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4) If all thegi are prestratified then gis prestratified.
5) If all thegi are transversal then gis transversal.
7.2
Whitney
n-categories
We are now ready to define a Whitney n-category.
Definition 7.15. FixnPNY t8u.A Whitney n-category A is a presheaf
P restratnÑSet,
whose restriction to the subcategoryStratn is a sheaf. We refer to the elements
of ApXq as the X-shaped morphisms orX-morphisms of A. In particular we refer to the setApptqassociated to the point as the objects of A.
Recall that a category with one object is a monoid, that a bicategory with one object is a monoidal category and that a bicategory with one object and one morphism is a commutative monoid. By analogy we define the following:
Definition 7.16. Ak-tuply monoidal Whitneyn-category A is Whitneypn kq- category whereApXq 1 is a one element set whenever dimpXq k.
We give three classes of examples of Whitney n-categories:
Example 7.17 (Representable Whitney Categories). For any stratified space Y PP restratn there is a Whitneyn-category given by the presheaf
ReppYq P restratnp, Yq,
whereP restratnp, Yq HomP restratnp, Yq.
Proposition 7.13 and Remark 7.14 immediately tell us that given a covering sieve Xi
fi
ÝÑX and compatible family of stratified mapsgi:Xi ÑY there is a unique
amalgamation, i.e. a prestratified mapg:XÑY.Thus,ReppYqis a sheaf when restricted to Stratn.
Example 7.18(Transversal Homotopy Whitney Categories). In the discussion [3], John Baez asked the following question. Can one assign to any Whitney stratified manifold a fundamental n-category with duals? In answer to this we propose the following. LetM be a Whitney stratified manifold with a basepoint
plying in some open stratum. We associate toM the Whitneypn kq-category Ψk,n kpMq,where givenXPStratn k,
Ψk,n kpMqpXq
$ & %
Germs of transversal maps g:XÑM such that whenever
SX and dimS k,thenSg1ppq , . - L
. Hereis the equivalence relation given by homotopy through transversal maps relative to all strataS X with dimS n k.
Given prestratifiedf :XÑY we definefrgs rgfs. Thengf is transversal to all strata of M and rgfs depends only on the morphism in P restratn k
represented by f. The verification that this restricts to a sheaf on Stratn k is
similar to that for representable presheaves,mutatis mutandis.
The condition that gpSq p whenever dimS k means that this is a k-tuply monoidal Whitneyn-category.
We call Ψk,k npMq thepk, n kq-transversal homotopy Whitney category ofM.
This notation clashes somewhat with that for ‘ordinary’ transversal homotopy categories in Chapter 6, but we will resolve this later by showing the two are equivalent, at least in some cases.
These categories are functorial for sufficiently nice maps between Whitney strat- ified manifolds. Specifically, they are functorial forweakly stratified normal sub- mersionsh:M ÑN, i.e. weakly stratified maps such that the induced mappings NpS Ñ NhppqhpSq of normal spaces to strata are always surjective. Whenever
h:M ÑN is a weakly stratified normal submersion andg:XÑM is transver- sal then the compositehg:XÑN is transversal. So we can define a map
Ψk,n kMpXq ÑΨk,n kNpXq:rgs ÞÑ rhgs.
Since composition on the left and right commute this specifies a natural transfor- mation of presheaves and so there is a functor Ψk,n kM ÑΨk,n kN of Whitney
categories.
Example 7.19 (The Whitney Category of Framed Tangles). Given k, n P N,
we define the Whitneypn kq-categorynT angkf r of framed tangles, where given XPStratn k and its ambient manifoldM,we define a tangle to be represented
by a codimension kframed, closed submanifold of a neighbourhood of X inM, which intersects each stratum ofX transversally. Tangles are equivalent if they agree in some open neighbourhood of X. For short we will say,
nT angf rk pXq "
GermsT of M at transversal intersections of X with codimension kframed submanifolds of M.
* L .