main result is a kind of classiﬁcation o f constructible stacks , ana logous to the description of constructible shea ves by exit paths. Main Theorem (Theorem 7.14) . L et ( X , S ) b e a top ologically stratiﬁed space. There is a 2-category E P ≤ 2 ( X , S ), in tro duced in section 7, suc h that the 2 -category of S -constructible stac ks on X is equiv alen t to the 2-catego r y of 2- f unctors 2F unct( E P ≤ 2 ( X , S ) , Cat ). The app earance of 2 -categories in this theorem is an application of a w ell-kno wn philosophy of G rothendiec k [9]. It is a mo diﬁcation of theorems in [18] and [2 2 ], where it w as sho wn that lo c al ly c onstant stacks on X corresp ond to represen ta tions of higher gr oup oids , namely the group oid of p o ints, pat hs, homotopies, homotopies betw een homotopies, a nd s o on in X . E P ≤ 2 ( X ) is a 2-truncated, stratiﬁed v ersion of this: the ob jects are the po in ts of X , the morphisms a r e exit pa t hs, and the 2-mo r phisms are homotopy classes of ho motopies b etw een exit paths. (Once again, we require a tameness condition on our homo t o pies and also our homotopies b et w een homotopies.) Example 1.6. Let P 1 and S b e as in example 1.4. Then E P ≤ 2 ( P 1 , S ) is equiv alent to a 2-category with tw o ob jects, lab eled b y ∞ and 0 as b efore, and one a r r ow from ∞ to 0 4 DA VID TREUMAN N represen ted b y an exit path α . The g roup of homotopies from α to itself is Z , generated by a homoto py that rotates α aro und the 2-sphere once. It follo ws that an S - constructible stac k on P 1 is giv en by a pair C ∞ and C 0 of categories, a functor α : C ∞ → C 0 , and a natural automorphism f : α → α . F or the stac k of S - constructible pervers e shea v es, C 0 is the category of vec tor spaces, C ∞ is Deligne’s category described in example 1.5, α is the for g etful functor α : ( V , W , m, n ) 7→ W , and f is the map 1 W − mn , whic h is in v ertible b y assumption. 1.4. Not at ion and con v en tions. R denotes the real n um bers. F or a, b ∈ R with a ≤ b we use ( a, b ) to denote the op en in terv al and [ a, b ] to denote the closed in terv al b et w een a and b . W e use [ a, b ) and ( a, b ] to denote ha lf -op en interv als. If L is a compact space then C L denotes the op en cone on L , that is, the space L × [0 , 1) /L × { 0 } . A d -cov er o f a space X is a collection of op en subse ts of X that co v ers X and that is closed under ﬁnite inters ections. F or us, a “2-category” is a strict 2- category in the sense that comp osition of 1-morphisms is strictly ass o ciative . On the other hand we use “2-functor” to r efer to morphisms of 2- categories that only preserv e comp osition of 1-morphisms up to isomor phism. W e will r efer to sub-2-categories of a 2-category C as simply “sub categories of C .” F or more basic deﬁnitions and prop erties of 2-categories see app endix B. All our stac ks are stac ks o f categories. W e write Prest( X ) f o r the 2-category o f prestac ks on a space X and St( X ) ⊂ Prest( X ) fo r the full sub category of Prest( X ) whose ob jects are stac ks. W e use St lc ( X ) ⊂ St( X ) to denote the full sub category o f lo cally constan t stacks on X , whic h w e in tro duce in se ction 2. When S is a top ological stratiﬁcation (deﬁnition 3.1) of X w e write St S ( X ) ⊂ St( X ) for the full sub cat ego ry of S -constructible stack s on X , whic h w e introduce in section 3. F or more basic deﬁnitions and prop erties of stac ks see app endix A. 2. Locall y const ant st ack s In this section w e in tro duce lo cally constant stac ks of categories. A stac k is called c onstant if it equiv alent to t he stac kiﬁcation of a constan t prestack , and lo c al ly c onstant if this is true in the c harts of an op en co v er. Our main ob jectiv e is to give an equiv a lent deﬁnition that is easier to c hec k in practice: on a lo cally contractible space, a stac k C is lo cally constant if and only if the restriction functor C ( U ) → C ( V ) is an equiv alence of categories whenev er V and U are con tractible. This is theorem 2.9. W e also dev elop some basic prop erties of lo cally constan t stac ks, including a base-c hange result (theorem 2.11) and the homotopy inv aria nce of the 2-category of lo cally constan t stac ks (t heorem 2.7). 2.1. Constan t stacks. Let C b e a small category . On any space X w e hav e the constant C -v alued prestac k, and its stack iﬁcation. W e will denote the prestack by C p ; X , and it s stac kiﬁcation b y C X . Example 2.1. Let X b e a lo cally con tractible space, or more generally any space in whic h eac h p oint has a fundamen tal system of neigh bo rho o ds ov er which each lo cally constan t sheaf is constan t. If C is the category of se ts, then C X is naturally equiv alent to t he stac k LC X of lo cally constan t shea v es. That is, the map C p ; X → LC X that tak es a set E ∈ C = C p ; X ( U ) EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 5 to the constant sheaf o v er U with ﬁb er E induces an equiv alence C X → L C X . I ndeed, it induces an equiv alence on stalks by the lo cal con tractibilit y o f X . Prop osition 2.2. L et X b e a l o c al ly c ontr actible sp ac e and C b e a sm al l c ate gory. (1) If F and G ar e obje cts in C X ( X ) , then the sh e af Hom ( F , G ) : U 7→ Hom( F | U , G | U ) is lo c al ly c onstant o n X . (2) L et U ⊂ X b e an op en set. F or e ach p oin t x ∈ U , the r estriction functor fr om C = C p ; X ( U ) to the stalk of C X at x is an e quivalenc e of c ate gories. (3) L et U ⊂ X b e a n op en set. Supp ose that U is c ontr actible. Then for e ach p oin t x ∈ U , the r estriction functor C X ( U ) → C X,x is an e quivalenc e of c ate gories. Pr o of. The map of prestac ks Hom : C op p ; X × C p ; X → Sets p ; X induces a map of stac ks C op X × C X → Sets X ∼ = LC X . The ob ject in LC X ( X ) a sso ciated to a pair ( F , G ) ∈  C op X × C X  ( X ) is exactly the sheaf Hom ( F , G ). This pro v es the ﬁrst assertion. The second a ssertion is trivial. T o prov e the third assertion, note that C X ( U ) → C X,x is alw a ys ess en tially surjectiv e, since the equiv a lence C p ; X ( U ) → C X,x factors throug h it. W e therefore only ha ve to sho w that C X ( U ) → C X,x is fully faithful. F or ob jects F and G of C X ( U ), we ha v e just see n that Hom ( F , G ) is lo cally constan t on U . Since U is con- tractible Hom ( F , G ) is constan t, and so Hom C X ( U ) ( F , G ) = Hom ( F , G )( U ) → Hom ( F , G ) x ∼ = Hom( F x , G x ) is a bijection. This completes the pro of.  2.2. Lo cally constan t stacks . Deﬁnition 2.3. A stac k C on X is called lo c al ly c on s tant if there exists a n op en co v er { U i } i ∈ I of X suc h that C | U i is equiv alent to a constant stack . Let St lc ( X ) ⊂ St ( X ) denote the full sub cat ego ry of the 2-category of stac ks on X whose ob jects are the lo cally constan t stac ks. Prop osition 2.4. L et X b e a lo c al ly c ontr actible sp ac e and let C b e a lo c al ly c onstant stack on X . (1) L et U ⊂ X b e an op en set, and let F , G ∈ C ( U ) . The she af Hom ( F , G ) is lo c al ly c onstant on U . (2) Every p oin t x ∈ X has a c o ntr actible neighb orh o o d V such that the r estriction map C ( V ) → C x is an e quivale nc e of c ate gories. Pr o of. Assertion 1 f o llo ws directly from assertion 1 of prop osition 2.2, and assertion 2 follows directly from assertion 3 of prop osition 2.2.  Prop osition 2.5. L et X and Y b e top olo gic al sp ac es, and let f : X → Y b e a c ontinuous map. Supp ose C is lo c al ly c onstant on Y . Th e n f ∗ C is lo c al ly c onstant on X . Pr o of. If C is constan t ov er the op en sets U i of Y , t hen f ∗ C will b e constant o v er the op en sets f − 1 ( U i ) of X .  The homotopy in v ariance of St lc ( X ) is a consequence of the following base-c hange result: Prop osition 2.6. L et X and Y b e top olo gic al sp ac es, and let f : X → Y b e a c ontinuous map. L et g den o te the map ( id, f ) : [0 , 1] × X → [0 , 1] × Y , and let p : [0 , 1] × X → X and 6 DA VID TREUMAN N q : [0 , 1 ] × Y → Y b e the natur al pr oje ction maps. [0 , 1] × X g / / p   [0 , 1] × Y q   X f / / Y L et C b e a lo c al ly c onstant stack on [0 , 1] × Y . Then the b ase change map f ∗ q ∗ C → p ∗ g ∗ C is an e quiva l e nc e of stacks on X . Pr o of. Let us ﬁrst prov e the f ollo wing claim: ev ery p oint t ∈ [0 , 1] has a neigh b orho o d I ⊂ [0 , 1] suc h that C  [0 , 1] × Y  → C  I × Y  is an equiv alence of catego r ies. There is an op en cov er of [0 , 1] × Y of the fo rm { I α × U β } suc h that C is constan t ov er eac h chart I α × U β . By b y basic prop erties of the in terv a l, C is constant o v er [0 , 1] × U β , and for an y subin terv a l I ⊂ [0 , 1] the restriction map C  [0 , 1] × U β  → C ( I × U β ) is an equiv alence of categories. This implies t he claim. Let y be a p oin t in Y , and let t b e a p oint in [0 , 1]. Let { U } b e a fundamental system of neigh b orho o ds of y . Ac cording to the claim, w e ma y pick for eac h U an op en set I U ⊂ [0 , 1] suc h that the restriction functor C  [0 , 1] × U  → C ( I U × U ) is an equiv alence of categories. W e ma y c ho ose the I U in suc h a w a y that the op en sets I U × U ⊂ [0 , 1] × Y form a fundamen ta l system of neigh bor ho o ds of ( t, y ). It follo ws that the natural r estriction functor on stalks ( q ∗ C ) y → C ( t,y ) is an equiv alence. No w let x be a po in t in X . W e hav e natural equiv alences ( f ∗ q ∗ C ) x ∼ = ( q ∗ C ) f ( x ) ∼ = C ( t,f ( x )) ( p ∗ g ∗ C ) x ∼ = ( g ∗ C ) ( t,x ) ∼ = C ( t,f ( x )) This completes the pro of.  Theorem 2.7 ( Ho mo t op y in v ariance) . L et X b e a top olo gic al sp ac e, and let π den ote the pr oje ction map [0 , 1] × X → X . Then π ∗ and π ∗ ar e inverse e quivalen c es b etwe en the 2- c ate gory of lo c al ly c onstant stacks on X , and the 2-c ate gory of lo c al ly c onstant stacks on [0 , 1] × X . Pr o of. Let C b e a lo cally constant stack on X and let D b e a lo cally constant stac k on I × X . Let x ∈ X , le t i x denote the inclusion map { x } ֒ → X , let j x denote the inclusion map [0 , 1] ∼ = [0 , 1] × x ֒ → [0 , 1] × X , and let p denote the map [0 , 1] → { x } . By prop o sition 2.6, the natural ma p ( π ∗ π ∗ C ) x = i ∗ x π ∗ π ∗ C → p ∗ j ∗ x π ∗ C is an equiv a lence. But p ∗ j ∗ x π ∗ C ∼ = p ∗ p ∗ i ∗ x C = p ∗ p ∗ ( C x ). T he natural map C x → ( π ∗ π ∗ C ) x ∼ = p ∗ p ∗ ( C x ) coincides with the adjunction map C x → p ∗ p ∗ ( C x ). Since p ∗ C x is constant, C x → p ∗ p ∗ ( C x ) is an equiv alence b y prop osition 2.2 . It follo ws that C → π ∗ π ∗ C is an equiv a lence of stack s. No w let ( t, x ) ∈ [0 , 1] × X . Th ere is an equiv a lence ( π ∗ π ∗ D ) ( t,x ) ∼ = ( π ∗ D ) x . Once again prop osition 2.6 provides an equiv alence ( π ∗ D ) x ∼ = p ∗ j ∗ x D = j ∗ x D  [0 , 1]  . The lo cally constan t stac k j ∗ x D is c onstan t on [0 , 1], so j ∗ x D  [0 , 1]  ∼ = D ( t,x ) b y prop osition 2.2. It follo ws that π ∗ π ∗ D → D is an equiv alence of stac ks, completing the pro of.  EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 7 Corollary 2.8. L et X and Y b e top olo gic al sp ac es, and let f : X → Y b e a h omotopy e quivalen c e. (1) The 2-functor f ∗ : St lc ( Y ) → St lc ( X ) is an e quivalenc e o f 2-c ate gories. (2) L et C b e a lo c al ly c onstant stack o n Y . Then the na tur al functor C ( Y ) → f ∗ C ( X ) is an e quiva l e nc e of c ate gories. Pr o of. Let g : Y → X b e a homoto py in v erse to f , and let H : [0 , 1] × X → X b e a homotopy b et w een g ◦ f and 1 X . Let π denote the pro jection map [0 , 1] × X → X , and for t ∈ [0 , 1] let i t denote the map X → [0 , 1] × X : x 7→ ( t, x ). Then i ∗ t ∼ = π ∗ b y theorem 2.7. It follows that i ∗ 0 ∼ = i ∗ 1 , and tha t ( H ◦ i 0 ) ∗ ∼ = ( H ◦ i 1 ) ∗ . But H ◦ i 0 = g ◦ f and H ◦ i 1 = 1 X , so f ∗ ◦ g ∗ ∼ = 1 St lc ( X ) . Similarly using a homotopy G : I × Y → Y we may construct a n equiv a lence g ∗ ◦ f ∗ ∼ = 1 St lc ( Y ) . This pro v es the ﬁrst a ssertion. T o prov e the second assertion, let ∗ denote the trivial category . By (1), Hom St( Y ) ( ∗ , C ) ∼ = Hom St( X )  ∗ , f ∗ C  . But C ( Y ) ∼ = Hom( ∗ , C ) and f ∗ C ( X ) ∼ = Hom( ∗ , f ∗ C ). This completes the pro of.  Theorem 2.9. L et X b e a lo c al ly c ontr actible sp ac e, and let C b e a stack of c ate gories on X . The fol lo wing ar e e quivalent: (1) C is lo c al ly c on stant. (2) If U and V ar e two o p en subsets of X with V ⊂ U , and the inclusion map V ֒ → U is a homotopy e quivalenc e , then the r estriction functor C ( U ) → C ( V ) is an e q uivalenc e of c ate gories. (3) If U and V ar e any two c ontr ac tible op en subsets of X , and V ⊂ U , t hen C ( U ) → C ( V ) is an e quival e n c e of c ate gories. (4) Ther e ex i s ts a c ol le ction { U i } of c ontr actible op en subsets of X such that e ach p oi n t x ∈ X has a fundamental system of neighb orho o ds of the form U i , and s uch that C ( U i ) → C ( U j ) is an e quiva l e nc e of c ate gories w h enever U j ⊂ U i . Pr o of. Supp ose C is lo cally constant, a nd let U and V b e as in condition (2). Then corollary 2.8 implies that the restriction functor C ( U ) → C ( V ) is an equiv alence of categories, so condition (1) implies condition (2). Clearly condition (2) implies condition (3), and condition (3) implies condition (4). Let us sho w condition (4) implies condition (1). Supp ose C satisﬁes condition (4). T o sho w C is lo cally constant it is enough to show its restriction to each of the distinguished charts in { U i } is constant. Let U ⊂ X b e such a c hart. Since eac h p oint x ∈ U has a fundamen tal system of neigh bo rho o ds { V } from { U i } , and since C ( U ) → C ( V ) is an equiv a lence for eac h V , the map C ( U ) → C x is an equiv a lence for eac h U . It follow s that the natural map from the constant stac k  C ( U )  U to C | U is an equiv alence on stalks, and therefore an equiv alence.  2.3. Direct images and base cha nge. Let f b e a con tin uous map b et w een lo cally con- tractible spaces. As an application of theorem 2.9, w e may giv e easy pro ofs of some basic prop erties of the direct ima g e f ∗ of lo cally constant stac ks. Prop osition 2.10. L et X and Y b e lo c al ly c ontr actible sp ac es. L e t f : X → Y b e a lo c al ly trivial ﬁb er bund le, or mor e ge n er al ly a Serr e ﬁbr ation. L et C b e a lo c al ly c onstant stack on X . Then f ∗ C is lo c al ly c onstant on Y . 8 DA VID TREUMAN N Pr o of. By theorem 2.9, it suﬃ ces to show that the restriction functor f ∗ C ( U ) → f ∗ C ( V ), whic h is equal to the r estriction functor C  f − 1 ( U )  → C  f − 1 ( V )  , is an equiv alence of categories whenev er V ⊂ U ⊂ X a re op en sets and U a nd V are con tractible. Since f is a Serre ﬁbration, the inclusion map f − 1 ( V ) ֒ → f − 1 ( U ) is a homotopy equiv a lence, and the prop osition follo ws f r o m corollary 2.8.  Theorem 2.11. L et X and S b e lo c al ly c o ntr actible sp ac es. L et p : X → S b e a lo c al ly trivial ﬁb er bund le , or mor e gener al ly a Serr e ﬁbr ation. L et T b e another lo c al ly c ontr actible sp ac e, a nd let f : T → S b e any c o n tinuous function. Set Y = X × S T , and let g : Y → X and q : Y → T denote the pr o je ction ma p s. Y g / / q   X p   T f / / S L et C b e a lo c al ly c onstant stack on X . The b as e-change map f ∗ p ∗ C → q ∗ g ∗ C is a n e quivalen c e of stacks. Pr o of. The statemen t is lo cal o n T and S , so w e ma y assume b oth T and S are con tractible. Since p and q are Serre ﬁbrations the stac ks f ∗ p ∗ C and q ∗ g ∗ C are lo cally constan t, and therefore constan t. T o sho w that the base-c hange map is an equiv alence of stacks it is enough to show that the f unctor f ∗ p ∗ C ( T ) → q ∗ g ∗ C ( T ) is an equiv alence of categories. W e hav e q ∗ g ∗ C ( T ) = g ∗ C ( Y ), wh ic h b y corollary 2.8 is equiv alen t to C ( X ). F urthermore, corollary 2.8 sho ws that f ∗ p ∗ C ( T ) ∼ = p ∗ C ( S ) = C ( X ). This completes the pro of.  3. Constructible st ack s In this section w e intro duce constructible stac ks. Fir st w e review the top ologically strat- iﬁed spaces of [7 ]. The deﬁnition is inductiv e: r o ughly , a stratiﬁcation S of a space X is a decomp osition into pieces called “strata ,” suc h that the decomp osition lo oks lo cally lik e the cone on a simpler (low er-dimensional) stratiﬁed space. A stac k o n X is called “ S -constructible” if its restriction to eac h stratum is lo cally constant. This deﬁnition is somewhat un wieldy , and w e giv e a more usable criterion in theorem 3.13, analog o us to the- orem 2.9 for lo cally constan t stac ks: a stac k is S -constructible if and only if the restriction from a “ conical” op en set to a smaller conical op en set is an equiv alence of catego ries. This criterion is a consequenc e of a stratiﬁed-homotop y inv aria nce statemen t (coro lla ry 3 .1 2). 3.1. T op ologically stratiﬁed spaces. Deﬁnition 3.1. Let X b e a paracompact Ha usdorﬀ space. A 0-dimensional topolo gical stratiﬁcation of X is a homeomorphism b et w een X and a coun table discrete set of p oin ts. F or n > 0, an n -dimensional top olo gic al str atiﬁc ation of X is a ﬁltration ∅ = X − 1 ⊂ X 0 ⊂ X 1 ⊂ . . . ⊂ X n = X of X b y closed subsets X i , such tha t for eac h i and fo r each p oint x ∈ X i − X i − 1 , there exists a neigh b o rho o d U of x , a compact Hausdorﬀ space L , an ( n − i − 1)-dimensional top ological EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 9 stratiﬁcation ∅ = L − 1 ⊂ L 1 ⊂ L 2 ⊂ . . . ⊂ L n − i − 1 = L of L , and a homeomorphism C L × R i ∼ = U that tak es eac h C L j × R i homeomorphically to U ∩ X j . He re C L = [0 , 1 ) × L/ { 0 } × L is the op en cone on L if L is non-empt y; if L is empt y then let C L b e a one- p oin t space. A ﬁnite dimensional top olo gic al ly str atiﬁe d sp ac e is a pair ( X, S ) where X is a pa r acompact Hausdorﬀ space and S is an n - dimensional top ological stra t iﬁcation of X for some n . Let ( X , S ) b e a top ologically stra t iﬁed space with ﬁltration ∅ = X − 1 ⊂ X 0 ⊂ X 1 ⊂ . . . ⊂ X n = X Note the following immediate consequen ces of the deﬁnition: (1) If X i − X i − 1 is not empt y , then it is an i -dimensional top o lo gical manifold. (2) If U ⊂ X is op en then t he ﬁltra t ion U − 1 ⊂ U 0 ⊂ U 1 ⊂ . . . of U , where U i = U ∩ X i , is a top ological stratiﬁcation. W e will call the connected comp onents of X i − X i − 1 , or unions of them, i -dimensional str ata . W e will call the neigh bo rho o ds U homeomorphic to cones “conical neighborho o ds”: Deﬁnition 3.2. Let ( X, S ) b e an n -dimensional top ologically stratiﬁed space. An op en set U ⊂ X is called a c onic al op en subset of X if U is homeomorphic to C L × R i for some L as in deﬁnition 3.1. Remark 3.3. By deﬁnition, ev ery point in a top ologically stratiﬁed space has an conical neigh b orho o d C L × R k . One of the quirks of top olog ical stratiﬁcations (a s opp osed to e.g. Whitney stratiﬁcations) is that the space L is not uniquely determined up to homeomor- phism: there ev en exist non-homeomorphic manifolds L 1 and L 2 suc h that C L 1 ∼ = C L 2 (see [17]). The following deﬁnition, from [7], is what is usually meant by “stratiﬁed map.” Deﬁnition 3.4. Let ( X , S ) a nd ( Y , T ) b e top ologically stratiﬁed spaces. A con tin uous map f : X → Y is str atiﬁe d if it satisﬁes the following tw o conditions: (1) F or a n y connected comp onen t C of any stratum Y k − Y k − 1 , the set f − 1 ( C ) is a union of connected comp onen ts of strata of X . (2) F or eac h p oint y ∈ Y i − Y i − 1 there ex ists a neigh bo rho o d U of x in Y i , a topo lo gically stratiﬁed space F = F k ⊃ F k − 1 ⊃ . . . ⊃ F − 1 = ∅ and a ﬁltration-preserving homeomorphism F × U ∼ = f − 1 ( U ) that commute s with the pro j ection to U . W e need a m uc h bro ader deﬁnition: Deﬁnition 3.5. Let ( X , S ) and ( Y , T ) b e top ologically stratiﬁed spaces. A contin uous map f : X → Y is called str atum-pr ese rv i ng if for each k , and each connected comp onen t Z ⊂ X k − X k − 1 , the image f ( Z ) is contained in Y ℓ − Y ℓ − 1 for some ℓ . 10 DA VID TREUMAN N Deﬁnition 3.6. Let ( X , S ) and ( Y , T ) b e top olo g ically stratiﬁed spaces, and let f and g b e t w o stratum-preserving maps from X to Y . W e say f a nd g a re homotopic r elative to the str atiﬁc ations if there exists a homotop y H : [0 , 1] × X → Y b et w een f and g suc h that the map H ( t, − ) : X → Y is strat um- preserving fo r ev ery t ∈ [0 , 1]. A sligh tly irritating feature of this deﬁnition is that the space [0 , 1] × X cannot b e stratiﬁed without t reating the b oundar y comp onen ts { 0 } × X and { 1 } × X diﬀeren tly . W e ma y tak e care of this b y using the o p en in terv al: if ( X , S ) a nd ( Y , T ) are top ologically stratiﬁed spaces, then w e ma y endow (0 , 1 ) × X with a top olog ical stratiﬁcation by setting  (0 , 1) × X  i = (0 , 1) × X i − 1 . Note the follo wing (1) Let H : [0 , 1] × X → Y be a stratiﬁed homoto p y . The restriction of this map to (0 , 1) × X is stratum-preserving. (2) Let f and g b e tw o stratum-preserving maps. Then f and g are ho mot opic relativ e to the stratiﬁcations if and only if there exists a stratum-preserving map H : (0 , 1) × X → Y suc h that f ( − ) = H ( t 0 , − ) and g ( − ) = H ( t 1 , − ) for some t 0 , t 1 ∈ (0 , 1 ) . Deﬁnition 3.7. Let ( X , S ) and ( Y , T ) b e top ologically stratiﬁed spaces. Let f : X → Y b e a stratum-preserving map. Call f a str a tiﬁe d homotopy e quivalenc e if there is a stratum- preserving map Y → X suc h that the comp osition g ◦ f is stratiﬁed ho mo t o pic to the iden tit y map 1 X of X , and f ◦ g is stratiﬁed ho motopic t o the iden tit y map 1 Y of Y . Note tha t a “stratiﬁed homotop y equiv alence” f need not b e a stratiﬁed map in the sense of deﬁnition 3.4, but o nly stratum-pr eserving. 3.2. Constructible stac ks. Deﬁnition 3.8. Let ( X, S ) b e a top olo gically stratiﬁed space a nd let C b e a stac k on X . C is called c ons tructible with respect to S if, for eac h k , i ∗ k C is lo cally constan t on X k − X k − 1 , where i k : X k − X k − 1 ֒ → X denotes the inclusion o f the k -dimensional stratum in to X . Let St S ( X ) de note the full sub category of the 2-category St( X ) of s tac ks on X whose ob jects are the S -constructible stac ks. The pullbac k of a constructible stac k is constructible: Prop osition 3.9. L et ( X , S ) and ( Y , T ) b e two top olo g i c al ly str atiﬁe d sp ac es. L et f : X → Y b e a str atum-pr eserving map. If C is a T -c onstructible stack on Y , then f ∗ C is S -c onstructible on X . Pr o of. W e hav e to sho w that f ∗ C is lo cally constan t on X k − X k − 1 . It is enough to sho w it is lo cally constan t on each connected comp onen t. Let C b e a comp onent of X k − X k − 1 , and let i : C → X be the inclusion. Then i ∗ f ∗ C ∼ = ( f ◦ i ) ∗ C . But f ◦ i : C → Y factors through j : Y ℓ − Y ℓ − 1 → Y for some ℓ , so i ∗ f ∗ C is obtained from pulling bac k j ∗ C on Y ℓ − Y ℓ − 1 to C . By prop osition 2 .5, this is lo cally constant on C .  Prop osition 3.10. L et ( X, S ) b e a top olo gic al ly str atiﬁe d sp ac e, and let C b e a c onne c te d str atum of X . L et i : C ֒ → X denote the inclusion map. L e t p : (0 , 1) × C → C and q : (0 , 1) × X → X denote the p r oje ction maps, a nd let j denote the in clusion map ( id, i ) : EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 11 (0 , 1) × C ֒ → X . (0 , 1) × C j / / p   (0 , 1) × X q   C i / / X Endow (0 , 1) × X with a top olo gic al str atiﬁc ation by setting  (0 , 1) × X  k = (0 , 1) × X k − 1 . L et C b e a stack on (0 , 1 ) × X c on structible w i th r esp e ct to this str atiﬁc ation. Then the b ase-c h ange map i ∗ q ∗ C → p ∗ j ∗ C is an e quivale n c e of stacks . Pr o of. Let x b e a p oin t in X and let t be a p oin t in (0 , 1). As in t he pro of of prop o sition 2.6, w e may sho w that the natural map ( q ∗ C ) x → C ( t,x ) is an equiv alence of categories. If x lies in the stratum C , t hen w e ha v e equiv a lences of categories: ( i ∗ q ∗ C ) x ∼ = ( q ∗ C ) x ∼ = C ( t,x ) ( p ∗ j ∗ C ) x ∼ = ( j ∗ C ) ( t,x ) ∼ = C ( t,x ) The base-c ha ng e map comm utes with t hese, proving the prop osition.  Theorem 3.11 (Homoto p y inv ariance) . L et ( X, S ) b e a top o lo gic al ly str atiﬁe d sp ac e. Endow (0 , 1) × X with a top olo gic al str atiﬁc ation T by se tting  (0 , 1) × X  i = (0 , 1) × X i − 1 . L et π b e the str a tiﬁe d pr oje ction m ap (0 , 1) × X → X . The adjoint 2-functors π ∗ and π ∗ induc e an e quivalenc e b etwe en the 2-c ate gory of S -c onstructible s tack s on X and the 2-c ate gory of T -c ons tructible stacks on (0 , 1) × X . Pr o of. W e hav e to sho w the maps C → π ∗ π ∗ C and π ∗ π ∗ D → D are equiv alences, where C is a constructible stack on X and D is a constructible stack on (0 , 1) × X . F or eac h k , let i k denote the inclusion map X k − X k − 1 ֒ → X . T o prov e that C → π ∗ π ∗ C is an equiv alence it suﬃce s to sho w t ha t i ∗ k C → i ∗ k π ∗ π ∗ C is a n equiv alence fo r each k . L et j k denote the inclusion (0 , 1) × ( X k − X k − 1 ) ֒ → (0 , 1) × X , and let p k denote the pro jection map (0 , 1) × ( X k − X k − 1 ) → X k − X k − 1 . (0 , 1) ×  X k − X k − 1  j k / / p k   (0 , 1) × X π   X k − X k − 1 i k / / X By prop osition 3.10, w e hav e an equiv alence i ∗ k π ∗ π ∗ C ∼ = p k ∗ p ∗ k i ∗ k C . Since i ∗ k C is lo cally constant on X k − X k − 1 , the map i ∗ k C → p k ∗ p ∗ k i ∗ C is an equiv a lence b y theorem 2.7. T o sho w π ∗ π ∗ D → D is an equiv alence, it is enough to sho w tha t for eac h k the map j ∗ k π ∗ π ∗ D → D is an equiv alence. By prop osition 3 .1 0 w e hav e j ∗ k π ∗ π ∗ D ∼ = p ∗ k i ∗ k π ∗ D ∼ = p ∗ k p k ∗ j ∗ k D , and since j ∗ k D is lo cally constan t the map p ∗ k p k ∗ j ∗ k D → j ∗ k D is an equiv alence of stac ks by theorem 2.7.  Corollary 3.12. L et ( X , S ) and ( Y , T ) b e top olo gic al ly str atiﬁe d sp ac es, and let f : X → Y b e a str atiﬁe d homotopy e quivalenc e. (1) The 2-functor f ∗ : St T ( Y ) → St S ( X ) is an e quivale n c e of 2- c ate gories. 12 DA VID TREUMAN N (2) L et C b e an S -c onstructible stack o n X . The functor C ( X ) → f ∗ C ( Y ) is an e quiva- lenc e o f c ate gories. Pr o of. A pro of iden tical to the one of corollary 2.8 giv es b oth statemen ts.  Theorem 3.13. L et ( X , S ) b e a top olo gic al ly str atiﬁe d sp ac e and let C b e a stack on X . The fol lowing ar e e quiva l e nt: (1) C is c onstructible with r esp e ct to the str a tiﬁc ation. (2) If U and V ar e two op en subsets of X with V ⊂ U , and if the inclusion map V ֒ → U is a str atiﬁe d homotopy e quivalenc e, then the r estriction functor C ( U ) → C ( V ) is an e quivalen c e of c a te gories. (3) Whenever U an d V ar e c onic al op en subsets of X such that V ⊂ U and the in c lusion map V → U is a str atiﬁe d hom o topy e quivalenc e, the r estriction functor C ( U ) → C ( V ) is an e quivalenc e of c ate gories. If C satisﬁ e s these c onditions then the natur al functor C ( U ) → C x is a n e quivalenc e of c ate gories whenever U is a c onic al op en neigh b orho o d of x . Pr o of. Supp ose C is constructible, and let U and V b e as in (2) . Then C ( U ) → C ( V ) is an equiv alence b y corolla ry 3.1 2, so (1) implies (2). Clearly (2) implies (3). Supp ose now that C satisﬁes the third condition. Let Y = X k − X k − 1 b e a stratum, and let i : Y ֒ → X denote t he inclusion map. Let { U } b e a collection of conical op en sets in X that cov er Y , and such that each U ∩ Y is closed in U . T o show that i ∗ C is lo cally constant on Y it is enough to sho w that j ∗ ( C | U ) is constan t on Y ∩ U , where j denotes the inclusion Y ∩ U → U . F or eac h ǫ > 0 let C ǫ L ⊂ C L denote the set [0 , ǫ ) × L/ { 0 } × L , and let B ǫ ( v ) denote the ball of ra dius ǫ a round v ∈ R k . Let { U i } b e the collection of op en subsets of X of the form C ǫ L × B δ ( v ) under the homeomorphism U ∼ = C L × R k . Whe nev er U i and U j are o f this form and U j ⊂ U i , it is easy to directly construct a stratiﬁed homotopy inv erse to the inclusion map U j ֒ → U i ; th us b y assumption the restriction functor C ( U i ) → C ( U j ) is an equiv alence. F or eac h y ∈ Y ∩ U t he U i con taining y form a fundamen tal system o f neigh b orho o ds of y , so the functor C ( U ) → C y ∼ =  j ∗ C  y is an equiv alence. It follo ws that j ∗ C is equiv alen t to the constan t sheaf on Y ∩ U with ﬁb er C ( U ). This completes the pro of .  3.3. Direct images. Prop osition 3.14. L et ( X , S ) and ( Y , T ) b e top olo gic al ly str atiﬁe d sp ac es, and l e t f : X → Y b e a str atiﬁe d map (se e deﬁnition 3 .4). L et C b e an S - c onstructible stack on X . Then f ∗ C is T -c onstructible on Y . Pr o of. Let y b e a p oint of Y . Let U ∼ = R k × C L b e a conical neigh b orho o d of y , and let V ⊂ U b e a smaller conical neigh b orho o d suc h t ha t the inclusion map V ֒ → U is a stratiﬁed homotopy equiv alence. W e may assume U is small enough so tha t there exists a top ologically stratiﬁed space F and a stra t um- preserving homeomorphism f − 1 ( U ) ∼ = F × U that commutes with the pro jection to U . Then the inclusion map f − 1 ( V ) ֒ → f − 1 ( U ) is a stratiﬁed homotop y equiv alence: if φ : U → V is a homotop y in verse , then a homotop y in v erse to f − 1 ( V ) ֒ → f − 1 ( U ) is giv en b y ( id , φ ) : F × U → F × V . By prop o sition 3 .12 and theorem 3.1 3 it follow s that f ∗ C is constructible.  EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 13 4. Example: the st ack of pe r verse shea ves Let ( X , S ) b e a top ologically stratiﬁed space. Let D b S ( X ) denote the b ounded constructible deriv ed category o f ( X, S ). D b S ( X ) is the full sub category of the b o unded deriv ed category of shea v es of ab elian groups on X whose ob jects are the cohomologically constructible com- plexes of shea v es o n X ; that is, the complexes whose cohomology sheav es are constructible with resp ect to the stratiﬁcation of X . F or details and references see [7]. W e no te the follo wing: Lemma 4.1. L et ( X , S ) b e a top olo gic al ly str atiﬁe d s p ac e, let T b e the induc e d str atiﬁc ation on (0 , 1) × X , and let π : (0 , 1) × X → X denote the pr oje ction map. The pul lb ack f unc tor π ∗ : D b S ( X ) → D b T  (0 , 1) × X  is an e quivalenc e of c ate gories. Pr o of. A constructible sheaf F on a top ologically stratiﬁed space U of the form U = R k × C L has the prop ert y that H i ( U ; F ) = 0 for i > 0. W e may use this to sho w that R i π ∗ F v anishes for F constructible on ( 0 , 1) × X and i > 0. Indeed, R i π ∗ F is the sheaﬁﬁcation of the presheaf U 7→ H i  (0 , 1) × U ; F )  and since ev ery p oint of X has a fundamen tal system of neigh b orho o ds of the form R k × C L the stalks of this presheaf v a nish; it follo ws that R i π ∗ F v anishes. Th us F → R π ∗ π ∗ F is a quasi-isomorphism for ev ery sheaf F on X , and π ∗ R π ∗ F → F is a quasi-isomorphis m for ev ery constructible sh eaf on (0 , 1) × X , completing the pro of.  If C is a connected stratum of X let i C denote the inclusion map i C : C ֒ → X . Let D b lc ( C ) denote the sub category of D b ( C ) whose ob jects a r e the complexes with lo cally constant cohomology shea v es. Recall the four functors R i C, ∗ , i C, ! : D b lc ( C ) → D b S ( X ) and R i ! C , i ∗ C : D b S ( X ) → D b lc ( C ), and recall the follo wing deﬁnition from [1]: Deﬁnition 4.2. Let ( X , S ) b e a top ologically stratiﬁed space, and let p : C 7→ p ( C ) b e an y function from connected strata of ( X, S ) to Z . F or each connected stratum C , let i C denote the inclusion C ֒ → X . A p erverse she af of p erversity p on X , constructible with resp ect to S , is a complex K ∈ D b S ( X ) suc h that (1) The cohomology shea ves o f i ∗ C K ∈ D b ( C ) v anish ab o v e degree p ( C ) for eac h C . (2) The cohomology shea ves o f R i ! C K ∈ D b ( C ) v a nish b elow degree p ( C ) for each C . Let P ( X , S, p ) denote the full subcategory of D b S ( X ) whose o b jects are t he p erv erse shea ves of p erv ersity p . Ev ery o p en set U ⊂ X inherits a stratiﬁcation from X , and w e ma y form the category D b S ( U ). This deﬁnes a prestac k on X : there is a restriction functor D b S ( U ) → D b S ( V ) deﬁned in the ob vious w ay whenev er V ⊂ U are op en sets in X . It is easy to see that if P is a p erv erse sheaf on U then its restriction to V is also a p erve rse sheaf. W e obtain a prestac k U 7→ P ( U, S, p ). W rite P X,S,p for this prestac k. The follo wing t heorem is a result of [1]: Theorem 4.3. L et X b e a top olo gic al ly str atiﬁe d sp ac e with str atiﬁc ation S . L et p b e any function fr om c onne cte d str a ta of X to inte gers. The pr estack P X,S,p is a stack. W e may easily pro v e, using the criterion in theorem 3.13: Theorem 4.4. L et ( X , S ) b e a top olo gic al ly str atiﬁe d sp ac e. L et p b e any function fr om c onne cte d str ata of X to inte ge rs . The stack P X,S,p is c ons tructible. 14 DA VID TREUMAN N Pr o of. Let U and V b e open s ets in X , and s upp ose V ⊂ U and that the inclusion map V ֒ → U is a strat iﬁed homotop y equiv alence. By lemma 4.1, the restriction map D b S ( U ) → D b S ( V ) is an equiv alence of categories. It follo ws that P ( U ) → P ( V ) is a lso an equiv alence. Th us P is constructible b y t heorem 3.13.  5. The fundament al 2-groupoid and 2-monodrom y In this section w e review the unstratiﬁed v ersion of our main theorem 1.3: we in tro duce the fundamen tal 2-group oid π ≤ 2 ( X ) of a space X and prov e that the 2-catego ry of lo cally constan t stac ks St lc ( X ) is equiv alent to the 2 -category of Cat -v alued functors on π ≤ 2 ( X ). Let us call the latter ob jects “2-mono drom y functors,” and write 2Mon( X ) for the 2-category of 2- mono dromy functors F : π ≤ 2 ( X ) → C at . W e deﬁne a 2-functor N : 2Mon( X ) → St lc ( X ) and pro v e that it is essen tially fully faithful and essen tially surjectiv e. The most imp ortant ingredien t is an analog for π ≤ 2 ( X ) of the classical v an Kamp en theorem; t his is t heorem 5.6. The results of this section are essen tially contained in [9] and [18]. 5.1. The fundamen tal 2-group oid. Let X b e a compactly gene rated Hausdorﬀ space, and let x and y b e t w o po in ts of X . A Mo or e p ath from x to y is a pair ( λ, γ ) where λ is a nonnegativ e real num b er a nd γ : [0 , λ ] → X is a path with γ (0) = x and γ ( λ ) = y . L et us write P ( x, y ) for the space of Moo re paths fro m x to y , giv en the compact-o p en top ology . W e hav e a concatenation map P ( y , z ) × P ( x, y ) → P ( x, z ) deﬁned by the form ula ( λ, γ ) · ( κ, β ) = ( λ + κ, α ) where α ( t ) =  β ( t ) if t ≤ κ γ ( t − κ ) if t ≥ κ If w e giv e the pro duct P ( y , z ) × P ( x, y ) the Kelly top ology (the categorical pro duct in the category of compactly generated Hausdorﬀ spaces), this concatenation map is con tin uous. It is strictly asso ciativ e and the constan t paths from [0 , 0] are strict units. Deﬁnition 5.1. Let π ≤ 2 ( X ) denote the 2 -category whose ob jects are p oints o f X , and whose hom categories Ho m( x, y ) are the fundamen tal group o ids of the spaces P ( x, y ). (The discussion ab o v e show s that this is a strict 2-catego ry .) Remark 5.2. The 2-morphisms in π ≤ 2 ( X ) are techn ically equiv alence classes of paths [0 , 1] → P ( x, y ). A path [0 , 1] → P ( x, y ) b etw een α and β is giv en b y a pair ( b, H ) where b is a map [0 , 1] → R ≥ 0 , and H is a map from the closed region in [0 , 1] × R ≥ 0 under the graph of b : b EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 15 H is required to tak e the to p curv e to y , the b ottom curv e to x , a nd to map the left and righ t in terv als in to X by α and β . It is inconv enien t and unnecessary to kee p trac k of t he function b : there is a reparameterization map from Mo ore paths to ordinary (length 1) paths whic h tak es ( λ, γ ) to the pa th t 7→ γ ( λ · t ). This map is a homotop y equiv alence, so it induces an equiv alence of fundamen tal group oids. Thus , 2- morphisms f rom α to β ma y b e represen ted b y homoto py classes of maps H : [0 , 1 ] × [0 , 1] → X with the pro p erties (1) H (0 , u ) = α ( s · u ), where s is the length of the path α . (2) H (1 , u ) = β ( t · u ), where t is the length of the path β . (3) H ( u , 0) = x and H ( u, 1) = y . 5.2. Two-m ono drom y and lo cally c onstan t stacks. Deﬁnition 5.3. Let X b e a compactly generated Hausdorﬀ space. Let 2 Mon( X ) denote the 2 - category of 2-functors from π ≤ 2 ( X ) to the 2- cat ego ry of categories: 2Mon( X ) := 2F unct  π ≤ 2 ( X ) , Cat  Let U ⊂ X b e an op en set. The inclus ion morphism U ֒ → X induces a strict 2- f unctor π ≤ 2 ( U ) → π ≤ 2 ( X ); let j U denote this 2-functor. If F : π ≤ 2 ( X ) → Cat is a 2-mono drom y functor on X set F | U := F ◦ j U . Deﬁnition 5.4. Let X b e a compactly generated Hausdorﬀ space. Let N : 2Mon( X ) → Prest( X ) denote the 2-functor whic h assigns to a 2- mono dromy functor F : π ≤ 2 ( X ) → Cat the prestack N F : U 7→ 2lim ← − π ≤ 2 ( U ) F | U Our goa l is t o prov e that when X is lo cally con tractible N giv es a n equiv alence of 2- categories b et w een 2Mon( X ) and St lc ( X ); this is theorem 5.7. 5.3. A v an Kampen theorem for t he fundamen tal 2-groupoid. Let X b e a compactly generated Hausdorﬀ space. Let { U i } i ∈ I b e a d -co v er of X . (By this w e just mean t ha t { U i } i ∈ I is an op en cov er of X closed under ﬁnite in tersections; then I is partially ordered by inclusion. See app endix A.) An ideal v an Kamp en theorem w ould state that the 2-categor y π ≤ 2 ( X ) is the direct limit (or “direct 3-limit”) of the 2-categories π ≤ 2 ( U i ). W e do not wish to dev elop the relev a n t deﬁnitions here. Instead, we will relate the 2 -category π ≤ 2 ( X ) to the 2-categories π ≤ 2 ( U i ) b y studying 2-mono drom y functors. W e will deﬁne a 2-cat ego ry 2Mon  { U i } i ∈ I  of “2-mono dro m y functors on the d -cov er,” and o ur v an K amp en theorem will state that this 2-category is equiv alen t to 2Mon( X ). If U is an op en subset of X , the inclusion morphism U ֒ → X induces a 2- functor π ≤ 2 ( U ) → π ≤ 2 ( X ). Let us denote b y ( − ) | U the 2-functor 2Mon( X ) → 2Mon( U ) obtained by comp osing with π ≤ 2 ( U ) → π ≤ 2 ( X ). Deﬁnition 5.5. Let { U i } i ∈ I b e a d - co v er of X . A 2 -mono dr omy functor on { U i } i ∈ I consists of the follow ing data: (0) F or eac h i ∈ I , a 2-mono drom y functor F i ∈ 2Mon( U i ). (1) F or eac h i, j ∈ I with U j ⊂ U i , an equiv alence of 2-mono drom y functors F i | U j ∼ → F j . 16 DA VID TREUMAN N (2) F or eac h i, j, k ∈ I with U k ⊂ U j ⊂ U i , a n isomorphism b etw een the comp osite equiv alence F i | U j | U k ∼ → F j | U k ∼ → F k and the equiv alence F i | U k ∼ → F k suc h t ha t the follo wing condition holds: (3) F or eac h i, j, k , ℓ ∈ I w ith U ℓ ⊂ U k ⊂ U j ⊂ U i , the tetrahedron comm utes: F i | U j | U k | U ℓ / / & & M M M M M M M M M M M M   F j | U k | U ℓ   = F i | U j | U k | U ℓ / /   F j | U k | U ℓ   x x q q q q q q q q q q F k | U ℓ / / F ℓ F k | U ℓ / / F ℓ The 2-mono dromy functors on { U i } i ∈ I form the ob jects of a 2-category in a natural w a y . If F is a 2 - mono dromy functor o n X then we may form a 2-mono dro my functor on { U i } i ∈ I b y setting F i = F | U i , a nd taking a ll the 1- morphisms and 2- morphisms to b e iden tities. This deﬁnes a 2-functor 2Mon( X ) → 2Mon( { U i } ); let us denote it b y r es . Theorem 5.6 (v an Ka mp en) . L et X b e a c omp actly gener ate d Hausdorﬀ sp ac e, and let { U i } i ∈ I b e a d -c over of X . The natur a l 2-functor r es : 2Mon( X ) → 2Mon( { U i } i ∈ I ) is an e quivalen c e of 2-c ate gories. W e will pro v e t his in section 5.5. L et us ﬁrst use this result to deriv e our 2-mono dromy theorem. 5.4. The 2-mono drom y theorem. Theorem 5.7. L et X b e a c omp actly gen e r ate d Hausdorﬀ sp ac e, and let F b e a 2-mono dr omy functor on X . The pr e stack N F is a s tack . F urthermor e, if X is lo c al ly c ontr actible, the stack N F is lo c al ly c on stant, e ach stalk c ate gory ( N F ) x is natur al ly e quivale nt to F ( x ) , and the 2-functor N : 2Mon( X ) → St lc ( X ) is an e quivalenc e o f 2-c ate gories. Pr o of. Let G b e another 2-mono dromy f unctor on X , and let N ( G, F ) b e the prestac k U 7→ Hom 2Mon( U ) ( G | U , F | U ). It is useful to show that N ( G, F ) is a stac k; we o bt a in that N F = N ( ∗ , F ) is a stac k as a sp ecial case. Let U ⊂ X b e an open set, and let { U i } i ∈ I b e a d - co v er of U . T o see that the natural functor N ( G, F )( U ) → 2lim ← − I N ( G, F )( U i ) is an equ iv alence of categories no t e that 2lim ← − I N ( G, F )( U i ) is equiv a len t to the category of 1- morphisms from r es ( G | U ) to r es ( F | U ). Here r es ( G | U ) and r es ( F | U ) denote the 2-mono drom y functors on the d -cov er { U i } induced b y the 2-mono drom y functors G | U and F | U on U . By theorem 5.6 , r es induces an equiv alence on hom catego r ies. Th us N ( G, F ) is a stac k. Let U and V b e con tractible op en subse ts of X with V ⊂ U . Then b oth π ≤ 2 ( V ) and π ≤ 2 ( U ) are trivial, so π ≤ 2 ( V ) → π ≤ 2 ( U ) in an equiv alence. It f ollo ws that N ( G, F )( U ) → N ( G, F )( V ) is an equiv alence of categories. If X is lo cally con tractible then by theorem 2.9 N ( G, F ) is lo cally constan t. In fa ct if U is contractible and x ∈ U , the triviality of the 2-category π ≤ 2 ( U ) sho ws that N ( G, F )( U ) is naturally equiv alent to the category of functors from G ( x ) to F ( x ), th us t he stalk N ( G, F ) x is equiv alent to F unct( G ( x ) , F ( x )). EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 17 No w supp ose X is lo cally con tractible, and let us show that N : 2Mon( X ) → St lc ( X ) is essen tially fully faithful: w e hav e to sho w that Hom 2Mon( X ) ( G, F ) → Hom St( X ) ( N G, N F ) is an equiv a lence of categories. In fact we will sho w tha t the morphism o f stac ks N ( G, F ) → Hom ( N G, N F ) is an equiv a lence. (Here Hom( N G, N F ) is the stac k on X that tak es an o p en set U to the category of 2-natural transformations Hom ( N G ( U ) , N F ( U )).) It suﬃces to sho w that eac h of the functors N ( G, F ) x → Hom ( N G, N F ) x b et w een stalks is an equiv alence of categories; b oth these categories are naturally equiv a len t to the category of functors F unct  G ( x ) , F ( x )  . Finally let us show that N : 2Mon( X ) → St lc ( X ) is essen tially surjectiv e. F or eac h 1-category C , if F is the constan t C -v alued 2-mono drom y functor on X then N F is the constan t stac k with ﬁb er C : the obvious map from the constan t prestac k C p ; X to N F induces an equiv alence on stalks. Thus ev ery constan t stac k is in the essen tial imag e of N . L et C b e a lo cally constan t stac k on X , and let { U i } i ∈ I b e a d -co v er of X ov er whic h C trivializes. Then w e ma y form a 2-mono drom y functor on the d -cov er as follows : for eac h i ∈ I w e ma y ﬁnd an F i (a constant functor) and an equiv alence N F i ∼ = C | U i ; then for eac h i, j ∈ I w e ma y form the comp o site equiv alence F i | U j ∼ = C | U i | U j = C | U j ∼ = F j ; etc. By theorem 5.6, this descends to a 2- mono dromy functor F on X , and N F is equiv alent to C . This completes the pro of.  5.5. The pro of of the v an Kamp en theorem. Before pro ving theorem 5.6 let us discuss homotopies in more detail. W e wish to sho w that an y 2 - morphism in π ≤ 2 ( X ) may b e f actored in to smaller 2-morphisms, where “small” is in terpreted in terms of an op en co v er of X . Deﬁnition 5.8. Let X b e a compactly generated Hausdorﬀ space. Let { U i } i ∈ I b e a d - co v er of X . A homotop y h : [0 , 1] × [0 , 1] → X is i -elementary if there is a subin terv al [ a, b ] ⊂ [0 , 1] suc h that h ( s, t ) is indep enden t of s so long as t / ∈ [ a, b ], and suc h tha t the image of [0 , 1] × [ a, b ] ⊂ [0 , 1] × [0 , 1] under h is contained in U i . If a homoto p y h is i -elemen tary for some unsp eciﬁed i ∈ I the n w e will simply call h elementary . Let X and { U i } b e as in the deﬁnition. Let x, y ∈ X be p oints, α, β ∈ P ( x, y ) b e Mo o re paths, and let h : [0 , 1] × [0 , 1] → X b e a homotopy from α to β . (See remark 5.2.) Supp ose w e hav e paths γ 0 , γ 1 , α ′ and β ′ , and a homotopy h ′ : α ′ → β ′ , suc h that α = γ 1 · α ′ · γ 0 , β = γ 1 · β ′ · γ 0 , and h = 1 γ 1 · h ′ · 1 γ 0 . h ′ α ′ β ′ γ 0 γ 1 Then h is an i -elemen tary homo t op y if and only if the image of h ′ lies in U i . An y i - elemen ta ry homotopy ma y b e written as γ 1 · h ′ · γ 0 for some γ 0 , h ′ , γ 1 . Prop osition 5.9. L et X b e a c om p actly gener a te d Hausdorﬀ sp ac e, and let { U i } i ∈ I b e a d -c ov er of X . L et α and β b e two Mo or e p a ths fr om x to y , and let h : [0 , 1] × [0 , 1] → X b e 18 DA VID TREUMAN N a homotopy fr om α to β . Then ther e is a ﬁnite list α = α 0 , α 1 , . . . , α n = β of Mo or e p aths fr om x to y , and of homotopies h 1 : α 0 → α 1 , h 2 : α 1 → α 2 , . . . h n : α n − 1 → α n such that h is homo topi c to h n ◦ h n − 1 ◦ . . . ◦ h 1 , and such that e ach h i is elementary. Pr o of. Pic k a con tin uous triangulation of [0 , 1 ] × [0 , 1] with the prop erty that eac h tr iangle is mapp ed by h in to one of the U i . Let n b e the n um b er of triangles, and supp ose w e hav e constructed a n appro priate f actorization whenev er the square may b e triangulated with few er than n triangles. Pic k an edge along { 0 } × [0 , 1]; this edge is incid en t with a unique triangle σ , as in the dia gram σ ❅ ❅   W e ma y ﬁnd a ho meomorphism η b et w een the complemen t of σ in this square with another square suc h that the comp o sition [0 , 1] × [0 , 1] η ∼ = closure  [0 , 1] × [0 , 1] − σ  → X ma y be triangulated with n − 1 triangles. Let us denote this comp osition b y g . O n t he other hand it is clear how to parameterize the union of σ and { 0 } × [0 , 1] b y an elemen ta r y homotop y: ❅ ❅   − → Let us write k : [0 , 1] × [0 , 1] → X for the comp o sition of this para meterization with h . Now k is an elem en tary homotop y and g ma y b e factored in to elemen tary homotopies b y induction. The mapping cylinders on the ho meomorphism η and the par a meterization of σ ∪ { 0 } × [0 , 1] form a homotopy b et w een h and g ◦ k .  W e a lso need a notio n of elemen tary 3- dimensional homotop y . Deﬁnition 5.10. Let X b e a compactly generated Ha usdorﬀ space, and let { U i } i ∈ I b e a d -co v er o f X . Let x, y ∈ X , α, β ∈ P ( x, y ), and let h 0 , h 1 : [0 , 1] × [0 , 1] → X be homotopies from α to β . A homotopy t 7→ h t b et w een h 0 and h 1 is called i -elem entary if there is a closed rectangle [ a, b ] × [ c, d ] ⊂ [0 , 1] × [0 , 1] suc h that (1) h t ( u, v ) is indep enden t of t for ( u , v ) / ∈ [ a, b ] × [ c, d ] (2) F or eac h t , h t ([ a, b ] × [ c , d ]) ⊂ U i . EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 19 Prop osition 5.11. L et X , { U i } i ∈ I , x, y , α, β b e as in deﬁnition 5.10. L e t h and g b e ho- motopies fr om α to β . Supp ose that h an d g ar e homotopic. Then ther e is a se quenc e h = k 0 , k 1 , . . . , k n = g of homotopies fr om α to β such that k i is homotopic to k i +1 via a n elementary homotopy. Pr o of. Note that the homotopy b et w een h and a factorization h m ◦ . . . ◦ h 1 constructed in prop ostion 5.9 is giv en by a sequence of elemen tary 3- dimensional homotopies. T h us we ma y assume that h is of the fo rm h m ◦ . . . ◦ h 1 , where eac h h i is an elemen tary homotopy , and that g is of the form g ℓ ◦ . . . ◦ g 1 where eac h g i is an elemen tary homotop y . By induction we ma y reduce to t he case where m = ℓ = 1 so that h and g are b oth elemen tary . Supp ose that H : [0 , 1] × [0 , 1] × [0 , 1] is a homotop y b et w een h and g . W e may triangulate [0 , 1] × [0 , 1] × [0 , 1] in suc h a w a y that eac h simplex σ is carried b y H into one of the c harts U i . W e may use these simplices to factor H j ust as in prop osition 5.9.  No w w e ma y pro v e theorem 5.6. T o sho w that r es : 2Mon( X ) → 2 Mon  { U i }  is an equiv- alence of 2-categories it suﬃces to show that r es is essen tially f ully faithful and essen tially surjectiv e. This is the conte n t of the following three prop ositions. Prop osition 5.12. L et X b e a c omp a ctly gener ate d Hausdorﬀ sp ac e, and let { U i } i ∈ I b e a d -c ov er of X . L et F and G b e two 2-mono d r omy functors on X . Th e functor Hom( F , G ) → Hom  r es ( F ) , r es ( G )  induc e d by r es is ful ly faithful. Pr o of. Let n and m b e tw o 2-natura l transformations F → G , and let φ and ψ b e tw o mo diﬁcations n → m . Since we ha v e φ = ψ if and only if φ x = ψ x for eac h x ∈ X , and since φ x = r es ( φ ) x,i and ψ x = r es ( ψ ) x,i whenev er x ∈ U i , w e ha v e r es ( φ ) = r e s ( ψ ) if and o nly if φ = ψ . This prov es that the functor induced b y r es is faithful. Let n and m b e a s b efore, and no w let { φ x,i } b e a 2- morphism b et w een r es ( n ) and r es ( m ). The 2- mo r phisms φ x,i : n ( x ) → m ( x ) ar e necessarily indep enden t of i , since whenev er x ∈ U j ⊂ U i w e ha v e φ x,j ◦ 1 n ( x ) = 1 m ( x ) ◦ φ x,i . W rite φ x = φ x,i for this common v alue. Since { φ x,i } is a 2-morphism in 2Mon  { U i } i ∈ I  , ev ery path γ : x → y whose image is con tained in one o f the U i induces a comm utativ e diagram n ( x ) n ( γ ) / / φ x   n ( y ) φ y   m ( x ) m ( γ ) / / m ( y ) It follow s that this diagram comm utes fo r ev ery path γ , sinc e ev ery γ may b e written as a concatenation γ N · . . . · γ 1 of paths γ k with the pro p ert y that for eac h k there is a n i suc h that the image of γ k is con tained in U i . Thus , x 7→ φ x is a 2-morphism n → m , and r es  { φ x }  = { φ x,i } , so the f unctor induced b y r es is full.  Prop osition 5.13. L et X , { U i } i ∈ I , F , and G b e as in pr op osition 5.12. The functor Hom( F , G ) → Hom  r es ( F ) , r es ( G )  induc e d by r es is essential ly surje ctive. Pr o of. Let { n i } b e a 1-morphism r es ( F ) → r es ( G ). F o r eac h i a nd eac h x with x ∈ U i w e are giv en a functor n i ( x ) : F ( x ) → G ( x ), and for eac h j with x ∈ U j ⊂ U i w e a re giv en a natural 20 DA VID TREUMAN N isomorphism ρ ij ; x : n i ( x ) ∼ → n j ( x ) which mak es certain diagrams comm ute. In particular, if w e ha v e x ∈ U k ⊂ U j ⊂ U i , then ρ j k ; x ◦ ρ ij ; x = ρ ik ; x . Let us tak e n ( x ) := lim − → i ∈ I | U i ∋ x n i ( x ) Since the limit is ﬁltered and each n i ( x ) → n j ( x ) is an isomorphism, the limit exis ts and all the nat ur a l maps n i ( x ) → n ( x ) are isomorphisms. T o sho w that { n i } is in the essen tial image o f r es w e will extend t he assignmen t x 7→ n ( x ) to a 1-morphism F → G . T o do this w e need to deﬁne an isomorphis m n ( γ ) : G ( γ ) ◦ n ( x ) ∼ → n ( y ) ◦ F ( γ ) for ev ery path γ starting at x and ending at y . In case the image of γ is en tirely con tained in U i for some i , deﬁne n ( γ ) to b e the comp osition G ( γ ) ◦ n ( x ) ∼ ← G ( γ ) ◦ n i ( x ) n i ( γ ) → n i ( y ) ◦ F ( γ ) ∼ → n ( y ) ◦ F ( γ ) By naturality of the morphisms ρ ij ; x , this map is indep endent o f i . F o r general γ w e may ﬁnd a factorizat io n γ = γ 1 · . . . · γ N where eac h γ k is contained in some U ℓ , a nd deﬁne n ( γ ) = n ( γ 1 ) n ( γ 2 ) . . . n ( γ k ). Let x and y b e p oints in X , let α and β b e tw o paths from x to y , and let h b e a homotop y from α to β . T o sho w that the assignmen ts x 7→ n ( x ) and γ 7→ n ( γ ) form a 1-morphism F → G , w e ha v e to sho w that n ( α ) and n ( β ) mak e the follo wing square comm ute: n ( y ) ◦ F ( α ) n ( y ) F ( h ) / / n ( α )   n ( y ) ◦ F ( β ) n ( β )   G ( α ) ◦ n ( x ) G ( h ) n ( x ) / / G ( β ) ◦ n ( x ) By prop osition 5.9 w e may assum e h is elemen tary . An elemen ta ry homot o p y may be factored as h = 1 γ · h ′ · 1 δ where the imag e of h ′ lies in U i , so we ma y a s w ell assume the image of h lies in U i . In that case the diagram ab o v e is equiv alen t to n i ( y ) ◦ F | i ( α ) n i ( y ) F | i ( h ) / / n i ( α )   n i ( y ) ◦ F | i ( β ) n i ( β )   G | i ( α ) ◦ n i ( x ) G | i ( h ) n i ( x ) / / G | i ( β ) ◦ n i ( x ) whic h commute s b y assumption. (Here F | i and G | i denote the restrictions o f F and G to π ≤ 2 ( U i ).) The na tural isomorphisms n i ( x ) → n ( x ) assem ble to an isomorphism b etw een r es ( n ) and { n i } , completing the pro of.  Prop osition 5.14. L et X b e a c omp ac tly gener ate d Hausdorﬀ sp ac e, and let { U i } b e a d -c ov er of X . The natur a l 2-functor r es : 2Mon( X ) → 2Mon  { U i }  is essen tial ly surje c tive . Pr o of. Let { F i } b e an ob ject o f 2Mon  { U i } i ∈ I  . F or eac h p oint x ∈ X let F ( x ) denote the category F ( x ) := 2lim − → i ∈ I | x ∈ U i F i ( x ) EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 21 Since I is ﬁltered and eac h of the maps F i ( x ) → F j ( x ) is an equiv alence, the natural map F i ( x ) → F ( x ) is an equiv alence of categories for eac h i . W e wish to extend the assignmen t x 7→ F ( x ) to a 2-functor π ≤ 2 ( X ) → Cat . Let γ b e a path b etw een p oints x and y in X . If the image of γ is con tained in some U i then w e may form F ( γ ) : F ( x ) → F ( y ) by taking t he direct limit ov er i of the functors c i,γ : F ( x ) → F ( y ) , where c i,γ is the comp osition F ( x ) ∼ ← F i ( x ) F i ( γ ) → F i ( y ) ∼ → F ( y ) Whenev er U j ⊂ U i the natural transformat io n c i,γ → c j,γ induced by the comm utativ e square F i ( x ) F i ( γ ) / /   F i ( y )   F j ( x ) F j ( γ ) / / F j ( y ) is an isomorphism, and the limit is ﬁltered, so each of the maps c i,γ → F ( γ ) is an isomor- phism. No w fo r each path γ not necessarily contained in one c hart, pick a factorization γ = γ 1 · . . . · γ N with the prop erty that for eac h ℓ there is a k suc h that the image of γ ℓ lies in U k . If x ℓ and x ℓ +1 denote the endp oin ts of γ ℓ , let F ( γ ) : F ( x ) → F ( y ) b e the functor giv en b y the comp osition F ( x ) = F ( x 1 ) F ( γ 1 ) → F ( x 2 ) F ( γ 2 ) → . . . F ( γ N ) → F ( x N +1 ) = F ( y ) Supp ose h is a homotopy b etw een paths α and β with the prop erty that the imag es of α , β , a nd h lie in a single c hart U i . Then deﬁne a na t ural transformat io n F ( h ) : F ( α ) → F ( β ) to b e the comp osition F ( α ) ∼ ← F i ( α ) F i ( h ) → F i ( β ) ∼ → F ( β ) If h = 1 γ 1 · h ′ · 1 γ 0 is an elemen tary homotopy , suc h that the image of h ′ lies in some U i , deﬁne F ( h ) = 1 F ( γ 1 ) · F ( h ′ ) · 1 F ( γ 0 ) . If g is an arbitrar y homoto py , let g n ◦ g n − 1 · · ·◦ g 1 b e a comp osition of elemen tary homotopies tha t is homotopic to g , and deﬁne F ( g ) = F ( g n ) ◦ · · · ◦ F ( g 1 ). Th e g i exist b y prop osition 5.9, and the f orm ula for F ( g ) is indep enden t of the f actorization b y prop osition 5.11. W e may extend F to all elemen tary homotopies, since any elemen tary homotop y can b e written as 1 γ 1 · h ′ · 1 γ 0 where the image of h ′ lies in some U i ; it follows that if h and g are elemen ta ry ho mo t o pies that are themselv es homotopic by an eleme n tary homoto p y , then F ( h ) = F ( g ). By prop ositions 5.9 and 5.11 this is we ll-deﬁned. The maps F i ( x ) → F ( x ) assem ble to a map { F i } → r e s ( F ) in 2Mon  { U i }  . As eac h F i ( x ) → F ( x ) is an equiv alence b y construction, this sho ws that { F i } is equiv alen t to r es ( F ), so t ha t r es is essen tially surjectiv e.  22 DA VID TREUMAN N 6. Stra tified 2- trunca tions and 2-monodromy In this sec tion w e dev elop a n abstract ve rsion of our main theorem. W e in tro duce the notion of a str atiﬁe d 2-trunc ation . A stratiﬁed 2-truncation − → π ≤ 2 is a strict functorial assign- men t from top ologically stratiﬁed spaces to 2- categories satisfying a few axioms. W e show that these axioms guaran tee that the 2- category of 2-functors f rom − → π ≤ 2 ( X , S ) to Cat is equiv alen t to the 2-catego ry of S -constructible stac ks on X . Let Strat denote the categor y of t o p ologically stratiﬁed s paces a nd stra t um-preserving maps b etw een t hem. W e will consider functors f r o m St rat to the category (that is, 1 - category) of 2 - categories and strict 2-functors; w e will denote the latter category by 2c at . Th us, such a functor − → π ≤ 2 consists of (1) an a ssignmen t ( X, S ) 7→ − → π ≤ 2 ( X , S ) that take s a top olog ically stratiﬁed space to a 2-category . (2) an assignmen t f 7→ − → π ≤ 2 ( f ) that tak es a stratum-preserving map f : X → Y to a strict 2- functor − → π ≤ 2 ( f ) : − → π ≤ 2 ( X ) → − → π ≤ 2 ( Y ). suc h that for any pair of comp osable strat um- preserving maps X f → Y g → Z , we ha v e − → π ≤ 2 ( g ◦ f ) = − → π ≤ 2 ( g ) ◦ − → π ≤ 2 ( f ). A functor − → π ≤ 2 is called a str a tiﬁe d 2-trunc ation if it satisﬁe s the four axioms b elo w. Tw o of t hese axioms require some more discus sion, but w e will state them here ﬁrst somewhat imprecisely: Deﬁnition 6.1. Let − → π ≤ 2 b e a functor Strat → 2cat . W e will say that − → π ≤ 2 is a str atiﬁe d 2-trunc ation if it satisﬁes the follo wing axioms: (N) Normalization. If ∅ denotes the empty top o logically stratiﬁed space, t hen − → π ≤ 2 ( ∅ ) is the empty 2-catego r y . (H) Homotop y inv a riance. F or eac h to p ologically stratiﬁed s pace ( X, S ), the 2-functor − → π ≤ 2 ( f ) : − → π  (0 , 1) × X, S ′  → − → π ( X, S ) induced b y the pro jection map f : (0 , 1) × X → X is an equiv alence of 2- categories. Here S ′ denotes the stratiﬁcation on (0 , 1) × X induced b y S . (C) C ones. Roughly , for eac h compact top olo g ically stratiﬁed space L , − → π ≤ 2 ( C L ) ma y be iden tiﬁed with the cone on the 2-category − → π ≤ 2 ( L ). See section 6.1 b elo w. (vK) v an Kamp en. Ro ug hly , for ev ery top ologically strat iﬁed space X and ev ery d - cov er { U i } i ∈ I of X , the 2-category − → π ≤ 2 ( X ) is na turally equiv alen t to the direct limit ( o r “3-limit”) ov er i ∈ I of the 2-categories − → π ≤ 2 ( U i ). See section 6.2 b elo w. 6.1. Cones on 2-categories and axiom (C). If C is a 2- category , let  ∗ ↓ C  denote the 2-category whose ob jects are the ob jects of C together with one new ob ject ∗ , a nd where the hom categories Hom ∗↓ C ( x, y ) are as follows: (1) Hom( x, y ) = Hom C ( x, y ) if b oth x and y a r e in C . (2) Hom( x, y ) is the trivial category if x = ∗ . (3) Hom( x, y ) is the empt y catego r y if y = ∗ and x 6 = ∗ . Deﬁnition 6.2. Let − → π ≤ 2 : Strat → 2cat b e a functor satisfying a xioms (N) and (H) ab ov e. F or eac h compact top ologically stratiﬁed space L , let us endo w (0 , 1) × L and C L with the naturally induced top ological stratiﬁcation. Let us say that − → π ≤ 2 satisﬁes axiom EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 23 (C) if for each compact top ologically stratiﬁed space L t here is an equiv alence of 2-categor ies − → π ≤ 2 ( C L ) ∼ →  ∗ ↓ − → π ≤ 2 ( L )  suc h t ha t (1) the followin g square comm utes up to equiv alence of 2-functors: − → π ≤ 2  (0 , 1) × L  / /   − → π ≤ 2 ( C L )   − → π ≤ 2 ( L ) / /  ∗ ↓ − → π ≤ 2 ( L )  (2) The comp osition − → π ≤ 2  { cone p oin t }  → − → π ≤ 2 ( C L ) →  ∗ ↓ − → π ≤ 2 ( L )  is equiv a lent to the nat ur a l inclusion − → π ≤ 2  { cone p oin t }  ∼ = ∗ →  ∗ ↓ − → π ≤ 2 ( L )  6.2. Exit 2-mono drom y functors and axiom (vK) . Mora lly , the v an Kamp en axiom states that − → π ≤ 2 preserv es direct limits (at least in a diagram of op en immersions). W e ﬁnd it incon v enien t to deﬁne a direct 3-limit of 2-categories directly; we will instead form ulate it in t erms of catego ry-v alued 2-functors on the 2-categores − → π ≤ 2 ( X ), as in section 5.3. In this section, ﬁx a functor − → π ≤ 2 : Strat → 2cat . Deﬁnition 6.3. Let ( X , S ) b e a top olog ically stratiﬁed space. An exit 2 -mono dr omy functor on ( X , S ) with resp ect to − → π ≤ 2 is a 2- f unctor − → π ≤ 2 ( X , S ) → Cat . W rite 2Exitm( X, S ) = 2Exitm( X , S ; − → π ≤ 2 ) for the 2- category of exit 2-mono drom y functors o n ( X , S ) with respect to − → π ≤ 2 . Deﬁnition 6.4. Let ( X , S ) b e a top o logically stratiﬁed space. Let { U i } i ∈ I b e a d -cov er of X . Endo w eac h U i with the top ological stratiﬁcation S i inherited from S . An exit 2-mono dr omy functor on { U i } i ∈ I , with resp ect to − → π ≤ 2 consists of the follo wing data: (0) F or eac h i ∈ I a 2-mono dro m y functor F i ∈ 2Exitm( U i , − → π ≤ 2 ). (1) F or eac h i, j ∈ I with U j ⊂ U i an equiv alence of exit 2-mono drom y functors F i | U j ∼ → F j . (2) F or each i, j, k ∈ I with U k ⊂ U j ⊂ U i an isomorphism b etw een the comp osite equiv alence F i | U j | U k ∼ → F j | U k ∼ → F k and the equiv alence F i | U k ∼ → F k suc h t ha t the follo wing condition holds: (3) F or eac h i, j, k , ℓ ∈ I w ith U ℓ ⊂ U k ⊂ U j ⊂ U i , the tetrahedron comm utes: F i | U j | U k | U ℓ / / & & M M M M M M M M M M M M   F j | U k | U ℓ   = F i | U j | U k | U ℓ / /   F j | U k | U ℓ   x x q q q q q q q q q q F k | U ℓ / / F ℓ F k | U ℓ / / F ℓ W rite 2Exitm  { U i } i ∈ I , − → π ≤ 2  for the 2-category of exit 2-mono drom y functors on { U i } . Let X b e a top ologically strat iﬁed space, and let { U i } i ∈ I b e a d -co v er of X . Denote b y r es the natural strict 2-functor 2Exitm( X ) → 2Exitm  { U i } i ∈ I  . Deﬁnition 6.5. L et − → π ≤ 2 b e a 2-functor Strat → 2cat . W e sa y that − → π ≤ 2 ( X ) satisﬁes axiom (vK) if r es : 2Exitm( X ) → 2Exitm  { U i } i ∈ I  is an equiv alence of 2 -categories for ev ery top ologically stratiﬁed space X and ev ery d - co v er { U i } i ∈ I of X . 24 DA VID TREUMAN N 6.3. The exit 2-monodromy theorem. In this section ﬁx a stratiﬁed 2-tr uncatio n − → π ≤ 2 . Deﬁnition 6.6. Let ( X , S ) b e a topolo gically stratiﬁed sp ace. F or eac h op en se t U ⊂ X , let S U denote the induced stratiﬁcation of U and let j U denote the inclusion map U ֒ → X . Let N : 2Exitm( X , S ) → Prest( X ) denote the 2- functor whic h assigns to a n exit 2-mono drom y functor F : − → π ≤ 2 ( X , S ) → Cat the prestac k N F : U 7→ 2lim ← − − → π ≤ 2 ( U,S U ) F ◦ − → π ≤ 2 ( j U ) W e wish to pro v e that N is an equiv alence of 2Exitm( X , S ) on to the 2- category St S ( X ). W e need a preliminary result ab out constructible stac ks on cones. Deﬁnition 6.7. Let ( L, S ) b e a top ologically stratiﬁed space. Let  Cat ↓ St S ( L )  denote the 2 - category whose ob jects are triples ( C , C , φ ), where (1) C is a 1-category . (2) C is a constructible stac k on L (3) φ is a 1-morphism C L → C , where C L denotes the constan t stac k o n L . If ( L, S ) is a compact top ologically stra t iﬁed space, let S ′ denote the induced stratiﬁcatio n on (0 , 1) × L a nd S ′′ the induced stratiﬁcation on C L . There is a 2-functor St S ′′ ( C L ) → ( Cat ↓ St S ′ ((0 , 1) × L )) ∼ = ( Cat ↓ St S ( L )) whic h asso ciates to a stack C the triple  C ( X ) , C | (0 , 1) × L , φ  , where φ is the eviden t restriction map. Deﬁnition 6.8. Let ( L, S ) b e a top o lo gically stra t iﬁed space. Let  Cat ↓ 2Exitm( L )  denote the 2-category whose ob jects are triples ( C , F , φ ) where (1) C is a 1-category . (2) F is an exit 2-mono dr o m y functor on L . (3) φ is a 1-morphism from t he constan t C -v alued functor to F . Note that the equiv alence − → π ≤ 2 ( C L ) ∼ →  ∗ ↓ − → π ≤ 2 ( L )  giv es a n equiv alence ( Cat ↓ 2Exitm( L )) ∼ → 2Exitm( C L ) . Prop osition 6.9. L et L b e a c omp act top olo gic al ly s tr atiﬁe d sp ac e, a nd let C L b e the op e n c one on L . The 2-functor St S ′′ ( C L ) →  Cat ↓ St S ( L )  is an e quivalenc e of 2-c ate gories. F urthermor e, the squar e 2Exitm( C L, S ′′ ) N / /   St S ′′ ( C L )    Cat ↓ 2Exitm( L, S )  N / /  Cat ↓ St S ( L )  c omm utes up to an e quivalen c e of 2-functors. EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 25 Pr o of. The 2-functor St S ′′ ( C L ) →  Cat ↓ St S ′  (0 , 1) × L  is inv erse to the 2-functor that tak es an ob ject ( C , C , φ ) to the unique stac k giv en by the form ula U 7→  C if U is of the form C ǫ L = [0 , ǫ ) × L/ { 0 } × L C ( U ) if U does not con tain the cone p oin t  Theorem 6.10. L et ( X, S ) b e a top olo gic al ly str atiﬁe d sp ac e, and let F b e an exit 2- mono d r omy functor on ( X , S ) . The pr estack N F is an S -c onstructible stack, an d the 2- functor N : 2Exitm( X, S ) → St S ( X ) is an e quivale n c e of 2- c ate gories. Pr o of. W e will follo w the proo f o f theorem 5.7 . Let G b e another exit 2-mono dromy functor on X , and once again let N ( G, F ) b e the prestac k U 7→ Ho m 2Exitm( U ) ( G | U , F | U ). As in t he pro of of theorem 5.7, the v an Kamp en prop ert y of − → π ≤ 2 (axiom (vK)) implies N ( G, F ) is a stac k. By the homoto p y a xiom (H), − → π ≤ 2 ( V ) → − → π ≤ 2 ( U ) is an equiv alence of 2-categories whenev er V ⊂ U are op en sets and V ֒ → U is a lo o se stratiﬁed homotop y equiv alence. It follo ws t ha t the stac ks N ( G, F ) are constructible b y theorem 3.13. In particular N F is a constructible stack. T o see t ha t N : 2Exitm ( X , S ) → St S ( X ) is es sen tially fully f a ithful, it suﬃces to sho w that N ( G, F ) → Hom ( N G, N F ) is an equiv a lence of stack s, and w e may c hec k this on stalks. W e will induct on the dimension of X : it is clear that this morphism is an equiv alence of stac ks when X is 0- dimensional, so supp ose w e ha v e pro v en it a n equiv alence for X of dimension ≤ d . Let x ∈ X and le t U b e a conical neighbor ho o d of x . The morphism N ( G, F ) x → Hom ( N G, N F ) x is equiv alen t to the morphism N ( G, F )( U ) → Hom ( N G | U , N F | U ), and b y the stratiﬁed homoto p y equiv alence U ≃ C L w e ma y as we ll assume U = C L . Let T denote the stratiﬁcation on L . By prop o sition 6 .9, we hav e to sho w that the 2-f unctor  Cat ↓ 2Exitm( L, T )  →  Cat ↓ St T ( L )  is an equiv alence, but this map is induced b y N : 2Exitm( L, T ) → St T ( L ) whic h is an equiv alence by induction. Finally let us sh ow that N : 2Exitm( X, S ) → St S ( X ) is esse n tially surjectiv e. Again let us induct on the dimension of X . Let C b e a constructible stac k on X . The restriction of C to a conical op en set U ∼ = R d × C L is in the essen tial image of N : 2Exitm( U, S U ) → St S U ( U ) b y induction and prop osition 6.9. W e may ﬁnd a d - cov er { U i } i ∈ I of X g enerated b y conical op en sets, so that f or eac h i there is an F i ∈ 2Exitm( U i ) suc h that C | U i is equiv alen t to N F i . These F i assem ble to an exit 2- mono dromy functor on the d -co v er, wh ic h b y axiom (vK) comes f rom an exit 2 -mono dromy functor F on X with N F ∼ = C . This completes the pro of.  7. Exit p a ths in a stra tified sp ace In this section w e iden tify a particular stratiﬁed 2-truncation: the exit-path 2-category E P ≤ 2 . If ( X, S ) is a top ologically stratiﬁed space, then the ob jects of E P ≤ 2 ( X , S ) are the p oin ts of X , the morphisms are Mo ore paths with the “ exit prop ert y” describ ed in the in tro duction, and the 2-morphisms are homotopy classes of homotopies b et w een exit paths, sub j ect to a tameness condition. The purp ose of this section is to giv e a precise deﬁnition of the functor E P ≤ 2 , and to c hec k the axioms 6.1. 26 DA VID TREUMAN N Deﬁnition 7.1. L et X b e a top ologically stratiﬁed space. A path γ : [ a, b ] → X is called an exit p ath if for each t 1 , t 2 ∈ [ a, b ] with t 1 ≤ t 2 , the p oin t γ ( t 1 ) is in the closure of the stratum con taining γ ( t 2 ); equiv alently , if the dimension of the stratum con taining γ ( t 1 ) is not larger than the dimension of the stratum containing γ ( t 2 ). F o r eac h pair of p oints x, y ∈ X let E P ( x, y ) denote the subspace of t he space P ( x, y ) of Mo o re paths (section 5.1) with the ex it prop ert y , starting at x and ending at y . Remark 7.2. If w e wish to emphasize the space X we will sometimes write E P ( X ; x, y ) for E P ( x, y ). 7.1. T ame homotopies. L et ( X , S ) b e a top ologically stratiﬁed space. Let us call a map [0 , 1] n → X tame with respect to S if there is a con tin uous triangulation o f [0 , 1] n suc h that the in terior of ev ery simplex maps into a stratum of X . Note t hat the composition of a tame map [0 , 1] n → X with a stratum- preserving map ( X , S ) → ( Y , T ) is again t a me. If x a nd y ar e tw o p oin ts of X , call a path h : [0 , 1] → E P ( x, y ) tame if the associated homotop y [0 , 1] × [0 , 1] → X is tame with resp ect to S . (See remark 5.2 for how to asso ciate an ordinary “square” homotop y to a homoto p y b etw een Mo ore paths.) Finally if H : [0 , 1] × [0 , 1] is a homotop y b et w een paths h and g in E P ( x, y ), w e call H tame if the associated map [0 , 1] × [0 , 1] × [0 , 1] → X is tame with resp ect to S . Deﬁnition 7.3. Let ( X, S ) be a topolo gically stratiﬁed space, and let x and y b e p oints of X . Let tame ( x, y ) b e the gr o up oid whose ob jects are the p oints of E P ( x, y ) and whose hom sets Ho m tame ( x,y ) ( α, β ) are ta me homotop y classes of tame paths h : [0 , 1] → E P ( x, y ) starting at α and ending at β . The concatenation map E P ( y , z ) × E P ( x, y ) → E P ( x, z ) takes a pair of tame ho motopies h : [0 , 1] → E P ( x, y ) and k : [0 , 1] → E P ( y , z ) to a tame homoto p y k · h : [0 , 1] → E P ( x, z ), and this giv es a w ell-deﬁned f unctor tame ( y , z ) × tame ( x, y ) → tame ( x, z ). It follows w e ma y deﬁne a 2-category: Deﬁnition 7.4. Let ( X , S ) b e a top olog ically stratiﬁed space. L et E P ≤ 2 ( X , S ) denote the 2-category whose ob jects are p oin ts of X and whose hom categories Hom E P ≤ 2 ( X,S ) ( x, y ) are the g r o up oids tame ( x, y ). Remark 7.5. The tameness condition is necessary for our pro of of the v an K a mp en pro p ert y of E P ≤ 2 ( X , S ) (whic h follows the pro of giv en in section 5.5) – it allow s us to sub divide our ho mo t o pies indeﬁnitely . W e can deﬁne a similar 2-category E P naiv e ≤ 2 ( X , S ) whose hom categories are the fundamen tal group oids π ≤ 1  E P ( x, y )  . I b elieve t his 2-category to b e naturally equiv alen t to E P ≤ 2 ( X , S ), a nd that E P naiv e ≤ 2 could b e used in place of E P ≤ 2 in our main theorem. T o pro ve t his one w ould hav e to sho w that the natural functor tame ( x, y ) → π ≤ 1  E P ( x, y )  is an equiv a lence of group oids. I hav e b een unable to obtain suc h a “ t a me appro ximation” result. 7.2. The exit path 2-category is a stratiﬁed 2-tr uncation. As a stratum-preserving map f : ( X, S ) → ( Y , T ) preserv es tameness of maps [0 , 1] n → X , it induces a functor f ∗ : tam e ( x, y ) → tame  f ( x ) , f ( y )  and a strict 2-functor f ∗ : E P ≤ 2 ( X , S ) → E P ≤ 2 ( Y , T ). Th us, E P ≤ 2 is a functor Str at → 2cat . The remainder of this sec tion is dev oted to sho wing that E P ≤ 2 satisﬁes the axioms 6.1 fo r a stratiﬁed 2-truncation. EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 27 Theorem 7.6. The 2-functor E P ≤ 2 : Strat → 2cat satisﬁes axioms (N) and (H) of 6.1. Pr o of. Clearly E P ≤ 2 ( ∅ ) is empt y , so E P ≤ 2 satisﬁes axiom (N). Let us now v erify axiom (H). Let ( X, S ) b e a topo lo gically stratiﬁed sp ace, and let π : (0 , 1) × X → X denote the pro j ection map. The 2-functor π ∗ : E P ≤ 2  (0 , 1) × X  → E P ≤ 2 ( X ) is clearly essen tia lly surjectiv e. Let ( s, x ) and ( t, y ) b e tw o p oints in (0 , 1) × X . T o sho w that π ∗ is essen tially fully faithful w e hav e to sho w that tame  ( s, x ) , ( t, y )  → tame ( x, y ) is an equiv alence of g r oup oids. In fact this map is equiv alen t to the pro jection tame ( s, t ) × tame ( x, y ) → tame ( x, y ), and the group oid tame ( s, t ) coincides with the funda- men tal group oid π ≤ 1  E P ( s, t )  as (0 , 1) has a single stratum. Since E P ( s, t ) is con tractible, this group oid is equiv alen t to the trivial group oid, so the pro jection tame ( s, t ) × tame ( x, y ) → tame ( x, y ) is an equiv alence.  Theorem 7.7. The 2-functor E P ≤ 2 : Strat → 2cat satisﬁes axiom ( C ) of 6.1 Pr o of. Let ( L, S ) b e a compact top ologically stratiﬁed space. Let C L b e the op en cone on L , and let ∗ ∈ C L b e the cone po in t. W e ha v e to sho w tha t for eac h x ∈ C L , the gr o up oid tame ( ∗ , x ) is equiv alent t o the t r ivial group oid. This is clear when x is the cone p oin t, so supp ose x = ( u, y ) ∈ (0 , 1) × L ⊂ C L . Let tame ′ ⊂ tame ( ∗ , x ) denote the full subgroup oid whose ob j ects are the exit paths α of Moo re length 1 (i.e. α : [0 , 1 ] → C L ) with α ( t ) 6 = ∗ for t > 0. Ev ery exit path γ ∈ tame ( x, y ) is clearly tamely homotopic to one in tame ′ ; it follo ws that tame ′ is equiv alent tame ( ∗ , x ). Let W ⊂ E P ( ∗ , x ) b e the subspace of exit paths α with Mo ore length 1 and with α ( t ) 6 = ∗ for t > 0. W is homeomorphic to the space of paths β : (0 , 1] → (0 , 1) × L with the prop ert y that β is an exit path, that β (1) = x , and that for all ǫ > 0, there is a δ > 0 suc h tha t β − 1  (0 , ǫ ) × L  ⊃ (0 , δ ). This space ma y b e expressed as a pro duct W ∼ = W 1 × W 2 , where (1) W 1 is the space of paths α : (0 , 1] → (0 , 1) with α (1) = u and ∀ ǫ ∃ δ suc h that α − 1 (0 , ǫ ) ⊃ (0 , δ ) (2) W 2 is the space of paths β : (0 , 1] → L with β ( 1 ) = y and β has the exit prop ert y . The ﬁrst factor W 1 is contractible via κ t : W 1 → W 1 , where κ t ( α )( s ) = t · s · u + (1 − t ) · α ( s ). The second factor is con tractible via µ t : W 2 → W 2 where µ t ( β )( s ) = β ( t + s − ts ). These con tractions preserv e tameness, and therefore t hey induce an equiv alence b et w een tame ′ and the tr ivial group o id.  Finally w e ha v e to pro v e that E P ≤ 2 satisﬁes the v an Kamp en axiom. L et us ﬁrst discuss elemen ta ry tame homotopies, analogous to the elemen tary homo t opies used in the pro of of the v an Kamp en theorem for π ≤ 2 in section 5.5. Deﬁnition 7.8. Let ( X , S ) b e a top o logically stratiﬁed space, a nd let { U i } i ∈ I b e a d - co v er of X . Let x and y b e p oin ts of X , and let α and β b e exit paths from x to y . A ho motop y h : [0 , 1] × [0 , 1] → X b etw een α a nd β is i -elem entary if there is a subin terv al [ a, b ] ⊂ [0 , 1] suc h that h ( s, t ) is independen t of s so long as t / ∈ [ a, b ], and suc h tha t the image of [0 , 1] × [ a, b ] ⊂ [0 , 1] × [0 , 1 ] under h is con tained in U i . Remark 7.9. Elemen tary homotopies b etw een exit paths ma y be pictured in the same w ay as o rdinary ho mo t o pies, as in ﬁgure 5.5. 28 DA VID TREUMAN N Prop osition 7.10. L et ( X , S ) b e a top olo gic al ly str atiﬁe d sp ac e, a n d let { U i } i ∈ I b e a d -c ov er of X . L et α and β b e exit p a ths fr om x to y , an d let h : [0 , 1] × [0 , 1] → X b e a hom otopy fr om α to β . Then ther e is a ﬁn i te list α = α 0 , α 1 , . . . , α n = β of exit p aths fr om x to y , and of homotopies h 1 : α 0 → α 1 , h 2 : α 1 → α 2 , . . . , h n : α n − 1 → α n such that h is ho m otopic to h n ◦ . . . ◦ h 1 , and such that e ach h i is elementary. Pr o of. As in the pro of of propo sition 5.9, it suﬃces to ﬁnd a suitable triangulation of [0 , 1] × [0 , 1]. In our case a triangula t io n is “ suitable” if eac h triangle is mapp ed in to one of the c harts U i , and if furthermore for eac h triangle σ we may order the v ertices v 1 , v 2 , v 3 in such a w ay t hat h carries the half-op en line segmen t v 1 v 2 − v 1 in to a stratum X k , and the third- op en triang le v 1 v 2 v 3 − v 1 v 2 in to a stratum X ℓ . In that case w e may ﬁnd a parameterization g : [0 , 1] × [0 , 1 ] → σ o f σ with the prop ert y that for eac h t the path [0 , 1] → { t } × [0 , 1] → σ → X has the exit prop ert y in X . W e ma y ﬁnd a triangulation with these prop erties b y pic king a triangulation that is ﬁne enough with resp ect to { U i } , and taking its barycen tric sub division.  W e a lso may discuss elemen tary 3- dimensional homotopies b et w een homotopies b et w een exit paths, and a vers ion of prop osition 5.1 1 holds. Deﬁnition 7.11. Let ( X , S ) be a top ologically stratiﬁed space, and let { U i } i ∈ I b e a d -cov er of X . Let x, y ∈ X , α, β ∈ E P M ( x, y ), and let h 0 , h 1 : [0 , 1 ] × [0 , 1] → X b e ho motopies from α to β . A homotop y t 7→ h t b et w een h 0 and h 1 is called i -ele m entary if there is a closed rectangle [ a, b ] × [ c, d ] ⊂ [0 , 1] × [0 , 1] suc h that (1) h t ( u, v ) is indep enden t of t for ( u , v ) / ∈ [ a, b ] × [ c, d ]. (2) F or eac h t , h t  [ a, b ] × [ c, d ]  ⊂ U i . Prop osition 7.12. L et ( X , S ) , { U i } , x, y , α , β b e as in deﬁnition 7.11. L e t h and g b e homotopies fr om α to β . Supp ose that h and g ar e homotopic. Then ther e is a se quenc e h = k 0 , k 1 , . . . , k n = g of homo topi e s fr om α to β such that, for e ach i , k i is homotopic to k i +1 via an e l e m entary homotopy. Pr o of. Similar to prop osition 5 .11.  Theorem 7.13. The functor E P ≤ 2 : Strat → 2cat satisﬁes axiom (vK). Pr o of. Let X b e a top o logically stratiﬁed space, and let { U i } i ∈ I b e a d -cov er of X . W e ha v e to sho w that r es : 2Exitm( X ) → 2 Exitm  { U i }  is an equiv alence o f 2-categories. It suﬃces to sho w that r es is es sen tially fully faithful and essen tially surjectiv e. The proof s of these facts in the unstratiﬁed case – prop ositions 5.12, 5.13, and 5.14 – ma y b e follow ed almost v erbat im to obtain the same results, after substituting prop o sitions 7.10 and 7.12 for prop ositions 5.9 and 5.11.  7.3. Pro of of the main theorem. W e may now prov e the main theorem stated in the in tro duction. Theorem 7.14. The 2-functor E P ≤ 2 is a str atiﬁe d 2-trunc a tion . Be c ause of this, for a ny top olo gic al ly str atiﬁe d sp ac e ( X, S ) the 2-c ate gory of S -c onstructible stacks on X is natur al ly e quivalen t to the 2-c ate gory of C at - value d 2-functors on E P ≤ 2 ( X , S ) . EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 29 Pr o of. That E P ≤ 2 satisﬁes the axioms of a stratiﬁed 2- truncation is the con ten t of theorems 7.6, 7.7, and 7.13. The conclusion that t he main theorem holds is implied by theorem 6.10.  A cknow le dgements: I w ould lik e to thank Mark Goresky , Andrew Sno wden, and Zhiw ei Y un for many helpful commen ts and conv ersations. I would especially lik e to thank m y advisor, Bob MacPherson. This pap er is adapted from a Ph.D. dissertation written under his direction. Appendix A. St acks The w ord “stac k” has at at least tw o diﬀeren t and related meanings in mathematics. Ma yb e most frequen tly it refers to some kind of geometric ob ject that represen ts a g r oup oid- v alued, rather than a set-v alued, functor. But it ma y also refer to a sheaf of categories, where the sheaf structure and axioms hav e b een mo diﬁed to tak e a ccoun t of the “t w o-dimensional” nature of categories – that is, t o tak e a ccoun t of the fa ct tha t categories a re most naturally view ed as the o b jects of a 2 -category . In this app endix w e dev elop some basic prop erties of stac ks in the second sense. W e will pro ceed in a w ay that emphasizes the similarit y with sheav es. It requires gen- eralizing the ba sic deﬁnitions of category theory , su c h as limits and a djoin t functors, to 2-categories. W e summarize what w e need from the theory of 2-categories in app endix B. A.1. Prestacks. Deﬁnition A.1. A pr estack on a 2- category I is a 2-functor C : I op → Cat . A prestac k o n a top ological space X is a pres tac k on the partially ordered set of op en subsets of X , regar ded as a 2 -category whose ob jects are the o p en sets, whose 1-morphisms are the inclusion maps, and with trivial 2-morphisms. In detail, a prestac k C on a space X consists of: (0) An assignmen t that tak es an op en set U to a category C ( U ). (1) A con tra v ariant assignmen t that tak es a n inclusion V ⊂ U to a restriction functor C ( U ) → C ( V ) (2) F or eac h triple of op en subsets W ⊂ V ⊂ U an isomorphism b etw een C ( U ) → C ( V ) → C ( W ) and C ( U ) → C ( W ) (3) Suc h that the tetrahedron a sso ciat ed to eac h quadruple Y ⊂ W ⊂ V ⊂ U comm utes: C ( U ) / / # # G G G G G G G G G   C ( V )   = C ( U ) / /   C ( V )   { { w w w w w w w w w C ( W ) / / C ( Y ) C ( W ) / / C ( Y ) W rite Pres t ( I ) for the 2-category of prestac ks on a 2-category I , and Prest( X ) for the 2-category o f prestac ks on a space X . Deﬁnition A.2. Let C b e a prestac k o n X . The stalk of C at x ∈ X is the category C x = def 2lim − → U | U ∋ x C ( U ) 30 DA VID TREUMAN N Remark A.3. Let I b e a 1-category . The 2-category of 2-functors I op → Cat is equiv a len t to the 2-category of so- called ﬁb er e d c ate gories ov er I ([10]). The theory of stack s is usually ([10], [6 ], [24]) dev eloped using ﬁb ered categories rather than 2-functors. A.2. Stac ks. A stack is a prestac k on X that satisﬁes a kind of sheaf condition. W e ﬁnd it con v enient to phrase this condition in terms o f 2-limits o v er an op en co v er; in order to mak e this precise w e need our op en co v ers to b e closed under ﬁnite in tersections. W e will call t hese “descen t co v ers” or d -co v ers. Deﬁnition A.4. A d -c over of a space U is a subset I ⊂ O p en( U ) of the set of op en subse ts of U that is closed under ﬁnite inte rsections, and that cov ers U . Let C b e a prestac k on X . If U is an op en subset of X a nd { U i } i ∈ I is a d -co v er of U , then the restriction functors C ( U ) → C ( U i ) assem ble to a functor C ( U ) → 2lim ← − I C ( U i ) Deﬁnition A.5. Let C be a prestac k on a space X . Then C is a stack if for each op en set U ⊂ X and eac h d -co v er { U i } i ∈ I of U , the restriction functor C ( U ) → 2lim ← − I C ( U i ) is an equiv alence of categories. L et St( X ) ⊂ Prest( X ) denote the full sub catego ry of the 2-category o f prestac ks on X whose ob j ects a re stac ks. Theorem A.6. (1) L et P and C b e pr es tack s on X . Supp ose C is a stack. The pr estack Hom ( P , C ) o n X that takes an op en se t U to the c ate gory Hom( P | U , C | U ) is a stack. (2) L et C b e a stack on X . L et c and d b e two obje cts of C ( X ) . The pr eshe af of hom sets U 7→ Hom C ( U ) ( c | U , d | U ) is a she a f . (3) L et C and D b e two stack s on X . The m ap φ : C → D has a left adjoint (r esp . has a right adjoint, r esp. is an e quivalenc e) if a nd only if the maps φ x : C x → D x on stalks al l have left a djoints (r es p . al l have right adjoints, r esp. ar e al l e quivalenc es). (4) The inclusion 2-functor St( X ) → Prest( X ) has a right adoint, c al le d stac kiﬁcation . Denote the stackiﬁc ation of P by P † . The adjunction mo rphism P → P † induc es an e quivalen c e on stalks. A.3. Op erations on stac ks. Deﬁnition A.7. Let X and Y b e top o logical spaces, and let f : X → Y b e a contin uo us map. If C is a prestac k on X let f ∗ C denote the prestack on Y that asso ciates to an open set U the category f ∗ C ( U ) := C  f − 1 ( U )  . W e call f ∗ C the pushforwar d o f C . It is easy to v erify that f ∗ deﬁnes a strict 2-functor Prest( X ) → Prest( Y ) , a nd that if C is a stac k then f ∗ C is also. The deﬁnition for the pullbac k of a stac k is more complicated – it requires a direct 2 - limit ov er neighborho o ds, f o llo w ed by stac kiﬁcation: Deﬁnition A.8. Let X and Y b e top o logical spaces, and let f : X → Y b e a contin uo us map. If C is a prestac k on Y , let f ∗ p C b e the prestac k on X that asso ciates to an op en set EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 31 U ⊂ X the category f ∗ p C = 2lim − → V | V ⊃ f ( U ) C ( V ) Let f ∗ C denote the stac kiﬁc ation of the prestack f ∗ p C . Example A.9. If x ∈ X and i : { x } ֒ → X denote the inclusion map, the prestac k i ∗ p C on { x } coincides with the stalk C x . Th us, if f : X → Y , w e hav e an equiv alence of stalk categories  f ∗ p C  x ∼ = C f ( x ) . Example A.10. If U ⊂ X is op en, and j : U ֒ → X denotes the inclusion map, w e ha ve j ∗ p C ( V ) = C ( V ) when V ⊂ U is another op en set. If C is a stac k then j ∗ p C is also. W e often denote j ∗ C b y C | U . Prop osition A.11. L e t X and Y b e top olo gic al sp ac es, and let f : X → Y b e a c ontinuous map. The 2-functor f ∗ : St( X ) → St( Y ) is right 2-adjoint to the 2- functor f ∗ : St( Y ) → St( X ) . A.4. Descen t for stac ks. Let X b e a top ological space, and let { U i } i ∈ I b e a d -co v er of X . Supp ose w e are giv en the following data: (0) F or eac h i ∈ I a stac k C i on U i . (1) F or eac h i, j ∈ I with U j ⊂ U i , an equiv alence of stac ks C i | U j ∼ → C j . (2) F or eac h i, j, k ∈ I with U k ⊂ U j ⊂ U i , a n isomorphism b etw een the comp osite equiv alence C i | U j | U k ∼ → C j | U k ∼ → C k and C i | U k ∼ → C k . (3) Suc h that for each i, j, k , ℓ ∈ I with U ℓ ⊂ U k ⊂ U j ⊂ U i , the tetrahedron comm utes: C i | U j | U k | U ℓ / / & & L L L L L L L L L L L L   C j | U k | U ℓ   = C i | U j | U k | U ℓ / /   C j | U k | U ℓ   x x r r r r r r r r r r C k | U ℓ / / C ℓ C k | U ℓ / / C ℓ W e will abuse terminolo g y and refer to suc h data as a stack on the d -c over { U i } . Stac ks on { U i } form the ob jects of a 2-category St( { U i } ) in the natural w a y . If C is a stac k o n X then C i := C | U i and the iden tit y 1 - and 2-morphisms form a stac k on t he d -cov er { U i } , and the a ssignmen t C 7→ {C i := C | U i } forms a strict 2-functor in a natural w a y . Theorem A.12. The natur al r estriction 2-functor St( X ) → St( { U i } ) is an e quivalenc e of 2-c ate gories. Remark A.13. The 2-category St( { U i } ) may b e in terpreted a s an in v erse limit (or “ in v erse 3-limit”) of the 2 - categories St ( U i ). Th ere is a sense then in whic h the theorem means stacks form a 2-stack . See [4]. Appendix B. 2-ca tegories In this app endix we summarize some o f the theory of 2- categories, and ﬁx our con v en t io ns. Deﬁnition B.1. A strict 2-c ate gory C consists of (1) a collection O b( C ) of obje cts 32 DA VID TREUMAN N (2) for eac h pa ir x, y ∈ O b( C ) a category Hom C ( x, y ) (3) for eac h triple x, y , z ∈ Ob( C ), a c omp osition functor Hom C ( y , z ) × Hom C ( x, y ) → Hom C ( x, z ). The comp osition functors are assumed to satisfy asso ciativity and to ha v e units in the strict sense: for each ob ject x ∈ C , t here is an ob ject 1 x of Hom( x, x ) such that fo r eac h y , Hom( x, y ) ◦ 1 x − → Hom( x, y ) and Hom( y , x ) 1 x ◦ − → Hom( y , x ) are the iden tit y functors, and suc h that the follow ing diagram comm utes Hom( z , w ) × Hom( y , z ) × Hom( x, y ) / /   Hom( z , w ) × Hom( x, z )   Hom( y , z ) × Hom( x, y ) / / Hom( x, w ) for eac h x, y , z , w . Ob jects of Hom C ( x, y ) are called 1-m orphisms of C , and morphisms of Hom C ( x, y ) are called 2-morphisms of C . W e will also use the follo wing terminology: Deﬁnition B.2. A (2 ,1)-c ate gory is a 2-category C all of whose 2 - morphisms are inv ertible. Remark B.3. If cat denotes the cartesian closed category whose ob jects are categories, and whose morphisms a r e functors, then a strict 2-category is a cat -enric hed category in the sense of [12]. If gp d denotes the full sub category of cat whose ob jects ar e group oids, then a (2 , 1)-category is a gpd -enric hed category . Example B.4. There is a 2-catego ry Cat whose ob jects are categories, and where the usual categories of functors and natural transformatio ns are the hom categories. Note that w e use a diﬀeren t sym b ol for the 2 - category Cat than for the 1-category cat . Cat is the more natura l ob ject. Remark B.5. There is a more natural notion of we ak 2 - category , where t he asso ciativit y diagram ab ov e is required to comm ute only up to isomorphis m, and these isomorphisms are required to satisfy some equations of their o wn. Ev ery w eak 2-catego ry is equiv alent in the appropriate sense to a strict one. Moreov er, the 2-categories encoun tered in this pap er are either strict already (suc h as the 2- cat ego ry of stac ks or of prestack s) or else may b e easily made strict by a tr ick (suc h as the fundamen tal 2-group oid and the exit-path 2-category). W e hav e therefore decided to deve lop this pap er in terms of strict 2-categories. Deﬁnition B.6. Let C b e a 2- category . The o p p osite 2-c ate gory C op is the 2-category with the same ob jects as C , with Hom C op ( x, y ) = Hom C ( y , x ), and with the eviden t comp osition functor. Remark B.7. One could also deﬁne a kind o f “opp osite 2-categor y” by rev ersing only t he 2-morphisms, or by r evers ing b oth 1- and 2- morphisms. W e will not need these v ariatio ns and so w e w o n’t intro duce notation for them. EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 33 B.1. Two- dimensional comp osit ion in a 2-category. Let C b e a 2-category . Let x and y b e ob jects in C . If α , β , and γ are three 1-mor phisms in C , a nd f : α → β and g : β → γ are 2-mor phisms, then w e may of course form a third 2-morphism g ◦ f : α → β by taking the comp osition of g and f in the 1-category Hom C ( x, y ). There is another direction that w e may comp ose 2-morphisms. Let x , y and z be three ob jects of C , let α and β b e t w o 1-morphisms from x to y and let γ and δ b e tw o 1-mor phisms from y to z . Let f : α → β and h : γ → δ be 2-morphisms. Then w e may form a new 2-morphism h ⋆ f : γ ◦ α → δ ◦ β by applying the functors γ ◦ ( − ) : Hom C ( x, y ) → Hom C ( x, z ) and ( − ) ◦ β : Ho m C ( y , z ) → Hom C ( x, z ). That is, let h ⋆ f denote the comp osite map γ ◦ α γ ◦ f → γ ◦ β h ◦ β → δ ◦ β No w, let C be a 2- category , and let x , y , and z b e ob jects of C . Let α , β , and γ b e 1-morphisms x → y , and let δ , ǫ , and ζ be 1-morphisms y → z . Let f : α → β , g : β → γ , h : δ → ǫ , and k : ǫ → ζ b e 2-mo r phisms. W e ha v e the following equation: ( k ⋆ g ) ◦ ( h ⋆ f ) = ( k ◦ h ) ⋆ ( g ◦ f ) In practice, this equation allows us to ignor e the diﬀerence b et w een ◦ and ⋆ for 2 - morphisms. In fact, it follows t ha t g iv en an y collection o f 2-morphisms that may b e comp o sed using ◦ and ⋆ , an y t w o comp ositions a gree. B.2. Adjoin ts and equiv alences within a 2-category. Deﬁnition B.8. Let C b e a 2- category . Suppose f is a 1-morphism b etw een ob jects x a nd y in C . A right ad joint to f is a triple ( g , η , ǫ ), where g is a 1-morphism y → x called the adjoint , η : 1 x → g f and ǫ : f g → 1 y are 2-morphisms called the adjunction morph i s ms , and the so-called “triangle iden t it ies” hold: the natural maps η g : g → g f g and g ǫ : g f g → g comp ose to 1 g , and the natural maps f η : f → f g f and ǫf : f g f → f comp o se to 1 f . Dually , ( f , ǫ, η ) is called a left adjoint of g . W e sometimes abuse notation by suppressing the adj unction morphisms ǫ and η . Prop osition B.9. L et C b e a 2 -c ate gory, and l e t f b e a 1-morphism in C . If f h a s a right (r esp. left) adjoint ( g , α, β ) , then ( g , α, β ) is unique up to unique is o morphism c ommuting with α and β . Deﬁnition B.10. Let C b e a 2-category , and let f : x → y b e a 1-morphism in C . The n f is called an e quivalenc e if it ha s a right a dj o in t g , and if the a djunction maps 1 → f g and g f → 1 are bo th isomorphisms. This is equiv alen t to requiring f to ha v e a left a dj o in t g with isomorphisms for adjunction maps. B.3. Comm utativ e diagrams in a 2-category. Recall that, in a 1-category , a commu- tativ e diag ram is a collection of ob jects a nd of morphisms b et w een them such that an y tw o paths of comp osable arrows in the diagram betw een ob jects x and y coincide. In a 2-category , w e sa y that a diagram of ob jects, morphisms, and 2-morphisms comm utes if for eac h pair of comp osable paths f 1 , f 2 , f 3 . . . and g 1 , g 2 , g 3 . . . b et w een ob jects x and y , an y t w o compo sable sequence s o f 2-mo r phisms from . . . f 3 ◦ f 2 ◦ f 1 to . . . g 3 ◦ g 2 ◦ g 1 coincide. 34 DA VID TREUMAN N Example B.11. Let C b e a 2-category . A c ommutative triangle in C is a triple x, y , z of ob jects, a triple x → y , y → z , and x → z of 1-morphisms, and an isomorphism b et w een the comp osite x → y → z and the map x → z . A c ommutative squar e is a t uple of ob jects w , x, y , z , tuple of 1-morphisms w → x , w → y , x → z , y → z , and a 2- isomorphism betw een the composites w → x → z and w → y → z . F or t yp esetting reasons, w e omit the picture of the isomorphism when w e draw a comm utativ e triangle or square: x / /   > > > > > > > > y   w / /   x   z y / / z Example B.12. Let C b e a 2-category . A c ommutative tetr ahe dr on in C is a tuple w , x, y , z of ob jects, a collection of 1-morphisms w → x , w → y , w → z , x → y , x → z , a nd a collection of 2-isomorphisms b et w een the comp o site w → x → y and w → y , the comp osite w → x → z and w → z , the comp osite w → y → z and w → z , and the comp osite x → y → z and x → z , suc h t hat the t w o isomorphisms b et w een the comp osite w → x → z and w → y → z coincide. W e often draw a commutativ e tetrahedron in the following manner: w / / @ @ @ @ @ @ @ @   x   = w / /   x   ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ y / / z y / / z Remark B.13. W e may go on deﬁning comm utativ e n - simplices for n > 3. When w e refer t o a comm utativ e n -simplex in C w e a re r eferring to a diagram in whic h all t he 2- morphisms are isomorphisms. This is most satisfying when C is a (2,1)-category: in that case t he collection of o b jects, 1 - morphisms, comm utativ e triangles, comm utativ e tetrahedra, etc. ass em ble to a simplicial set called the n e rve of t he (2,1)-category . (D eﬁning the nerv e of a general ( 2 ,2)-category is more subtle.) Example B.14. A prism with ve rtices a, b, c, x, y , z is a collection of arro ws a → b , a → c , b → c , a → x b → y , c → z , x → y , x → z , and y → z and a collection of 2-isomorphisms b et w een a → b → c and a → c , b et w een x → y → z and x → z , b et w een a → b → y and a → x → y , b et w een b → c → z and b → y → z , and b et w een a → c → z and a → x → z , as in the picture. ❄ ❄ ❄ ✲ P P P P P q ✟ ✟ ✟ ✯ ✲ P P P P P q ✟ ✟ ✟ ✯ a c b x z y The prism is called comm utativ e if the t w o isomorphisms b et w een a → c → z and a → x → y → z coincide. EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 35 B.4. 2-group oids. Deﬁnition B.15. A 2-gr oup o i d is a 2-category in whic h all the morphism s are equiv a lences and all the 2-morphisms are isomorphisms. Remark B.16. A 2-group oid with one ob ject is called a 2-gr oup . T o e ac h ob ject x in a 2-category we can asso ciate a 2-group Aut( x ) whose unique ob ject is x , whose mor phisms are the self-equiv alence of x , and whose 2-morphisms are isomorphisms b et w een these self- equiv alences. It is p ossible to describe 2-g roups in terms of more classical algebraic structures. The nerv e of a 2-gro up oid is a ﬁbran t simplicial set whose homotopy groups (in eac h connected comp onen t) v anish abov e dime nsion 2. It is p ossible to show that connected, po in ted sp aces X whose homotop y groups v anish ab ov e dimension 2 are classiﬁed up to homotop y by a group G (the fundamental g r o up o f X ), a comm utativ e group A (the second ho mo t o p y group of X ), and an elemen t in group cohomology H 3 ( G ; A ) (a P ostnik ov in v ariant of X ). All t his is more naturally enco ded in terms of a “crossed mo dule.” B.5. 2-functors betw een 2-categories. Deﬁnition B.17. Let C and D b e 2-categories. A 2-functor F : C → D is (1) An assignmen t that tak es an ob ject x ∈ C to an ob ject F x ∈ D . (2) A collection of functors F : Hom C ( x, y ) → Hom D ( F x, F y ). (3) F or eac h triple x, y , z of ob jects, a natural isomorphism µ x,y ,z b et w een the t w o w ays of comp o sing the square: Hom C ( y , z ) × Hom C ( x, y ) ( F, F ) / /   Hom D ( F y , F z ) × Hom D ( F x, F y )   Hom C ( x, z ) F / / Hom D ( F x, F z ) W e furthermore assume t hat F (1 x ) = 1 F x for eac h x , and that a certain diagram of µ s built from a quadruple w , x, y , z of ob jects comm utes. See [11] for details. Deﬁnition B.18. Let C and D b e 2-categories, and let F : C → D b e a 2-functor. F is called strict if all the coherence maps µ x,y ,z are iden tities. The data of a strict 2-functor is equiv alen t to the data of a cat -enric hed functor in the sens e of [1 2]. Remark B.19. W e will shortly in tro duce a notion of equiv alence for 2- functors (in f act w e will in tro duce a 2- category o f 2-functors); note tha t no t a ll 2 -functors are equiv alen t to strict ones. Man y authors reserv e the word “2- functor” for what we hav e called strict functors, and call 2-functors pse udofunctors (e.g. [8], [13]), tho ug h usually not in the more recen t literature. Remark B .20. There is a more general not io n of 2-functor where t he coherence maps µ are not required to b e in v ertible. They are often called “lax 2-functors.” There ar e lax v ersions of ma ny concep ts in 2-category theory , where one replaces an isomorphism in a deﬁnition with a map in one direction or another. F or our purp oses – that is, stac ks o f categories – the non- lax versions of all these concepts seem to b e the correct ones. 36 DA VID TREUMAN N Example B.21. Let C b e a 2-category and let Cat b e the 2-category of 1-cat ego ries. F or eac h ob ject x o f C there is a strict 2- functor Hom( x, − ) : C → Cat and a strict 2-functor Hom( − , x ) : C op → Cat . In fact, there is a strict 2-functor Hom : C op × C → Cat . Prop osition B.22. L et C an d D b e 2-c ate go ries, and let F : C → D b e a 2-functor. If f is a 1-morphis m in C , and f has a left (r esp. right) adjoint, then F f has a left ( r esp. ri g ht) adjoint in D . F urthermor e if f is an e quivalenc e in C then F f is an e quivalenc e in D . B.6. Comp osite 2-functors. Let C , D , and E b e 2-categories. Let F : C → D and G : D → E b e 2-functors. There is a c omp osi te 2- f unc tor GF = G ◦ F from C to E ; GF is deﬁned on ob jects, 1-morphisms, and 2- morphisms o f C in the eviden t w ay . T o complete the deﬁnition it is necessary to describe the coherence data µ x,y ,z for GF – t his is straightforw ard but we refer to [8] for details. B.7. The 2-category 2F unct( C , D ) . Let C and D b e 2-categories. F or ev ery pair of 2- functors F : C → D and G : C → D we may deﬁne a 1 - category 2Nat( F, G ). Ob jects of 2Nat( F , G ) are called 2-natur al tr an s formations , and morphisms are called m o diﬁc ations . Giv en three 2- functors F , G a nd H from C to D , a comp osition functor 2Nat( G, H ) × 2Nat( F , G ) → 2Nat( F, H ) is deﬁned . This composition is strictly asso ciative, and it has strict units, so this data deﬁnes a 2-category 2F unct( C , D ). W e deﬁne 2-natural transfor- mations here; more details may b e found in [11]: Deﬁnition B.23. Let C and D b e 2-categories. Let F and G b e tw o 2-f unctor s from C to D . A 2-natur al tr ansfo rm ation n from F t o G consists of (1) An assignmen t that tak es ob jects x of C to 1- morphisms n ( x ) : F x → Gx in D (2) An assignmen t tha t take s 1-morphisms f : x → y in C to isomorphisms n ( f ) : Gf ◦ n ( x ) ∼ = n ( x ) ◦ F f . suc h that , for eac h pair o f 1-morphisms x f → y a nd y g → z in C , a ce rtain prism with v ertices F x, F y , F z , Gx, Gy , G z comm utes. See [1 1 ] f or details. Prop osition B.24. L et C and D b e 2-c ate gories. L et n : F → G b e a 1-morphism in 2F unct( C , D ) . (1) n has a right (r esp . left) adjoint m : F → G if a n d only if e ach 1-morphism n ( x ) : F ( x ) → G ( x ) in D h a s a left (r esp. ri g ht) adoint. (se e deﬁnition B.8) (2) n is a n e quivalenc e if and only if e ach n ( x ) : F ( x ) → G ( x ) is an e quivalenc e in D . (se e de ﬁnition B.10) B.8. Adjoin t 2-functors b et w een 2-categories. Let C and D be 2-categor ies, and let F : C → D and G : D → C b e 2-functors. W e may form the tw o 2-f unctors Hom D ( F − , − ) : C op × D → Cat Hom C ( − , G − ) : C op × D → Cat Deﬁnition B .25. Let C and D b e 2-categories. A 2-a djunction from C to D is a pair of 2-functors F : C → D and G : D → C together with a 2- na tural equiv alence – i.e. an equiv- alence in the 2 -category 2F unct( C op × D , Cat ) – b et w een the tw o 2-functors Ho m( F − , − ) and Hom( − , G − ). W e say that F is lef t 2-adjoint to G and that G is right 2-adjoint to F . EXIT P A THS AND CONS TRUCT IBLE ST ACKS. 37 Remark B.26. This is another deﬁnition with lax generalizations. W e can apply the adjunction t o the iden tit y 2-functors F c → F c and Gd → Gd to obtain 2-natural transformations 1 C → GF and F G → 1 D . Deﬁnition B.27. Let C and D b e 2-categories. An e quivalenc e f rom C to D is a pair of adjoin t 2-functors F : C → D , G : D → C suc h that the 2-na tural transformations 1 C → GF and F G → 1 D are equiv alences. A 2-functor F : C → D is called esse n tial ly ful ly faithful if the functor Hom C ( x, y ) → Hom D ( F x, F y ) is an equiv a lence of categories for ev ery pair of ob jects x, y ∈ C . F is called essential ly surje ctive if for ev ery ob ject d ∈ D there is an ob ject c ∈ D suc h that F c and d are equiv alent in D . Prop osition B.28. L et C and D b e 2-c ate gories. L et F : C → D b e a 2-functor. The fol lowing ar e e quiva l e nt. (1) F is essential ly ful ly faithful a n d essential ly surje c tive. (2) F is p art of an e quivalenc e F : C → D , G : D → C . B.9. Direct and in v erse 2-limits of 1-categories. Let I be a 2- category , and let F : I → Cat b e a 2-functor. It is possible to deﬁne categories 2lim ← − I F a nd 2lim − → I F with the appropriate 2 - univ ersal property . W e will giv e these deﬁnitions here in the form that w e need them; in particular w e only deﬁne 2lim − → I F in case I is a ﬁltered p oset. Deﬁnition B .29. Let I b e a 2-cat ego ry and let F : I → Cat b e a 2-functor. Let 2lim ← − I F denote the category whose ob jects consist of the follow ing data: (1) An assignmen t that tak es an ob ject i ∈ I to an ob ject x i ∈ F ( i ). (2) An assignmen t that tak es a morphism f : i → j to an isomorphism F ( f )( x i ) ∼ = x j . W e require that for eac h commutativ e triangle α : g f ∼ = h with v ertices i, j, k in I , the diagram F ( g ) F ( f ) x i / / α   F ( g ) x j   F ( h ) / / x k comm utes. The mor phisms o f this category are collec tions of maps x i → y i comm uting with the maps F ( f )( x i ) → x j . Deﬁnition B.30. Let I b e a ﬁltered p oset, and let F : I → Cat b e a 2-functor. Let 2 lim − → I F denote the category whose ob jects are ` i ∈ I F ( i ), and whose morphisms Hom( x ∈ F ( i ) , y ∈ F ( j )) are elemen ts of the limit lim − → k ≥ i and j Hom F ( k ) ( x i | k , x j | k ) Here if ℓ ≤ k and x ∈ F ( ℓ ), the notation x | k denotes the image of x under the functor F ( ℓ ) → F ( k ) induced b y the unique morphism ℓ → k . 38 DA VID TREUMAN N Reference s [1] A. Beilinson, J. Be r nstein, and P . Deligne. F aisc e aux p ervers. Ast´ eris que 100. [2] T. Br aden. Perverse she aves on Gr assmannians. Canad. J. Math. 54 (20 02). [3] T. Br aden and M. Grinber g. Perverse she aves on r ank str atiﬁc ations. Duke Math. J. 96 (1999). [4] L. Br een. On the classiﬁc ation of 2-gerb es and 2-st acks. Ast´ er isque 22 5. (19 9 4). [5] S. Gelfand, R. MacP herson, and K. Vilo ne n. Perverse she aves and quivers. Duk e Math. J. 83 (19 9 6). [6] J . Gir aud. Cohomolo gie Nonab elienne. Springer - V erla g, 1971 . [7] M. Gor esky and R. Ma cPherson. 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