Whitney categories and the Tangle Hypothesis

WHITNEY CA TEGORIES AND THE T ANGLE HYPOTHESIS CONOR SMYTH AND JON WOOLF 1. Introduction W e prop ose a new notion of ‘ n -category with duals’, which w e call a Whitney n -category . There are t w o motiv ations. The first is to giv e a definition whic h makes the Baez–Dolan T angle Hypothesis [1] almost tautological. The T angle Hyp othesis is that, given suitable definitions of the terms in quotes, The ‘ n -category of framed co dimension k tangles’ is equiv alen t to the ‘free k -tuply monoidal n -category with duals on one ob ject’. This generalises Shum’s theorem [9] that the category of framed tangles in three dimensions is equiv alen t to the free tortile tensor category on one ob ject. In § 4 w e pro v e a version of the hypothesis by interpreting it as a statement about Whitney n -categories. There is of course a price to pay for obtaining a simple pro of of the T angle Hyp othesis, and that is that Whitney n –categories are a geometric, as opp osed to algebraic, theory of higher categories. Therefore to realise more fully the original conception one should relate Whitney categories to some more algebraic theory of higher categories. Sadly this is not something we understand ho w to do at this stage. The second motiv ation, in fact the original one for this work, is to giv e a definition whic h enables us to construct ‘fundamen tal n -categories with duals’ for each smo oth stratified space. The idea here, also due to Baez and Dolan, is that there should b e a v arian t of homotop y theory which detects aspects of the stratification of a stratified space. The inv arian ts will not b e group oids but rather more general categories with duals (a group oid is a category with duals with the additional prop erty that the dual of a morphism is an inv erse). They are obtained by restricting attention to maps in to the space whic h are transv ersal to all strata; the full construction, and the functoriality , of the in v arian ts is explained in § 3.2.3. The definition of Whitney category has a geometric fla v our, and is in tended for applications in smo oth geometry . W e b orro w heavily from the ideas of Morrison and W alker expressed in [8]. They promote the p oint of view that (1) it should be easier to define a notion of n -category with duals than of plain n -category; (2) one should consider higher morphisms of quite general shap es (not merely globules, simplices, or cub es); (3) rather than having a source and target, a morphism should hav e a ‘bound- ary’ encompassing b oth. T o emphasise the first p oint; Whitney categories are not a general theory of higher categories, but only a theory of ‘higher categories with duals’. This fragment of higher category theory app ears to b e simpler and more amenable to a geometric treatmen t. Despite Morrison and W alk er’s influence, our definition of Whitney Date : August 2011. This work w as made possible by the generous supp ort of the Lev erhulme T rust (Grant ref. F/00 025/AI). The second author would also lik e to thank the Newton Institute, Cambridge for their hospitalit y and support in April and Ma y 2011, whilst this pap er w as in preparation, and T om Leinster for several helpful conv ersations. 1 2 CONOR SMYTH AND JON WOOLF category is quite different from their definition of top ological or disk-lik e category . They giv e an inductiv e list of axioms, whereas w e define an n -category as a presheaf of sets on a category Prestrat n of stratified spaces and pr estr atifie d maps, whose restriction to the sub category Strat n of stratified spaces and str atifie d maps is a sheaf for a certain Grothendiec k top ology . The subscript n refers to the fact that w e consider only spaces of dimension ≤ n , and that the morphisms are homotop y classes of maps relative to the strata of dimension < n . Roughly , b y stratified space we mean a Whitney stratified space with cellular strata, b y a stratified map w e mean a smo oth map whose restriction to each stratum in the source is a lo cally- trivial fibre bundle o ver a stratum in the target, and by a prestratified map we mean one which b ecomes stratified after a p ossible sub division of the stratification of the source. The precise definitions, as well as the specification of the top ology on Strat n , are the sub ject of § 2. The definition of Whitney category app ears in § 3. W e consider Whitney cate- gories as a full subcategory n Whit of the preshea ves on Prestrat n . V arious formal prop erties follo w; n Whit is complete, cocomplete and there is a left adjoin t to the inclusion into the preshea ves, whic h asso ciates a Whitney category to any presheaf. W e also introduce a notion of equiv alence of Whitney n -categories — Definition 3.8 — generalising the description of an equiv alence of (ordinary) categories as a span of fully-faithful functors which are surjectiv e on ob jects. A t first sight the notion of Whitney category is quite remote from the usual no- tion of category . The in tuitive picture is as follo ws. The set A ( X ) asso ciated to the space X consists of the ‘morphisms in A of shape X ’. F or example the point-shaped morphisms A (pt) are the ob jects. All our spaces carry sp ecified stratifications, and these pla y an important rˆ ole. F or example the set associated to an interv al strat- ified by its endpoints and in terior is the 1-morphisms, whereas the set asso ciated to the sub divided interv al with a third p oint stratum in the interior is the set of pairs of comp osable 1-morphisms. This last assertion uses the fact that a Whitney category is a sheaf on Strat n . More generally , insisting that a Whitney category is a sheaf ensures that the set it assigns to a space X is determined by the sets it assigns to the (cellular) strata. One can think of X as a template for pasting diagrams, and the set assigned to X as the set of pasting diagrams in A modelled on this template. Prestratified maps b etw een spaces induce maps, in the opposite direction, b etw een the corresp onding sets. In particular, • the inclusion of the boundary induces a map taking a cell-shaped morphism to its ‘b oundary’, which pla ys the rˆ ole of source and target combined; • the map to a p oint induces a map taking an ob ject in A (pt) to the iden tit y morphism (of appropriate shap e) on that ob ject; • a sub division of a cell induces a map taking a pasting diagram mo delled on the sub divided cell to its comp osite. T o further clarify the relation consider the simplest case of Whitney 0-categories. Since Prestrat 0 con tains only 0-dimensional spaces, the only information here is the set of ob jects A (pt). More precisely the map A 7→ A (pt) induces an equiv alence b et w een the category of Whitney 0-categories and the category of sets. The next simplest case, n = 1, is treated in § 3.3, where w e show that the category of Whit- ney 1-categories and the category of small dagger categories are equiv alent. The sets A (pt) and A ([0 , 1]) are resp ectively the ob jects and morphisms of the dagger category corresp onding to A . This corresp ondence is our principal justification for considering Whitney n -categories as ‘ n -categories with duals’. Man y of our examples will b e k -tuply monoidal Whitney n -categories. By suc h w e mean a Whitney ( n + k )-category which is ‘trivial’ in dimensions < k , i.e. that assigns a one elemen t set to any space X with dim X < k . This slightly confusing WHITNEY CA TEGORIES AND THE T ANGLE HYPOTHESIS 3 terminology makes sense if one recalls that a monoid can be view ed as a one-ob ject category , a commutativ e monoid a one ob ject, one morphism bicategory and so on. In § 3.1 w e giv e a functorial pro cedure for asso ciating a genuine Whitney n -category Ω k A to a k -tuply monoidal one A , by considering the presheaf A ( S k × − ) where S k is the k -sphere stratified by a p oint and its complement. This is the analogue of the re-indexing pro cedure used to turn a one-ob ject category in to a monoid. Three classes of examples are discussed in § 3.2. Firstly , we show that repre- sen table preshea ves are Whitney categories. Secondly , w e define a k -tuply monoidal Whitney n -category n T ang fr k of framed tangles. The X -shap ed morphisms are the set of framed co dimension k submanifolds of X , transversal to all strata, consid- ered up to isotopies relative to strata of dimension < n + k . Interpreted in this framew ork the T angle Hyp othesis sa ys: The Whitney category n T ang fr k of framed tangles is equiv alent to the fr e e k -tuply monoidal Whitney n -category on one S k -morphism. W e pro v e this in § 4 by establishing an equiv alence betw een n T ang fr k and the Whitney ( n + k )-category represented by the sphere S k . This equiv alence arises from the P on trjagin–Thom collapse map construction whic h relates framed co dimension k tangles in X to maps X → S k . The Whitney category represented b y the sphere is, by the Y oneda Lemma, free on one S k -morphism, namely the identit y map of the sphere. The third class of examples is pro vided b y transv ersal homotop y theory: in § 3.2.3 w e explain ho w to asso ciate a transv ersal homotop y Whitney category Ψ k,n + k ( M ) to each based Whitney stratified manifold M . The X -shap ed morphisms are the set of transversal maps X → M considered up to homotop y relative to strata in X of dimension < n + k . W e also insist that the maps are ‘based’ in that strata in X of dimension < k are mapp ed to the basepoint. This mak es Ψ k,n + k ( M ) in to a k -tuply monoidal Whitney n -category . F or n = 0 and n = 1 these are closely related respectively to the transv ersal homotop y monoids and the transversal homotop y categories introduced in [13]. See § 3.2.3 and Example 3.13 for details of the resp ective relationships. The use of Whitney categories th us allows us to extend the definitions of [13] to arbitrary n , and provides a general framew ork for studying transversal homotop y theory . The transv ersal homotopy theory of spheres is also closely related to framed tangles. In § 4.1 we sho w that it is equiv alent to the Whitney category represen ted b y the sphere. Thus w e ha ve equiv alences of k -tuply monoidal Whitney n -categories Ψ k,n + k  S k  ' Rep  S k  ' n T ang fr k yielding three descriptions of the same ob ject whic h we can think of resp ectiv ely as homotop y-theoretic, algebraic (in the sense that the represen table Whitney category is free on one S k -morphism) and geometric. The final section § 4.2 contains some remarks ab out the T angle Hyp othesis for tangles with other normal structures, and the relationship of these with transversal homotop y theory of Thom spaces other than the sphere. Our examples and applications are in smo oth geometry (smooth tangles, transv er- sal homotopy theory , . . . ) so we ha ve developed a smo oth theory of n -categories with duals based on Whitney stratified spaces. This choice is not essen tial. Firstly , it is not clear that w e need the Whitney conditions; the theory could b e dev elop ed using the w eak er notion of smo oth spaces with manifold decomp ositions. Ho wev er, the Whitney conditions are required to obtain a go o d theory of stratified smooth spaces, for instance to ensure that transversal maps form an op en dense subset of all smo oth maps. Since transversalit y plays a cen tral rˆ ole it seems natural to 4 CONOR SMYTH AND JON WOOLF imp ose the Whitney conditions, particularly when considering transversal homo- top y theory . More generally , there seems no reason why one should not develop an analogous theory by starting instead with stratified PL spaces, or subanalytic ones or indeed any of a n umber of other c hoices. It would also be interesting to replace Whitney stratified spaces by a ‘combinatorial’ category , for instance b y symmet- ric simplicial sets. A b etter understanding of com binatorial versions of this theory seems the most lik ely w ay of building a bridge to Lurie’s theory of ( ∞ , n )-categories with adjoints, and his proofs of the T angle and Cob ordism hypotheses [5]. 2. Stra tified sp aces and maps 2.1. Whitney stratified spaces. A str atific ation of a smooth manifold M is a decomp osition M = S i ∈S S i in to disjoin t subsets S i indexed b y a poset S suc h that (1) the decomp osition is lo cally-finite, (2) S i ∩ S j 6 = ∅ ⇐ ⇒ S i ⊂ S j , and this o ccurs precisely when i ≤ j in S , (3) eac h S i is a lo cally-closed smo oth connected submanifold of M . The S i are referred to as the str ata and the partially-ordered set S as the p oset of str ata . The second condition is usually called the fr ontier c ondition . Nothing has been said about ho w the strata fit together from the p oin t of view of smo oth geometry . T o gov ern this w e imp ose further conditions, proposed b y Whitney [12] follo wing earlier ideas of Thom [11]. Supp ose x ∈ S i ⊂ S j and that w e hav e sequences ( x k ) in S i and ( y k ) in S j con v erging to x . F urthermore, suppose that the secan t lines x k y k con v erge to a line L ≤ T x X and the tangent planes T y k S j con v erge to a plane P ≤ T x M . (An in trinsic definition of the limit of secan t lines can b e obtained by taking the limit of ( x k , y k ) in the blow-up of M 2 along the diagonal, see [7, § 4]. The limit of tangent planes is defined in the Grassmannian Gr d ( T M ) where d = dim S j . The limiting plane P is referred to as a gener alise d tangent sp ac e at x .) In this situation w e require (Whitney A): the tangen t plane T x S i is a subspace of the limiting plane P ; (Whitney B): the limiting secant L is a subspace of the limiting plane P . Mather [7, Prop osition 2.4] sho w ed that the second Whitney condition implies the first. Nevertheless, it is useful to state both conditions b ecause the first is often what one uses in applications, whereas the second is necessary to ensure that the normal structure to a stratum is lo cally top ologically trivial, see for example [2, 1.4]. A Whitney str atifie d manifold is a manifold w ith a stratification satisfying the Whitney B condition. A Whitney str atifie d sp ac e is a closed union of strata X in a Whitney stratified manifold M . Examples ab ound, for instance an y manifold with the trivial str atific ation whic h has only one stratum is a Whitney stratified manifold. More interestingly , an y complex analytic v ariety admits a Whitney stratification [12], indeed any (real or complex) subanalytic set of an analytic manifold admits a Whitney stratification [4, 3]. A con tinuous map f : X → Y of Whitney stratified spaces is smo oth if it extends to a smooth map of the am bient manifolds. The notion of smoothness dep ends only on the germ of the ambien t space, in fact only on the equiv alence class of the germ generated b y em b eddings in to larger am bient spaces. By embedding the manifold M we ma y alwa ys assume that the am bient space of X is Euclidean. Definition 2.1. A str atifie d smo oth sp ac e X is the stable germ of a compact Whit- ney stratified subspace of some R k , where we stabilise b y the standard inclusions R k  → R k +1  → · · · . W e abuse notation b y using the same letter to denote the germ and the underlying Whitney stratified space. A smo oth map of str atifie d smo oth WHITNEY CA TEGORIES AND THE T ANGLE HYPOTHESIS 5 sp ac es is the stable germ of a smo oth map, where w e stabilise b y taking products with R . W e will restrict our atten tion to stratified spaces glued together from cells: a c el- lular str atifie d sp ac e is a stratified space X in whic h each stratum S is con tractible. Examples 2.2. (1) Let I b e the germ of R along the in terv al [0 , 1] stratified b y the endp oints and interior. Similarly let I n b e the germ of R n along [0 , 1] n stratified in the obvious fashion b y faces. (2) Let S n b e the sphere S n stratified b y a p oint, call it 0, and its complemen t and considered as a germ of R n +1 . 2.2. Stratified maps. W e are not interested in all smo oth maps, but only those whic h in teract nicely with the given stratifications. Definition 2.3. A smo oth map f : X → Y is a str atifie d submersion if for any stratum B of Y (1) the inv erse image f − 1 B is a union of strata of X and (2) for any stratum A ⊂ f − 1 B the restriction f | A : A → B is a submersion. Whether or not a smo oth map is stratified dep ends only up on the map of un- derlying spaces, and not on the germ. The comp osite of stratified submersions is a stratified submersion. Thom’s first isotopy lemma implies that the restriction f | f − 1 B : f − 1 B → B is top ologically a lo cally trivial fibre bundle. Con ven tion 2.4. F or ease of reading, in the sequel we refer to stratified smo oth spaces and stratified submersions simply as str atifie d sp ac es and str atifie d maps . A map is we akly str atifie d if it ob eys only the first condition of Definition 2.3. Stratified maps f , g : X → Y are homotopic thr ough str atifie d maps r elative to str ata of dimension < n if there is a smooth map germ h : X × [0 , 1] → Y suc h that (1) eac h h ( − , t ) : X → Y is stratified and (2) h ( x, t ) = h ( x, 0) for all t ∈ [0 , 1] and x in a stratum S ⊂ X with dim S < n . This is an equiv alence relation with the property that f ∼ g implies f ◦ h ∼ g ◦ h and h ◦ f ∼ h ◦ g for stratified h . The first implication uses the fact that a stratified map sends strata to strata of equal or lo wer dimension. Definition 2.5. Fix n ∈ N ∪ {∞} . Let Strat n b e the category whose ob jects are the compact cellular stratified spaces of dimension ≤ n and whose morphisms are homotopy classes of stratified maps relative to strata of dimension < n . In particular Strat ∞ is the category of stratified spaces and stratified maps b et w een them. The category Strat n is small; the ob jects are certain subsets of Euclidean spaces, and the morphisms certain maps b etw een these subsets. 2.3. The stratified site. In this section we sp ecify a Grothendieck top ology on Strat n so that it b ecomes a site. Recall that to do so w e m ust sp ecify a collection of co v ering siev es for each ob ject X , satisfying certain conditions. A siev e on X is a collection of morphisms with target X which is closed under precomp osition. First w e need the following lemma. Lemma 2.6. Supp ose f : X → Z ← Y : g ar e str atifie d maps. Then X × Z Y c an b e str atifie d by the fibr e pr o ducts of the str ata of X and Y so that (1) X × Z Y   / / Y g   X f / / Z 6 CONOR SMYTH AND JON WOOLF is a c ommuting diagr am of str atifie d maps. Mor e over, if X , Y and Z ar e c el lular then this str atific ation is c el lular. Pr o of. Consider X × Z Y = { ( x, y ) ∈ X × Y : f ( x ) = g ( y ) } ⊂ X × Y . W e equip this with the germ along this subset of the pro duct of the germs of X and Y . It is decomp osed into the subsets A × f ( A )= g ( B ) B where A ⊂ X and B ⊂ Y are strata. Each of these is a manifold b ecause f | A and g | B are transv ersal. This decomp osition satisfies the Whitney B condition: Supp ose ( a i , b i ) ∈ A × Z B and ( s i , t i ) ∈ S × Z T are sequences in X × Z Y with the sam e limit ( a, b ) ∈ A × Z B . The pro duct stratification of X × Y satisfies the Whitney B condition. Hence (when the limits exist in the ambien t tangent space) lim  i ∈ lim T ( s i ,t i ) ( S × T ) where  i is the secan t line b etw een ( a i , b i ) and ( s i , t i ). In fact since these pairs lie in the fibre pro duct, the limiting secant line lies in the subspace U = { ( v , w ) ∈ lim T ( s i ,t i ) ( S × T ) : d f ( v ) = dg ( w ) } . Clearly U ⊃ lim T ( s i ,t i ) ( S × Z T ); in fact they are equal. F or supp ose ( v i , w i ) ∈ T ( s i ,t i ) ( S × T ) is a sequence with limit ( v , w ). Then d f ( v i ) − dg ( w i ) → d f ( v ) − dg ( w ) = 0 . Since f is submersive onto the tangent space of f ( S ) = g ( T ) we can find v 0 i ∈ T s i S with d f ( v 0 i ) = d f ( v i ) − dg ( w i ) and v 0 i → 0. Then ( v i − v 0 i , w i ) ∈ T ( s i ,t i ) ( S × Z T ) and ( v i − v 0 i , w i ) → ( v , w ). Hence U ⊂ lim T ( s i ,t i ) ( S × Z T ) as claimed. Therefore the giv en decomp osition of the fibre pro duct satisfies the Whitney B condition, and the fibre product b ecomes a stratified space. It is easy to c heck that the maps in (1) are stratified. Supp ose that X, Y and Z are cellular. Then by considering the long exact sequences of homotop y groups induced respectively from the fibrations F  → T → g ( T ) and F  → S × Z T → S and using the fact that eac h of S, T and g ( T ) is con tractible w e see that S × Z T is w eakly con tractible. Since it is a smooth manifold it is homotopy equiv alen t to a CW complex, and so by Whitehead’s Theorem it is con tractible. Hence X × Z Y is cellular.  Remark 2.7. The stratified space X × Z Y is not in general a fibre pro duct in Strat ∞ (b ecause of the constraints on dimension there is no hop e that Strat n for n ∈ N will hav e pro ducts). F or example if Z = pt and X = Y = I then X 2 do es not hav e the required universal prop erty b ecause the inclusion of the diagonal is w eakly stratified but not stratified. Moreo v er, it is imp ossible to sub divide the stratification of X 2 so that it becomes a fibre pro duct; to do so one would require that the graph of every strictly monotonic and surjective function (0 , 1) → (0 , 1) w as a stratum. Hence w e ha v e the stronger statemen t that the category of stratified spaces and maps do es not hav e pro ducts in general. Despite not b eing a fibre product, many familiar properties hold, in particular there is an isomorphism W × X ( X × Y Z ) ∼ = W × Y Z in Strat ∞ . Definition 2.8. A stratified map f : Y → X trivial ly c overs a stratum A ⊂ X if f − 1 A is a single stratum and f | f − 1 A a diffeomorphism. Prop osition 2.9. Ther e is a Gr othendie ck top olo gy on Strat n in which a cov ering siev e on X is one such that for e ach str atum of X ther e is a map in the sieve trivial ly c overing that str atum. (In gener al a c overing sieve wil l c ontain many such maps.) WHITNEY CA TEGORIES AND THE T ANGLE HYPOTHESIS 7 Pr o of. W e verify the axioms for a top ology . Suppose S is a cov ering sieve on X and g : X 0 → X an y stratified map. Then the pullback siev e g ∗ S = { f 0 : Y 0 → X 0 | g f 0 ∈ S } should also be a cov ering siev e. Fix a stratum A 0 ⊂ X 0 . Supp ose that f : Y → X trivially cov ers the stratum g ( A ). By Lemma 2.6 there is a commutativ e diagram A 0 × X f − 1 ( g ( A 0 )) / /   X 0 × X Y / / f 0   Y f   A 0 / / X 0 g / / X. of stratified maps. The pre-image of A 0 under f 0 is the single stratum A 0 × X f − 1 ( g ( A 0 )) and the left hand vertical map is a diffeomorphism. The closure of the latter stratum is of dimension ≤ n , therefore is in the pullback siev e and trivially co v ers A 0 . Hence the pullback siev e is a co vering siev e for X 0 . Let S b e a co vering siev e on X , and let T be an y sieve on X . Supp ose that for eac h stratified map f : Y → X in S , the pullbac k siev e f ∗ T is a cov ering siev e on Y . W e m ust show that T is a co vering sieve on X . Fix a stratum A ⊂ X . Since S is a cov ering siev e for X we can find f : Y → X trivially co v ering A . Since f ∗ T is a cov ering siev e for Y w e can find g : Z → Y trivially cov ering f − 1 A . Then g f : Z → X is in the siev e S and trivially co vers A , so w e are done. Finally , we must v erify that the maximal sieve of all stratified maps with target X is a co vering sieve. This is immediate since the iden tity map is in the maximal siev e and trivially cov ers every stratum.  Ha ving defined a topology we may sp eak of shea ves on Strat n . Recall that these are preshea v es A suc h that elements of A ( X ) are given by matching families of elemen ts for an y cov ering siev e. More precisely , consider a co vering siev e S on X as a presheaf Y 7→ { f : Y → X | f ∈ S } . Then a presheaf A is a sheaf if and only if the map (2) A ( X ) → Nat ( S , A ) : a 7→ ( f 7→ f ∗ a ) is an isomorphism for each cov ering sieve S . A natural transformation η ∈ Nat ( S , A ) is a collection of elements a f ∈ A ( Y ) for each f : Y → X in the sieve S whic h ‘matc h’ in the sense that g ∗ a f = a f g for any g : Y 0 → Y . Here, and in the sequel, w e write g ∗ for A ( g ). In these terms, A is a sheaf if and only if each matching family has a unique amalgamation a ∈ A ( X ) suc h that a f = f ∗ a . 2.4. Prestratified maps. Stratified maps are rather rigid, and a more flexible notion is useful. A sub division X 0 of a stratified space X is a Whitney stratification of the underlying space of X eac h of whose strata is contained within some stratum of X . W e equip X 0 with the same stable germ. Definition 2.10. A smooth map f : X → Y is pr estr atifie d if it becomes stratified with resp ect to some sub division of the source X . Clearly an y stratified map is prestratified. If X 0 is a non-trivial sub division of X then the identit y X → X 0 is prestratified, but not vic e versa . Lemma 2.11. If f : X → Y and g : Y → Z ar e pr estr atifie d then so is the c omp osite g f : X → Z . Pr o of. Cho ose subdivisions X 0 of X and Y 0 of Y so that f : X 0 → Y and g : Y 0 → Z are stratified. Supp ose A is a stratum of X 0 . Then f ( A ) is a stratum of Y . F urther supp ose B is a stratum of Y 0 con tained in f ( A ). Then f | A : A → f ( A ) is a submersion and hence is transversal to B . So f − 1 ( B ) ∩ A is a submanifold of 8 CONOR SMYTH AND JON WOOLF A . The collection of these as A and B v ary through the strata of X 0 and Y 0 resp ectiv ely forms a decomp osition X 00 of X , subordinate to the stratification X 0 . The Whitney conditions for X 00 follo w from those for X 0 and for Y 0 . T o see this recall that w e need only verify the Whitney B condition. Supp ose x i ∈ f − 1 B 0 ∩ A 0 and y i ∈ f − 1 B 1 ∩ A 1 are sequences with common limit x ∈ f − 1 B 0 ∩ A 0 . When the limiting secant and tangen t plane exist, lim x i y i ∈ lim T y i A 1 b y Whitney B for X 0 . Now consider the image sequences f ( x i ) ∈ T and f ( y i ) ∈ T 0 . By Whitney B for Y 0 w e kno w that d f (lim x i y i ) = lim f ( x i ) f ( y i ) ∈ lim T f ( y i ) B 1 . Com bining these we see that lim x i y i ∈ lim T y i ( f − 1 B 1 ∩ A 1 ) as required. By con- struction the comp osite g f : X 00 → Z is stratified.  Definition 2.12. Fix n ∈ N ∪ {∞} . Let Prestrat n b e the category whose ob jects are the compact cellular stratified spaces of dimension ≤ n and whose morphisms are homotopy classes of prestratified maps relative to strata of dimension < n . The definition of homotopy used here is identical to that just prior to Definition 2.5, except that we replace stratified by prestratified throughout. Lik e Strat n this is a small category . 3. Whitney ca tegories Definition 3.1. Fix n ∈ N ∪ {∞} . A Whitney n -c ate gory A is a presheaf of sets on Prestrat n suc h that the restriction to Strat n is a sheaf. A functor b etw een Whitney n -categories is a map of presheav es, i.e. a natural transformation. Whit- ney n -categories and functors b etw een them form a full sub category n Whit of the preshea v es. W e refer to the elemen ts of A ( X ) as the morphisms of shap e X or as X - morphisms of A . W e also refer to the elements of the set A (pt) associated to a p oin t as the ob jects of A . A Whitney 0-category is completely determined b y the set A (pt). More precisely , the functor A 7→ A (pt) from the category of Whitney 0-categories to the category of sets is an equiv alence. In § 3.3 w e indicate wh y the category of Whitney 1-categories is equiv alen t to the category of small dagger categories and dagger functors. F ull details will app ear in [10], in which the case n = 2 is also treated; here there is an equiv alence b et w een (the categories of ) one-ob ject Whitney 2-categories and dagger rigid monoidal categories. Lemma 3.2. The c ate gory n Whit is c omplete. Pr o of. Limits are computed ob ject-wise, i.e. we set  lim i A i  ( X ) = lim i ( A i ( X )) where the right hand limit is computed in Sets . The result is a presheaf, indeed it is the limit in the category of presheav es. Using the fact that n Whit is a full sub category , it suffices to show that lim i A i is in fact a Whitney category . That it is follows from the fact that categories of sheav es are complete with the limits b eing computed ob ject-wise as ab ov e.  W e note some consequences. Firstly n Whit has fibre pro ducts. Secondly n Whit is a monoidal category under the cartesian pro duct of Whitney n -categories. Com- pleteness is also key to the next result. WHITNEY CA TEGORIES AND THE T ANGLE HYPOTHESIS 9 Theorem 3.3. The inclusion n Whit  → PreSh ( Prestrat n ) has a left adjoint ω . We r efer to ω ( A ) as the Whitney category asso ciated to A . Pr o of. W e use the adjoint functor theorem. Recall that this guarantees the exis- tence of the claimed left adjoint if (1) n Whit is complete; (2) the inclusion n Whit  → PreSh ( Prestrat n ) is contin uous; (3) for each A ∈ PreSh ( Prestrat n ) there is a collection f i : A → B i of mor- phisms to Whitney n -categories, indexed b y a set I , suc h that an y mor- phism A → B factors through some f i . The first tw o conditions follow from Lemma 3.2 ab o v e. It remains to v erify the third condition. T o do so w e sho w that the collection of quotient presheav es of A forms a set. The required { f i } can then b e tak en to be the subset of these quotien ts whose target is a Whitney category . A quotien t A → Q of the presheaf is determined by a (compatible) collection of surjections A ( X ) → Q ( X ) for each X ∈ Prestrat n . The maps Q ( Y ) → Q ( X ) in the quotient presheaf are completely determined b y the corresponding maps in A . Suc h surjections are indexed by equiv alence relations on the set A ( X ), which w e think of as subsets of A ( X ) 2 . So quotients of A can b e indexed by a subset of the pro duct of p ow er sets Y X ∈ Prestrat n 2 A ( X ) 2 (whic h exists as a set b ecause Prestrat n is small).  Remark 3.4. It would b e useful to hav e an actual construction of the left adjoint, p erhaps using a modified version of the double plus construction for sheafification. Ho w ev er, the construction of ω cannot be exactly lik e the latter b ecause, in con trast to the plus construction, ω cannot preserv e finite limits. If it did then it w ould follo w that n Whit w as a top os, but this is not the case. F or instance we will show in § 3.3 that 1 Whit is equiv alent to the category of dagger categories and functors, and the latter is not a top os (it has no subob ject classifier). Corollary 3.5. The c ate gory n Whit is c o c omplete. Pr o of. Recall that categories of presheav es are cocomplete (colimits are computed ob ject-wise). It follows that colim i A i ∼ = ω (colim i A i ) where the left hand colimit is computed in n Whit and the right hand one in PreSh ( Prestrat n ).  Prop osition 3.6. L et A b e a Whitney n -c ate gory and P ∈ Strat n with dim P = p . Then the assignments X 7→ A ( P × X ) and f 7→ (id × f ) ∗ : A ( P × X ) → A ( P × Y ) define a Whitney ( n − p ) -c ate gory which we denote A P . Pr o of. It is clear that A P is a presheaf on Prestrat n − p , so we need only chec k that it restricts to a sheaf on Strat n − p . Let { f i : X i → X } i ∈ I b e a cov ering siev e for X in Strat n − p . Then (3) { f i × id : P × X i → P × X } i ∈ I generates a co v ering siev e for P × X in Strat n whose elemen ts are the stratified maps to P × X factoring through one of these. The presheaf A P is a Whitney category if eac h matching family for (3) extends to a matching family for the sieve which 10 CONOR SMYTH AND JON WOOLF it generates. In other words w e must c heck that whenever w e ha v e a commuting diagram W g j   g i / / P × X i id × f i   P × X j id × f j / / P × X and a matc hing family { a i ∈ A ( P × X i ) } i ∈ I for the maps in (3) that g ∗ i a i = g ∗ j a j . Since A is a Whitney category it suffices to show that h ∗ g ∗ i a i = h ∗ g ∗ j a j for all h in some co vering siev e of W . W e construct a cov ering sieve with this prop erty as follo ws. F or each stratum S k ⊂ W there is an image stratum in P × X and a map id × f k : P × X k → P × X in (3) trivially cov ering it. Consider the commuting diagram W k / /   h k ' ' P P P P P P P P P P P P P P P P × ( X i × X X k )   ' ' P P P P P P P P P P P P W g i / / g j   P × X i id × f i   P × ( X j × X X k ) / / ' ' P P P P P P P P P P P P × X k id × f k ' ' O O O O O O O O O O O O P × X j id × f j / / P × X where we set W k = ( P × X k ) × P × X W and stratify the fibre pro ducts as in Lemma 2.6. (T o be precise, for the diagram to exist in Strat n w e must expunge an y strata of dimension > n . But this do es not effect the argument.) By construction h k : W k → W trivially cov ers the stratum S k . The collection of the h k for all strata S k in W th us generates a cov ering sieve of W , namely all those maps to W which factor through one of the h k . Since we hav e a matching family for the maps in (3) it follo ws from the diagram that b oth h ∗ k g ∗ i a i and h ∗ k g ∗ j a j agree with the pullback of a k from A ( P × X k ), and so they are equal. Therefore they agree on the co vering siev e for W constructed ab ov e, and so g ∗ i a i = g ∗ j a j as required.  Corollary 3.7. L et A b e a Whitney n -c ate gory and fix obje cts a, a 0 ∈ A (pt) . The assignment X 7→ { α ∈ A ( X × I ) : ı ∗ 0 α = p ∗ a, ı ∗ 1 α = p ∗ a 0 } , wher e ı t : X × t  → X × I is the inclusion and p : X → pt the map to a p oint, defines a Whitney ( n − 1) -c ate gory A ( a, a 0 ) . Pr o of. This is a sp ecial case of the pro of of Prop osition 3.6 ab o v e, with P = I , except that w e no w hav e b oundary conditions. That is w e are working with the sub-presheaf of A I consisting of elements α satisfying ı ∗ 0 α = p ∗ a and ı ∗ 1 α = p ∗ b . Since the amalgamation of a matching family of elemen ts with this prop erty also has this prop erty the argumen t go es through as before.  W e refer to A ( a, a 0 ) as the Whitney c ate gory of morphisms from a to a 0 . The construction is functorial: given F : A → B and a, a 0 ∈ A (pt) there is an induced morphism F ( a, a 0 ) : A ( a, a 0 ) → B ( F a, F a 0 ) of Whitney ( n − 1)-categories. WHITNEY CA TEGORIES AND THE T ANGLE HYPOTHESIS 11 In order to compare Whitney n -categories we need a notion of equiv alence. W e mo del it on the follo wing symmetric v ersion of equiv alence of ordinary small cate- gories: an equiv alence of A and B is giv en by a span A C F o o G / / B in which the functors F and G are fully-faithful and surjectiv e (not merely essen- tial ly surjectiv e) on ob jects. W e use this to make the follo wing inductive definition. Definition 3.8. F or n > 0 an n -e quivalenc e of Whitney n -categories is a span A C F o o G / / B of functors which are surjective on ob jects, i.e. the maps F (pt) : C (pt) → A (pt) and G (pt) : C (pt) → B (pt) are surjective, and whic h induce ( n − 1)-equiv alences A ( F c, F c 0 ) C ( c, c 0 ) F o o G / / B ( Gc, Gc 0 ) . for each pair c, c 0 ∈ C (pt). A 0-equiv alence is a span such that F (pt) and G (pt) are bijections. Prop osition 3.9. The notion of n -e quivalenc e is an e quivalenc e r elation on Whit- ney n -c ate gories. Pr o of. Reflexivity and symmetry are immediate. T ransitivit y follo ws from the fact that we can compose spans using the fibre product: D × B E { { x x x x x # # F F F F F D          # # G G G G G G G G G G E { { w w w w w w w w w w   @ @ @ @ @ @ @ A B C . W e claim that the outer ro of is an equiv alence whenev er the inner ro ofs are equiv- alences. W e use induction on n . The base case n = 0 is clear. Assume the result holds for ( n − 1)-equiv alences. Supp ose we are giv en a diagram as ab ov e. Ev alu- ating at a p oin t the solid arro ws are, by assumption, surjective. Hence so are the dotted ones. Therefore the induced maps ( D × B E )(pt) → A (pt) , C (pt) are surjec- tiv e. F or an y ob jects ( d, e ) and ( d 0 , e 0 ) in ( D × B E )(pt) there is an induced diagram of morphism categories. Using the fact that ( D × B E ) (( d, e ) , ( d 0 , e 0 )) ∼ = D ( d, d 0 ) × B ( b,b 0 ) E ( e, e 0 ) and the inductiv e h yp othesis we deduce that the outer span of the diagram of morphism categories is an ( n − 1)-equiv alence. Therefore the outer span of the original diagram is an n -equiv alence.  3.1. Monoidal Whitney categories. Recall that a category with one ob ject is a monoid, that a bicategory with one ob ject is a monoidal category and that a bicategory with one ob ject and one morphism is a commutativ e monoid. (The latter follows from the fact that if a set has tw o monoid structures  and ∗ with the distributive property ( a  b ) ∗ ( c  d ) = ( a ∗ c )  ( b ∗ d ) then  and ∗ agree, and are comm utative.) By analogy w e define a k -tuply monoidal Whitney n -c ate gory to be a Whitney ( n + k )-category A for which A ( X ) = 1 is a one element set whenev er dim X < k . 12 CONOR SMYTH AND JON WOOLF W e can obtain a b ona fide Whitney n -category from a k -tuply monoidal one as follo ws. Let Ω k A b e the Whitney n -category Y 7→ { a ∈ A  S k × Y  : a | 0 × Y = 1 } where w e write 1 for the pullback of the unique element in A (pt) under the unique map to a point. The p ro of that this is a Whitney category is the sp ecial case P = S k of Prop osition 3.6, but with an added ‘boundary’ condition (whic h does not effect the argument). One can define a monoidal structure on Ω k A by choosing a prestratified map µ : S k → S k ∨ S k (the w edge of the spheres iden tifying the p oint strata) whic h is degree one onto eac h lob e. There is then a unique dotted map suc h that Ω k A ( Y ) 2   / / _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Ω k A ( Y )   A ( S k × Y ) × A ( Y ) A ( S k × Y ) A  ( S k ∨ S k ) × Y  ( µ × id) ∗ / / A ( S k × Y ) comm utes. Uniqueness follo ws because the v ertical maps are injections. The identi- fication in the b ottom row comes from the fact that A is a sheaf on Strat n . This de- fines a binary operation Ω k A × Ω k A → Ω k A . The distinguished elemen t 1 ∈ Ω k A ( Y ) acts as a (weak) unit. Ho w ev er, w e prefer to consider k -tuply monoidal n -categories as special ( n + k )- categories rather than n -categories with additional structure. This has the virtue that a monoidal functor is then simply a functor b etw een ( n + k )-categories, rather than a functor ob eying an extra condition. 3.2. Examples. In this section w e discuss three different classes of examples of Whitney n -categories. 3.2.1. R epr esentable Whitney c ate gories. F or any stratified space X there is a rep- resen table Whitney n -category Rep ( X ) giv en b y the presheaf Rep ( X ) = Prestrat n ( − , X ) The only thing to v erify is that this restricts to a sheaf on Strat n . Supp ose S is a co v ering siev e on Y . The canonical map Rep ( X ) ( Y ) → Nat ( S , Rep ( X )) : f 7→ ( g 7→ f ◦ g ) is injective: If the classes of f , f 0 : Y → X differ as elements of Rep ( X ) ( Y ) then their restrictions to some stratum A ⊂ Y differ, and choosing g : Z → Y in the siev e trivially cov ering A the classes of f ◦ g and f 0 ◦ g differ. The canonical map is also surjective: Fix an element of Nat ( S , Rep ( X )), i.e. a compatible family { f g } of prestratified maps f g : Z → X for eac h g : Z → Y in the siev e S . F or eac h stratum A ⊂ Y w e can define a prestratified map f A : A → X by c ho osing g : Z → Y trivially cov ering A and considering the composite f A = f g ◦  g | g − 1 A  − 1 : A → g − 1 A → X . Compatibly of the family implies that f A is indep endent of the choice of such g . T ogether with the fact that we work with germs of smooth maps it also means that the f A patc h together to form a smo oth map f : Y → X . Moreov er, f is prestrati- fied b ecause this condition can be chec ked stratum-b y-stratum. Surjectivit y follo ws since, by construction, f 7→ { f g } . Remark 3.10. An y presheaf is a colimit of representable presheav es, in fact has a canonical suc h description, see for example [6, p40]. Since the left adjoint ω to the WHITNEY CA TEGORIES AND THE T ANGLE HYPOTHESIS 13 inclusion n Whit  → PreSh ( Prestrat n ) preserves colimits it follo ws that an y Whitney category is a colimit of representable ones, again in a canonical wa y . 3.2.2. F r ame d tangles. Giv en k , n ∈ N we define a Whitney ( n + k )-category of framed tangles by setting n T ang fr k ( X ) =  Germs of co dimension k framed submanifolds T ⊂ X transverse to eac h stratum   ∼ . The equiv alence relation ∼ is given b y ambien t isotopy , relative to all strata of dimension strictly less than n + k . By a framing we mean a homotopy class of trivialisations of the normal bundle, where the homotopy is the iden tity ov er in ter- sections of the tangle with strata of dimension < n + k . In particular if dim X < n + k then a framing is a fixed trivialisation. Remark 3.11. In the ‘classical’ case of 1-dimensional tangles in 3-dimensional space, corresponding to n = 1 and k = 2, the adjective framed is more commonly used in the knot theorists’ sense of a chosen non-v anishing section of the normal bundle. In these terms, what w e call a framed tangle w ould be instead a framed and orien ted tangle. Despite the unfortunate clash of terminology , we use the top olo- gists’ notion of framing since it generalises appropriately to higher dimensions. T o complete the definition we need to sp ecify the map induced by prestratified f : X → Y . W e define f ∗ : n T ang fr k ( Y ) → n T ang fr k ( X ) : T 7→ f − 1 T . Since f is prestratified and T transversal to all strata of Y the pre-image f − 1 T is a submanifold of co dimension k in X , also transversal to all strata. It inherits a framing given b y the isomorphisms N  f − 1 T  ∼ = f ∗ N T ∼ = f ∗  T × R k  ∼ = f − 1 T × R k . Homotopic maps giv e rise to isotopic framed submanifolds, hence f ∗ is w ell-defined. The v erification that this restricts to a sheaf is similar to the case of representable preshea v es. A matching family of germs of framed submanifolds of e ac h stratum amalgamates to form a germ of a framed submanifold of the en tire stratified space. Note that n T ang fr k is a k -tuply monoidal n -category , since only the empt y co di- mension k submanifold is transv ersal to the strata of a space X with dim X < k . As an example, in § 3.3, w e explain how to reco v er a more familiar v ersion of the category of framed tangles in the case n = 1 and k = 2. 3.2.3. T r ansversal homotopy the ory. Let M b e a Whitney stratified manifold with a generic basepoint p , i.e. p lies in some op en stratum of M . W e define ‘transversal homotop y Whitney categories’ of M built out of maps in to M which are transv ersal to all strata. T o b e precise a smooth map g : X → M from a stratified space in to M is tr ansversal to al l str ata of M if for eac h stratum of S ⊂ X the restriction g | S is transversal to the inclusion of eac h stratum of M . F or each k , n ∈ N w e asso ciate a Whitney ( n + k )-category Ψ k,n + k ( M ) to M by defining Ψ k,n + k ( M ) ( X ) =  T ransversal g : X → M suc h that whenever S ⊂ X and dim S < k then S ⊂ g − 1 ( p )   ∼ . Here ∼ is the equiv alence relation given by homotopy through transv ersal maps relativ e to all strata S ⊂ X with dim S < n + k . W rite [ g ] for the class of g : X → M . Giv en prestratified f : X → Y we define f ∗ [ g ] = [ g ◦ f ]. Then g ◦ f is transv ersal to all strata of M and [ g ◦ f ] dep ends only on the morphism in Prestrat n + k represen ted b y f . The verification that this restricts to a sheaf on Strat n + k is similar to that for 14 CONOR SMYTH AND JON WOOLF represen table preshea v es. The condition that g ( S ) = p whenev er dim S < k means that this is a k -tuply monoidal Whitney n -category . T ransversal homotop y Whitney categories are functorial for sufficiently nice maps b et w een Whitney stratified manifolds. Sp ecifically , they are functorial for we akly str atifie d normal submersions h : M → N , i.e. weakly stratified maps such that the induced mappings N p S → N h ( p ) h ( S ) of normal spaces to strata are alw a ys surjectiv e. 1 Whenev er h : M → N is a w eakly stratified normal submersion and g : X → M is transv ersal then the comp osite h ◦ g : X → N is transversal. So we can define a map Ψ k,n + k ( M ) ( X ) → Ψ k,n + k ( N ) ( X ) : [ g ] 7→ [ h ◦ g ] . Since comp osition on the left and righ t comm ute this sp ecifies a natural transforma- tion of presheav es, i.e. a functor Ψ k,n + k ( M ) → Ψ k,n + k ( N ) of Whitney categories. In the case n = 0 one can recov er the transversal homotopy monoids ψ k ( M ) defined in [13] by considering the asso ciated Whitney 0-category Ω k Ψ k,k ( M ). This is completely determined by its set of ob jects Ω k Ψ k,k ( M ) (pt) = Ψ k,k ( M ) ( S k ) , i.e. the set of homotop y classes of based transversal maps S k → M . This is the underlying set of the dagger monoid ψ k ( M ). The monoid structure can be reco vered b y the pro cedure outlined in § 3.1, and the dagger structure from the map induced b y a reflection of S k . In § 3.3 w e will sk etc h the analogous relation for n = 1 to the transv ersal homotop y categories defined in [13]. 3.3. Relation to ‘ordinary’ categories. On the face of it the definition of Whit- ney n -category seems rather remote from ‘ordinary’ category theory . In order for our definition of Whitney n -category to b e a reasonable notion of ‘ n -category with duals’ it should agree with the accepted definitions for small n . The case n = 0 is rather trivial, as 0-categories with duals and Whitney 0-categories are b oth simply sets. In this section w e discuss the more interesting n = 1 case. W e sketc h con- structions pro ducing a small dagger category from a Whitney 1-category and vic e versa . These are functorial and induce equiv alences betw een the category 1 Whit of Whitney 1-categories and the category of small dagger categories and dagger functors. F ull details will app ear in [10]. The case n = 2 in whic h there is a close relation betw een one-ob ject Whitney 2-categories and rigid dagger categories will also b e addressed there. 3.3.1. Whitney 1 -c ate gories. Let Dagger be the category of small dagger categories and dagger functors b etw een them, i.e. functors whic h commute with the dagger duals. Theorem 3.12. Ther e ar e functors 1 Whit D ' ' Dagger W f f giving an e quivalenc e of c ate gories. Sketch pr o of. Giv en a Whitney 1-category A we define a small dagger category D ( A ) with ob jects A (pt) and morphisms A ([0 , 1]). Source and target maps are induced from the inclusions ı 0 and ı 1 of 0 and 1 respectively into [0 , 1]. Iden tities on ob jects arise from the map induced from p : [0 , 1] → pt. Comp osition is induced 1 Such maps w ere termed ‘stratified normal submersions’ in [13] where the notion of stratified map was weaker. WHITNEY CA TEGORIES AND THE T ANGLE HYPOTHESIS 15 from c : [0 , 1] → [0 , 2] : t 7→ 2 t , where we stratify b y the integer p oin ts and their complemen t. Note that the sheaf condition implies that A ([0 , 2]) ∼ = A ([0 , 1]) × A (pt) A ([0 , 1]) is the set of comp osable pairs of morphisms. The dagger dual is induced from d : [0 , 1] → [0 , 1] : t 7→ 1 − t . The v arious equations — asso ciativity of comp osition, trivialit y of composition with an identit y , the equations satisfied b y the dagger dual and so on — arise from homotopies b etw een prestratified maps. Giv en a functor b etw een Whitney 1-categories, in other w ords a natural trans- formation η : A → B , we define a functor D ( η ) : D ( A ) → D ( B ) by a 7→ η pt ( a ) on ob jects and α 7→ η [0 , 1] ( α ) on morphisms. That this is a dagger functor follows from the naturality square A ([0 , 1]) η [0 , 1] / / d ∗   B ([0 , 1]) d ∗   A ([0 , 1]) η [0 , 1] / / B ([0 , 1]) . The ab ov e constructions define a functor D : 1 Whit → Dagger . In the other direction, supp ose D is a dagger category . W e define a Whitney 1-category W ( D ) by associating to a stratified space X a set of equiv alence classes of lab ellings of X by ob jects and morphisms in D . T o assign a lab elling we (1) c ho ose an orientation for eac h 1-dimensional stratum of X ; (2) lab el each 0-dimensional stratum b y an ob ject of D ; (3) lab el eac h (orien ted) 1-dimensional stratum b y a morphism of D compatibly with the ob jects lab elling the endp oint(s). Tw o suc h lab ellings are equiv alen t if they ha ve the same class under the equiv alence relation generated b y reversing the orien tation of a 1-dimensional stratum and replacing the lab elling morphism by its dagger dual. Giv en prestratified f : X → Y we define the map f ∗ : W ( D )( Y ) → W ( D )( X ) b y ‘pulling back’ labellings from Y to X . More precisely , w e lab el a 0-dimensional stratum in X by the ob ject lab elling its image, necessarily a 0-dimensional stratum, in Y . T o assign a lab el to each (oriented) 1-dimensional stratum in X it in fact suffices to describ e how to do so for the maps p : [0 , 1] → pt, d : [0 , 1] → [0 , 1] : t 7→ 1 − t and [0 , 1] → [0 , n ] : t 7→ nt . In these cases we assign resp ectively the identit y on the ob ject labelling the image point, the dagger dual of the morphism lab elling the 1-dimensional image s tratum and the n -fold comp osite of the morphisms lab elling the 1-dimensional image strata. One can sho w that W ( D ) is a presheaf on Prestrat 1 . The restriction to Strat 1 is a sheaf, essentially because lab ellings are ‘lo cal’. Giv en a dagger functor F : D → E one can map a D -lab elling of X to an E - lab elling by applying F to each lab el. When F is a dagger functor this resp ects the equiv alence relation on lab ellings and yields a natural transformation W ( F ) : W ( D ) → W ( E ). W e ha ve therefore defined a functor W : Dagger → 1 Whit . These constructions are in verse to one another. There is a natural isomorphism of dagger c ategories D → D W ( D ) which is the identit y on ob jects and takes a morphism f to the class of the lab elling of [0 , 1], with standard orientation, and lab el f . In the other direction, consider fixed X and orien t the 1-dimensional strata. F or each stratum S there is then a unique-up-to-homotopy characteristic map χ S : [0 , 1] dim S → X which is stratified and of degree one. Moreo v er, the sheaf condition implies that the map A ( X ) → W D ( A ) ( X ) taking a to the obvious 16 CONOR SMYTH AND JON WOOLF lab elling of X b y the χ ∗ S a is an isomorphism. These maps fit together to form a natural isomorphism A → W D ( A ).  Example 3.13. Let M b e a Whitney stratified manifold with generic basep oin t p , and let A = Ψ k,k +1 ( M ) be the transv ersal homotopy category defined in § 3.2.3. Using the construction in § 3.1 one obtains a Whitney 1-category Ω k A with Ω k A ( Y ) = { f ∈ Ψ k,k +1 ( M ) ( S k × Y ) : f (0 , y ) = p ∀ y ∈ Y } The ob jects of the dagger category D  Ω k A  are germs of based transversal maps S k → M and the morphisms are homotopy classes of germs of transversal maps S k × I → M , mapping 0 × I to the basep oint p , relativ e to the ends S k × { 0 , 1 } . This is equiv alent to the k th transversal homotopy category — confusingly also denoted Ψ k,k +1 ( M ) — defined in [13, § 4]. The only difference is that here we use germs of maps, whereas in [13] smoothness of composites was ensured b y imp osing stronger b oundary conditions. Example 3.14. Let A = 1 T ang fr 2 b e the Whitney 3-category of framed 1-dimensional tangles in codimension 2 of § 3.2.2. This is 2-tuply monoidal and one can extract a Whitney 1-category Ω 2 A , and from that a dagger category D  Ω 2 A  . The ob jects of the resulting dagger category are finite sets of framed p oin ts in the op en stratum of the sphere S 2 . The morphisms are isotop y classes, relative to the b oundary , of framed 1-manifolds in S 2 × I − 0 × I with (p ossibly empt y) b oundary in S 2 × { 0 , 1 } . This is (a version of ) the usual category of normally-framed tangles. 4. The T angle Hypothesis Consider the Whitney ( n + k )-category Rep  S k  represen ted by the stratified sphere S k . It is a k -tuply monoidal Whitney n -category: if dim X < k then any prestratified map X → S k m ust map X to the point stratum, so Rep  S k  ( X ) has exactly one element. The iden tit y map id : S k → S k determines a distinguished S k -morphism. Lemma 4.1. The Whitney ( n + k ) -c ate gory Rep  S k  is the fr e e k -tuply monoidal Whitney n -c ate gory on one S k -morphism. Pr o of. This follows from the Y oneda lemma. Given a k -tuply monoidal Whitney n -category A and an S k -morphism a ∈ A ( S k ) there is a unique functor of Whitney ( n + k )-categories with Rep  S k  → A : [ f : X → S k ] 7→ f ∗ a mapping the distinguished element id S k to a .  Prop osition 4.2. The k -tuply monoidal Whitney n -c ate gories n T ang fr k and Rep  S k  ar e e quivalent. Pr o of. W e use the Pon trjagin–Thom construction. Fix a generic p oint p ∈ S k . If f : X → S k is prestratified then it is transversal to p , b ecause it is submersive on to the op en stratum whenever f − 1 ( p ) 6 = ∅ . Th us the pre-image f − 1 ( p ) is (a stable germ of ) a framed co dimension k submanifold of X which is transv ersal to all strata. If f and g are homotopic relative to strata of dimension < n + k through prestratified maps then the pre-images f − 1 ( p ) and g − 1 ( p ) are isotopic relative to strata of dimension < n + k . The assignment [ f ] 7→ [ f − 1 ( p )] determines a functor F p : Rep  S k  → n T ang fr k . Con v ersely , given a codimension k framed submanifold T of X , transversal to all strata, we can construct a prestratified ‘collapse map’ f : X → S k so that T = f − 1 ( p ), a tubular neighbourho o d of T fibres o v er the op en stratum of S k and the WHITNEY CA TEGORIES AND THE T ANGLE HYPOTHESIS 17 complemen t of this neighbourho o d maps to the p oin t stratum. It follows that F p ( X ) is surjectiv e for an y X , in particular for X = pt. Hence F p is an ( n + k )-equiv alence if and only if the induced functor (4) Rep  S k  ([ f ] , [ g ]) → n T ang fr k ([ f − 1 ( p )] , [ g − 1 ( p )]) is an ( n + k − 1)-equiv alence for an y [ f ] , [ g ] ∈ Rep  S k  (pt). (Of course there is only one prestratified map pt → S k , and for this the preimage of p is empt y . Ho wev er, we wish to mak e an inductive argument and the boundary conditions will not alw a ys b e ‘trivial’ in this w ay , so w e do not make any assumptions ab out f and g at this p oin t.) Chec king whether we ha ve an ( n + k − 1)-equiv alence in (4) is very similar to c hec king whether F p is an ( n + k )-equiv alence. The difference is that now we consider prestratified maps I × X → S k on the one hand and framed tangles in I × X on the other, with b oundary conditions on { 0 , 1 } × X given by the maps f and g , and the preimage tangles f − 1 ( p ) and g − 1 ( p ) resp ectively . The existence of prestratified collapse maps, extending given ones on the b ound- ary , shows that the induced functor is surjective on ob jects. So we reduce to c hec king whether it induces an appropriate ( n + k − 2)-equiv alence. Proceeding inductiv ely , we reach the base cases. These concern the map of sets [ f ] 7→ [ f − 1 ( p )] from homotopy classes of transv ersal maps f : [0 , 1] n + k → S k , where the restriction of f to the boundary is some fixed map, ϕ sa y , to the set of isotop y classes of framed co dimension k tangles in [0 , 1] n + k , where the boundary tangle is ϕ − 1 ( p ). W e can alw a ys construct a prestratified collapse map such a tangle T , extending the given map on the b oundary . Indeed, the collapse map is unique up to homotopy through prestratified maps. Moreov er, giv en an isotop y h t : [0 , 1] n + k → [0 , 1] n + k relativ e to the boundary , and such that h t ( T ) is transv ersal to all strata for eac h t ∈ [0 , 1], w e can construct a family of collapse maps for the the tangles h t ( T ) yielding a homo- top y betw een a collapse map for T = h 0 ( T ) and a collapse map for h 1 ( T ). Hence in the base case there is a bijection betw een isotopy classes of framed tangles and homotop y classes of prestratified collapse maps (each with appropriate b oundary conditions). It follows that F p is an ( n + k )-equiv alence.  4.1. T ransv ersal Homotop y Categories of Spheres. A minor v ariant of this pro of of the T angle Hyp othesis relates categories of framed tangles to the transversal homotop y categories of spheres. More precisely , taking the pre-image of the point stratum 0 ∈ S k induces a functor F : Ψ k,n + k  S k  → n T ang fr k : [ f ] 7→ [ f − 1 (0)] . There are tw o differences from the functor F p . Firstly the rˆ oles of the basep oint and stratum ha v e b een switc hed: prestratified maps to S k are transv ersal to the generic basep oint p rather than to the stratum 0. Secondly , prestratified maps are submersiv e not just at p but onto the entire open stratum, whereas transversal maps to S k are only required to b e submersive at the p oin t stratum 0. Prop osition 4.3. The functor F is an ( n + k ) -e quivalenc e. Pr o of. The pro of is almost word-for-w ord the same as that of Prop osition 4.2, but with 0 replacing p , and with transv ersal maps to S k replacing prestratified maps. The key p oint is that one can construct transv ersal collapse maps for framed tangles, and that these are unique up to homotop y through suc h maps. See [13, App endix A] for details.  4.2. Other fla vours of tangles. Thus far w e ha ve considered only framed tangles, ho w ev er there are v arian ts of the T angle Hypothesis for orien ted tangles, unorien ted tangles and so on. T o make this more precise, fix a subgroup G ⊂ O k . The most 18 CONOR SMYTH AND JON WOOLF in teresting examples come from stable representations G ∗ → O ∗ . Then we can define a k -tuply monoidal Whitney n -category n T ang G k of co dimension k tangles whose normal bundles ha ve structure group reducing to G , or G -tangles for short. The framed case corresponds to taking G = 1, at the other extreme G = O k corresp onds to ‘plain’ tangles with no sp ecial normal structure. The group G acts on S k , considered as R k ∪ {∞} , by prestratified maps fixing the p oint stratum. Hence there is an induced action on A ( S k ) for any Whitney category A , and one ma y sp eak of G -inv ariant S k -morphisms. In these terms w e formulate the T angle Hyp othesis for G -tangles as sa ying that The Whitney category n T ang G k is equiv alent to the free k -tuply monoidal Whitney n -category on one G -inv arian t S k -morphism. Unfortunately it is not straightforw ard to mimic the pro of of the framed case. The difficult y is in finding X with the property A ( X ) ∼ = A ( S k ) G . The naiv e candidate is X = S k /G , but since the action is not free one should presumably consider instead the stack [ S k /G ]. Thus one is led to enlarging the category Prestrat n to include suitable stratified smo oth stacks. Rather than pursue this, we outline an alternativ e, more elementary , argumen t. Giv en a k -tuply monoidal Whitney n -category A and G -inv ariant a ∈ A ( S k ) we wish to construct a functor F : n T ang G k → A which maps the p oint G -tangle in S k to a , and moreov er to show that such a construction is essentially unique. Th us for each G -tangle T ∈ n T ang G k ( X ) we m ust construct an element F ( T ) ∈ A ( X ), in a canonical fashion. Begin by choosing a small disk-bundle neighbourho o d of T in X , such that the boundary meets only those strata whic h T do es, and meets these transv ersely . Next choose a cellular decomp osition of the submanifold T , sub dividing the stratification induced from X , for instance b y choosing a compatible triangulation. Decomp ose the disk-bundle neighbourho o d into product cells C × D k where C is a cell in T and D k the standard k -ball. Finally extend this to a cellular decomp osition of X sub dividing the original stratification, and denote the sub division by s : X → X 0 . Define an elemen t of A ( X 0 ) by giving a matching family for eac h cell of this decomp osition as follo ws. F or product cells C × D k in the disk-bundle neighbourho o d assign π ∗ a where π : C × D k → S k is the composite of second pro jection and collapse of the disk’s boundary . This assignmen t is forced by the requirement that the p oin t G -tangle in S k maps to a , and this gives rise to the uniqueness of F . F or cells outside the disk-bundle neigh b ourho o d assign the unit (i.e. the pullbac k of the unique element in A (pt) under the map to a point). The G -in v ariance of a ensures that the elemen ts assigned to cells in the disk-bundle neighbourho o d match. Composing via s ∗ : A ( X 0 ) → A ( X ) yields the required F ( T ). Figure 1 illustrates the construction. Of course there are many tec hnical issues. One must choose the disk-bundle neigh b ourho o ds carefully . The cleanest approac h is to define F on an auxiliary category of ‘ G -tangles with disk-bundle neighbourho o ds’ and then show that the forgetful functor from this to n T ang G k is an equiv alence. In addition one m ust sho w that F ( T ) is independent of the choice of cellular decomposition. This w ould follo w from the existence of common sub divisions, so one must include sufficien t h yp otheses to ensure this property , for instance b y fixing a PL structure and w orking with PL stratifications. Finally one needs to pro ve uniqueness. The connection with transv ersal homotopy theory go es through more easily . Let M G b e the Thom space of the univ ersal bundle on the classifying space of G - bundles. Stratify the Thom space b y the classifying space (em b edded as the zero section) and its complemen t. (In practice more care is required. One m ust c ho ose WHITNEY CA TEGORIES AND THE T ANGLE HYPOTHESIS 19 X C × D k C ! T Figure 1. The construction of F ( T ) from a G -tangle T in n T ang G k ( X ). T o a pro duct cell in the disk-bundle neigh b ourho o d of T assign π ∗ a ∈ A ( C × D k ) where π : C × D k → S k is the com- p osite of pro jection and collapse of the disk’s b oundary , and to other cells assign the unit in A ( C 0 ). Then comp ose to obtain an elemen t F ( T ) in A ( X ). a finite-dimensional smo oth manifold mo del for the classifying space, which suffices to classify G -bundles o ver manifolds of dimension ≤ n . Then one w orks with the ‘fat’ Thom space, as defined in [13, § 2], constructed from this mo del. The p oint is that to define transversal homotop y theory one needs to work with a Whitney stratified manifold.) 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