Todd Trimble
Surface diagrams

Contents

Idea

A surface diagram is a higher-dimensional version of a string diagram as formalized by Joyal and Street, with a view to geometrically representing forms of higher-dimensional algebra. Just as Joyal and Street use isotopy classes of planar string diagrams to represent morphisms in monoidal categories or 2-categories that are freely generated in some sense, the rough idea is that there should be a class of nn-dimensional geometric structures whose isotopy classes represent nn-cells in nn-categories which again are freely generated in some sense.

Making the notion of surface diagram precise and workable is a big project, with a lot of possible offshoots. A paper begun with Margaret McIntyre back in 1996, which was essentially abandoned in 1999 in a still-embryonic state, had begun by focusing on the case n=3n = 3, describing a class of so-called “progressive” labeled 3D surface diagrams, whose isotopy classes represent 3-cells in a Gray category freely generated from a Gray computad. This was to be a higher-dimensional analogue of a similar result of Joyal and Street on progressive labeled string diagrams, whose isotopy classes represent morphisms in a strict monoidal category freely generated from a tensor scheme, or more generally a strict 2-category freely generated from a 2-computad.

Eventually this project grew in ambition (and in technical difficulty, likely beyond the technical abilities of the authors to handle at the time): the new goal became to describe a notion of “progressive” labeled nn-dimensional diagrams, whose isotopy classes were intended to represent nn-cells in a “semi-strictified” nn-category freely generated from an nn-computad. This part of the project was certainly conjectural, as there was (and to my knowledge still is) no well-tested theory of semi-strictifications for general algebraic notions of nn-category. The conjecture however is that these progressive nn-diagrams naturally continue the series starting at n=1n=1 and up to n=3n=3, where one does have appropriate notions of semi-strict nn-category and where the conjecture is indeed a theorem. For example, in the case n=2n=2, semi-strict 2-categories are identified with strict 2-categories, and planar string diagrams are used to present free strict 2-categories generated by 2-computads. In the case n=3n=3, semi-strict 3-categories are identified with Gray categories, and 3D surface diagrams are used to present free Gray categories generated by Gray computads. We were in effect suggesting that nn-dimensional (progressive) surface diagrams give a clue as to how analogous semi-strictifications of algebraic nn-categories ought to behave, and therefore some structural geometric analysis of them should yield up useful algebraic insights.

I will attempt to describe in some detail how far the project got before it entered an apparently permanent state of publishing limbo; my own thoughts did go a little further however on how the geometry might go. In particular, my struggles made me very aware of the advantage of using some form of “tame topology” to get over some technical hurdles. But at this point I think it’s clear to me that the project would benefit from fresh eyes, fresh blood, and fresh ideas – not to mention healthy doses of technical expertise – and in that spirit I’ll simply offer up my past thoughts on the subject.

String Diagrams

I’ll start by describing planar string diagrams, with a view toward how things will go in higher dimension. Roughly speaking, planar string diagrams are graphs which are “smoothly embedded” in a Euclidean rectangle [a 1,b 1]×[a 2,b 2][a_1, b_1] \times [a_2, b_2], with some technical assumptions which enforce (1) globularity conditions, and (2) “tameness” conditions, so that string diagrams can be handled easily and decomposed into simple primitive components.

Let’s start with the so-called “progressive” planar string diagrams of Joyal-Street. “Progressive” here intuitively means that the diagrams progress up the page. More precisely, it means that the edges of the diagram never go horizontal: that projection onto the vertical axis, when restricted to an edge, has no critical points.

Definition

A progressive planar string diagram in the rectangle R=[0,1]×[0,1]R = [0, 1] \times [0, 1] is a filtration

=X 1X 0X 1X 2=R\emptyset = X_{-1} \subseteq X_0 \subseteq X_1 \subseteq X_2 = R

where each X kX_k is compact and each X kX k1X_k - X_{k-1} is a smoothly embedded submanifold of dimension kk, satisfying the following conditions:

  • The connected components of the manifolds X kX k1X_k - X_{k-1} are the strata of a Whitney stratification.

  • Globularity: X 1X_1 does not intersect the vertical edges of RR, X 0X_0 does not intersect the boundary R\partial R, and X 1X_1 meets some neighborhood of R\partial R in straight vertical lines.

  • Progressivity: The second projection (x,y)y(x, y) \mapsto y is a regular map when restricted to any 1-stratum (any edge) ee.

Some notes on this definition:

It should be obvious that we are contemplating a 2-categorical calculus of string diagrams, but so far the vertical composition as defined above (which is to be composition in local hom-categories) is not strictly associative. We need an appropriate notion of isotopy of string diagrams to rectify this. Let XX and YY be two (progressive) string diagrams in a rectangle RR that have the same domain and have the same codomain.

Definition

Let XX and YY be progressive string diagrams in a rectangle RR. A deformation from XX to YY is an isotopy

h:I×RRh: I \times R \to R

satisfying the following conditions:

  • hh is constant along some neighborhood of R\partial R (i.e., h(0,x)=h(s,x)h(0, x) = h(s, x) for all 0s10 \leq s \leq 1 and all xx in some neighborhood of R\partial R);

  • h(0,):RRh(0, -): R \to R is the identity;

  • Each h(s,):RRh(s, -): R \to R is a homeomorphism that maps each manifold X kX k1X_k - X_{k-1} diffeomorphically onto its image, for all 0s10 \leq s \leq 1, and the filtration defined by the sets h(s,X k)h(s, X_k) is a progressive string diagram;

  • h(1,):RRh(1, -): R \to R maps X kX k1X_k - X_{k-1} onto Y kY k1Y_k - Y_{k-1}.

There is an obvious notion of deformation-equivalence of string diagrams, denoted XXX \sim X'. The following statements have straightforward proofs:

Proposition

If XXX \sim X' and YYY \sim Y', then Y 1XY 1XY \circ_1 X \sim Y' \circ_1 X' whenever either (hence each) of the composites is defined.

Proposition

If XXX \sim X' and YYY \sim Y', then Y 0XY 0XY \circ_0 X \sim Y' \circ_0 X' whenever either (hence each) of the composites is defined.

Proposition

The operations 1\circ_1 and 0\circ_0 on string diagrams induce a strict 2-category structure on the globular set where

  • There is just one 0-cell,
  • 1-cells are Whitney stratifications of [0,1][0, 1] for which the 0-stratum does not meet the boundary {0,1}\{0, 1\},
  • 2-cells are deformation-equivalence classes of progressive string diagrams.

(It is trivial to see that 1-cells of this 2-category are classified up to (unique) isomorphism by the cardinality of the 0-stratum, so we can think of a 1-cell as a copy of some integer n0n \geq 0.)

In particular, the interchange equation which relates vertical composition to horizontal composition corresponds to a string diagram deformation, as typified by the deformation-equivalence between the following string diagrams

insertpicturehereinsert picture here

where the deformation slides one point of the 0-stratum past the other.

We can now state the basic theorem (essentially due to Joyal-Street, but reinterpreted by Street in terms of computads) on progressive string diagrams:

Theorem (Joyal-Street)

The 2-category of progressive string diagrams is 2-equivalent to the 2-category Free(1)Free(\mathbf{1}) freely generated from the terminal 2-computad 1\mathbf{1}.

The terminal 2-computad has exactly one 2-cell between any two morphisms in the free category generated from the terminal directed graph. Morphisms in the free category may be identified with integers n0n \geq 0, and the unique 2-cell mnm \to n in the terminal 2-computad corresponds to a deformation class of string diagrams of shape

insertpicturehereinsert picture here

with exactly one point in the 0-stratum, lying in the closure of every edge. This diagram is called a cone (of type (m,n)(m, n)).

The Joyal-Street theorem asserts in part that every progressive string diagram XX is deformation-equivalent to another, YY, that is built up as a vertical ( 1\circ_1) composite of string diagram layers, each layer being a whiskering of a cone with isomorphisms on either side (whiskerings being 0\circ_0-composites):

insertpicturehereinsert picture here

This is accomplished by first finding a mesh 0=y 0<y 1<<y n=10 = y_0 \lt y_1 \lt \ldots \lt y_n = 1 so fine that

In that case, inside each box [0,1]×[y i1,y i][0, 1] \times [y_{i-1}, y_i], there is a mesh 0=x 0<x 1<<x m=10 = x_0 \lt x_1 \lt \ldots \lt x_m = 1 so fine that each connected component of X 1([0,1]×[y i1,y i]X_1 \cap ([0, 1] \times [y_{i-1}, y_i] is contained in the interior of some sub-box [x j1,x j]×[y i1,y i][x_{j-1}, x_j] \times [y_{i-1}, y_i].

Now what we do is deform the string diagram XX so as to vertically straighten where the edges in X 1X_1 meet the sub-box boundaries. To this end, let ψ:[0,1][0,1]\psi: [0, 1] \to [0, 1] be a smooth function such that

Now define a progressive string diagram YY by

Y k={(x,y):(x,ψ(y))X k}Y_k = \{(x, y): (x, \psi(y)) \in X_k\}

Then the edges of Y 1Y_1 are nice and vertical where they meet the sub-box boundaries. Moreover, one can define a deformation from XX to YY,

h:I×[0,1] 2[0,1] 2h: I \times [0, 1]^2 \to [0, 1]^2

Free 2-categories via wreath products

For more general 2-computads CC, the 2-category Free(C)Free(C) may be constructed in terms of Free(1)Free(\mathbf{1}) and CC by a kind of wreath product, one which has a nice intuitive description if we replace Free(1)Free(\mathbf{1}) up to 2-equivalence by the 2-category of progressive string diagrams. The key point is that each progressive string diagram XX possesses an underlying computad U(X)U(X):

The source and target of an edge ee are the planar regions to the immediate left and right of ee, and the source and target of a vertex vv are obtained by viewing vv as the vertex of a local cone diagram surrounding vv, and reading off the domain and codomain of the local cone. The underlying 2-computad is invariant with respect to deformation-equivalence classes [X][X].

In that case, Free(C)Free(C) may be described up to 2-equivalence as a 2-category whose 2-cells are pairs ([X],ϕ:U(X)C)([X], \phi: U(X) \to C), where ϕ\phi is a 2-computad map.

Degrees of coincidence and sesquicategories

The space of progressive string diagrams is itself stratified by “degrees of coincidence”. Basically, we think of two 0-strata occupying the same horizontal slice as a coincidence: for most string diagrams this doesn’t occur. If this happens twice in the same diagram, or if three 0-strata occupy the same horizontal slice, we think of that as super-coincidental or as a double coincidence. And so on.

Definition

A progressive string diagram is generic if no two 0-strata have the same second coordinate. The sum

0y1max(card(X 0π 2 1(y))1,0)\sum_{0 \leq y \leq 1} max(card(X_0 \cap \pi_2^{-1}(y)) - 1, 0)

is called the degree of coincidence deg(X)deg(X) of a progressive string diagram XX (so XX is generic if deg(X)=0deg(X) = 0).

Observations:

3D Surface Diagrams

Now we move up a dimension, with a view toward a geometry for GrayGray-categories. We define a notion of progressive surface diagram in the 3-cube [0,1] 3[0, 1]^3, and show that suitable deformation classes of such surface diagrams are the 3-cells of a GrayGray-category freely generated from the terminal “GrayGray-computad”. Again, this implies a more general result that the free GrayGray-category generated from a Gray-computad CC is equivalent to a GrayGray-category whose 3-cells are deformation classes of CC-labeled surface diagrams.

However, every aspect of these results is a degree more complicated than in the analogous results for planar string diagrams: these aspects include notions of tameness, globularity, and progressivity, the notion of deformation, and the notion of “coincidence”. All of them (except perhaps “tameness”) correspond to analogous complications in the algebra, so we examine these first: what for example is a GrayGray-computad? What does the free GrayGray-category on a GrayGray-computad look like? Then we set about defining the surface diagrams we are interested in, and outline the ingredients of proving the results. In so doing, we will get a good idea of some of what to look for in a general theory of nn-dimensional surface diagrams.

The free GrayGray-category on a GrayGray-computad

For the most part we assume known the notion of a GrayGray-category. In outline, there is a symmetric monoidal closed structure on the category of strict 2-categories and strict 2-functors, where the internal hom [C,D][C, D] consists of strict 2-functors CDC \to D, pseudonatural transformations, and modifications. The tensor product is called the “Gray tensor product”. The resulting monoidal category is denoted GrayGray, and a GrayGray-category is simply a category enriched in GrayGray. Such a GrayGray-category is a 3-dimensional structure which can be considered as a “semistrict” tricategory, and the celebrated coherence theorem of Gordon-Power-Street assures us that every tricategory is (tri)equivalent to a GrayGray-category.

Next, let us consider what we should mean by a GrayGray-computad. The rough idea is that we have sets of kk-cells C kC_k (k=0,1,2,3k = 0, 1, 2, 3) together with source and target maps

s k1,t k1:C kM k1(C (k1))×M k1(C (k1))\langle s_{k-1}, t_{k-1} \rangle: C_k \to M_{k-1}(C^{(k-1)}) \times M_{k-1}(C^{(k-1)})

where M k1M_{k-1} is the monad with respect to an monadic underlying functor that, roughly speaking, takes the (k1)(k-1)-skeleton of a Gray-category to its underlying (k1)(k-1)-computad, and C (k1)C^{(k-1)} denotes a (k1)(k-1)-dimensional computad skeleton of CC. The source and target maps must satisfy certain globularity conditions.

Let’s now parse this. The underlying 0-skeleton of a GrayGray-category is of course just a set of 0-cells, as is a 0-computad, and M 0:SetSetM_0: Set \to Set is just the identity monad. Thus the first level of a GrayGray-computad is a function

s 0,t 0:C 1C 0×C 0\langle s_0, t_0 \rangle: C_1 \to C_0 \times C_0

and this gives us a directed graph. This directed graph is the 1-computad skeleton C (1)C^{(1)} of CC. Next, the underlying 1-skeleton of a GrayGray-category bears a category structure, so the appropriate monadic underlying functor is,

F 1:CatGraphF_1: Cat \to Graph

with corresponding monad the free category construction M 1M_1. The second level of a GrayGray-computad thus consists of source-target maps

s 1,t 1:C 2M 1(C (1))×M 1(C (1))\langle s_1, t_1 \rangle: C_2 \to M_1(C^{(1)}) \times M_1(C^{(1)})

and this gives a 2-computad. This 2-computad is the 2-skeleton C (2)C^{(2)} of CC.

Finally, the underlying 2-skeleton of a GrayGray-category is… what? It is not a 2-category, because in a GrayGray-category we do not have strict interchange equations relating the two ways of composing 2-morphisms, but instead Gray interchanges , which are 3-cell isomorphisms. The 2-skeleton is blind to these isomorphisms. The best we can say is that the 2-skeleton is not a 2-category but rather a sesquicategory. Sesquicategories do have underlying 2-computads however, so we have an appropriate monadic underlying functor

U 2:SesquiCat2Computad,U_2: SesquiCat \to 2-Computad,

with corresponding monad the free sesquicategory construction, M 2M_2. The third level of a GrayGray-computad thus consists of source-target maps

s 2,t 2:C 3M 2(C (2))×M 2(C (2))\langle s_2, t_2 \rangle: C_3 \to M_2(C^{(2)}) \times M_2(C^{(2)})

and this completes the data of a GrayGray-computad.

Each GrayGray-category GG has an underlying GrayGray-computad U(G)U(G), in a straightforward way. For the 0-cells, define U(G) 0U(G)_0 to be G 0G_0. For the 1-cells, let U(G) 1U(G)_1 be G 1G_1, equipped with source-target data data

(U(G) 1s,tU(G) 0×U(G) 0)=(G 1dom,codG 0×G 0)(U(G)_1 \stackrel{\langle s, t \rangle}{\to} U(G)_0 \times U(G)_0) = (G_1 \stackrel{\langle dom, cod \rangle}{\to} G_0 \times G_0)

For the 2-cells, using the fact that G (1)G^{(1)} is a 1-category, define the 2-cells U(G) 2U(G)_2 as the pullback

U(G) 2 M 1(G (1)) 1×M 1(G (1)) 1 ε 1×ε 1 G 2 dom,cod G 1×G 1\array{ U(G)_2 & \to & M_1(G^{(1)})_1 \times M_1(G^{(1)})_1 \\ \downarrow & & \downarrow \varepsilon_1 \times \varepsilon_1 \\ G_2 & \underset{\langle dom, cod \rangle}{\to} & G_1 \times G_1 }

where ε:M 1(G (1))G (1)\varepsilon: M_1(G^{(1)}) \to G^{(1)} is the canonical counit or evaluation from the free category on G (1)G^{(1)} to G (1)G^{(1)}. The top horizontal arrow of the pullback gives the source and target data on 2-cells of U(G)U(G).

For the 3-cells, using the facts that G (2)G^{(2)} is a sesquicategory, define the 3-cells U(G) 3U(G)_3 as the pullback

U(G) 3 M 2(G (2)) 2×M 2(G (2)) 2 μ 2×μ 2 G 3 dom,cod G 2×G 2\array{ U(G)_3 & \to & M_2(G^{(2)})_2 \times M_2(G^{(2)})_2 \\ \downarrow & & \downarrow \mu_2 \times \mu_2 \\ G_3 & \underset{\langle dom, cod \rangle}{\to} & G_2 \times G_2 }

where μ:M 2(G (2))G (2)\mu: M_2(G^{(2)}) \to G^{(2)} the is the canonical counit from the free sesquicategory. The top horizontal arrow of the pullback gives the source and target data on the 3-cells of U(G)U(G).

The underlying functor

U:GrayCatGrayComputadU: Gray-Cat \to Gray-Computad

is again monadic (theorem due to Batanin; see his paper in the Proceedings from the 1997 Workshop at Northwestern). Before embarking further with the study of surface diagrams, it is worthwhile to get a rough idea of the structure of the free GrayGray-category F(C)F(C) generated from a GrayGray-computad CC. The easy part is what the GrayGray-category skeleta F(C) (k)F(C)^{(k)} look like for k<3k \lt 3: they are just M k(C (k)M_k(C^{(k)}, where M 2(C (2))M_2(C^{(2)}) in particular is a free sesquicategory. From our discussion of string diagrams, the 2-cells there can be pictured as generic planar string diagrams labeled in the 2-computad C (2)C^{(2)}, at least up to equivalence.

The harder part is getting a clear picture of the 3-cells of F(C)F(C). They are built up from primitive 3-cells by taking formal compositions in 3 directions: across 0-cells, 1-cells, and 2-cells. The primitive 3-cells arise from two sources:

We can take the “magma” (so to speak) which arises by taking formal compositions of these primitive cells, and then divide out by all the equations asserted to hold in a GrayGray-category, including strict associativity equations, strict interchange equations relating composition across 1-cells to composition across 2-cells, cylinder equations relating 0-cell composition to 2-cell composition, and, most notably, the cubical or Yang-Baxter equation, which is the main coherence condition imposed on the Gray-interchange isomorphisms. The result is the free GrayGray-category on the GrayGray-computad.

It will come as no surprise that compositions across 0-cells, 1-cells, and 2-cells will correspond respectively to juxtaposing surface diagrams in cubes (by pasting along faces of cubes) in the xx-, yy-, and zz-directions. It will turn out that the equations we need to impose are handled by considering equivalence up to appropriate deformations of surface diagrams. We tackle all this in the next few sections.

Primitive 3D Surface Diagrams

We now focus attention on the free GrayGray-category F(1)F(\mathbf{1}) generated from the terminal GrayGray-computad 1\mathbf{1}, and consider surface diagrams for modeling the primitive 3-cells used to generate F(1)F(\mathbf{1}).

Conings

First, the primitive 3-cells coming from the 3-cells of 1\mathbf{1}. There is exactly one 3-cell between any two given 2-cells of 1\mathbf{1} that have the same source and have the same target. Now 2-cells of 1\mathbf{1} belong to a free sesquicategory, which we know are represented (up to equivalence) by generic string diagrams. Therefore, we are invited to contemplate two generic string diagrams XX, XX' which have the same source and have the same target, and form a surface diagram out of them to model a suitable primitive 3-cell XXX \to X'. (Never mind that we have not formally defined surface diagrams yet. The definition is technical; for the moment we are just giving geometric constructions and gaining intuition.)

Coning Construction: Let v=(12,12,12)v = (\frac1{2}, \frac1{2}, \frac1{2}) be the barycenter of the cube [0,1] 3[0, 1]^3. Embed XX at the bottom of the cube:

X 0X 1X 2=[0,1] 2[0,1] 2×{0}\emptyset \subseteq X_0 \subseteq X_1 \subseteq X_2 = [0, 1]^2 \cong [0, 1]^2 \times \{0\}

and XX' at the top [0,1] 2×{1}[0, 1]^2 \times \{1\}. Further, extend the line segments where XX meets the edges y=0y = 0 and y=1y = 1 to rays extending out to infinity (in the plane z=0z = 0), and similarly extend the line segments where XX' meets the edges y=0y = 0 and y=1y = 1 to rays extending out to infinity (in the plane). Then take the cone on the union of these extended diagrams by taking the union of all rays which connect points in X 1X 1X_1 \cup X_{1}' to vv. The closure of

Cone(X 1X 1)[0,1] 3Cone(X_1 \cup X_{1}') \cap [0, 1]^3

defines a (at most) 2-dimensional set C 2[0,1] 3C_2 \subseteq [0, 1]^3. Similarly,

Cone(X 0X 0)[0,1] 3Cone(X_0 \cup X_{0}') \cap [0, 1]^3

defines a (at most) 1-dimensional set C 1[0,1] 3C_1 \subseteq [0, 1]^3. The filtration

{v}=C 0C 1C 2C 3=[0,1] 3\emptyset \subseteq \{v\} = C_0 \subseteq C_1 \subseteq C_2 \subseteq C_3 = [0, 1]^3

defines a surface diagram, called the coning C(X,X)C(X, X'). This models a primitive 3-cell coming from the GrayGray-computad. (For technical reasons this construction will have to be modified slightly: we will want to have the cone meet the top and bottom faces (z=0z = 0 and z=1z = 1) orthogonally, by applying a suitable deformation. Otherwise, this construction gives the correct picture.)

Graphs of Isotopies

Next, suppose given generic string diagrams X:stX: s \to t, X:stX': s' \to t' (this time not necessarily having the same sources/targets). As discussed in the section on string diagrams, there is an isotopy (a deformation of string diagrams)

h:I×[0,1] 2[0,1] 2h: I \times [0, 1]^2 \to [0, 1]^2

which “performs a GrayGray-interchange”, i.e., deforms the diagram DD given by

(X 01 s) 1(1 t 0X)(X \circ_0 1_{s'}) \circ_1 (1_t \circ_0 X')

to the diagram DD' given by

(1 s 0X) 1(X 01 t)(1_s \circ_0 X') \circ_1 (X \circ_0 1_{t'})

and in fact it would be easy to write an actual standard formula for hh. The following picture should give the idea:

insertpicturehereinsert picture here

The graph of this isotopy is the filtered set YY whereby

Y k={(h(x,t),t):xX k1,t[0,1]}Y_k = \{(h(x, t), t): x \in X_{k-1}, t \in [0, 1]\}

There are no 0-strata in YY: Y 0=Y_0 = \emptyset. This surface diagram YY models a Gray 3-cell interchange isomorphism (and once again, it needs to be straightened a bit so that YY meets the top and bottom faces orthogonally).

General progressive 3D diagrams

As intimated earlier, we now want to contemplate the scope of all diagrams which can be put together from conings and from graphs of string diagram isotopies by juxtaposing (pasting) cubes along the 3 coordinate directions. Obviously we are only really interested in 3D diagrams up to some notion of deformation equivalence, and we would like some sort of general synthetic description of what these diagrams look like.

Definition

A progressive surface diagram in K=[0,1] 3K = [0, 1]^3 is a filtration XX by compact subspaces,

=X 1X 0X 1X 2X 3=K\emptyset = X_{-1} \subseteq X_0 \subseteq X_1 \subseteq X_2 \subseteq X_3 = K

such that each X kX k1X_{k} - X_{k-1} is a smoothly embedded kk-dimensional manifold with corners (with corner sets in K\partial K), satisfying the following conditions:

  • The connected components of the manifolds X kX k1X_k - X_{k-1} are strata of a Whitney stratification of KK;

  • Globularity: X 0X_0 does not meet the planes z=0z = 0 and z=1z = 1, X 1X_1 does not meet the planes y=0y = 0 and y=1y = 1, and X 2X_2 does not meet the planes x=0x = 0, x=1x = 1. For all sufficiently small ε>0\varepsilon \gt 0, the 1-strata and 2-strata meet the planes z=εz = \varepsilon and z=1εz = 1 - \varepsilon orthogonally, and the 2-strata meet the planes y=εy = \varepsilon and y=1εy = 1 - \varepsilon in lines parallel to the zz-axis.

  • Progressivity: The projection (x,y,z)(y,z)(x, y, z) \mapsto (y, z), when restricted to any stratum, is a regular map.

These conditions probably look a little peculiar (particularly the globularity condition, which is rather rigid), so it may be a good idea to give some intuition. Basically, we think of a progressive surface diagram in the 3-cube as a “movie” through progressive string diagrams in the 2-cube; each horizontal slice z=cz = c through the surface diagram is a frame or still in the movie. The globularity conditions ensure that nothing is happening in the movie (there is no motion) for the first and last ε\varepsilon seconds, and that within an ε\varepsilon-neighborhood of the boundary of the square, the picture of the string diagram z=cz = c in the unit square is always the same throughout the movie (0c10 \leq c \leq 1). This rigidity near the boundary still leaves plenty of room for action inside the square.

Proposition

Each slice of a progressive surface diagram XX is a progressive string diagram.

Proof

This is straightforward; we touch on the essential points. The progressivity condition on XX implies that the third projection π 3:(x,y,z)maptoz\pi_3: (x, y, z) \mapto z is regular on the submanifold X kX k1X_k - X_{k-1}, so that for any c[0,1]c \in [0, 1],

π 3 1(c)(X kX k1)X kX k1\pi_{3}^{-1}(c) \cap (X_k - X_{k-1}) \hookrightarrow X_k - X_{k-1}

is a transverse intersection which defines a submanifold. It is not hard to see that the connected components of these manifolds are the strata of a Whitney stratification of the slice (i.e., Whitney’s condition B holds). The progressivity condition on XX also implies that the projection (x,y)y(x, y) \mapsto y, applied to any stratum within the slice z=cz = c, is also regular, so the progressivity condition on the slice is satisfied. The globularity conditions have already been discussed.

Now we come to the essential points:

Regarding the second point, we are not yet claiming that NXN \cap X is an honest-to-god coning: first we would need that the top and bottom slices of NN describe generic string diagrams, and moreover there is the somewhat irksome globularity condition we would additionally need for this box NXN \cap X, which practically never occurs. However, we do claim that it is possible to deform XX to achieve such a thing.

Thus, the ultimate claim is that any progressive surface diagram XX is deformation-equivalent to a progressive surface diagram YY which can be partitioned into finitely many boxes NN, where each NYN \cap Y either is the graph of an isotopy between generic string diagrams or is a coning of generic string diagrams. This YY thus models a 3-cell in the free Gray-computad F(1)F(\mathbf{1}). Moreover, any choice of such YY in the deformation class of XX models the same 3-cell. Therefore, we may assign this 3-cell to the (class of) XX itself.

However, this really works only under a careful definition of deformation: the obvious geometric thing doesn’t work because of attendant subtleties in the algebra. We treat this next.

Deformations of progressive surface diagrams

At a first pass, we might try defining deformations by analogy with the string diagram case. Suppose XX and YY are progressive surface diagrams having strictly the same generic string diagram as domain and strictly the same generic diagram as codomain. A deformation from XX to YY should be a map h:I×KKh: I \times K \to K (remember K=[0,1] 3K = [0, 1]^3) such that

(The last condition is added on so that graphs of deformations become four-dimensional surface diagrams. It is needed for reasons of globularity.)

These purely geometric conditions are for our purposes necessary, but not quite sufficient: they allow certain noxious collisions between isotopies and 0-strata which we need to disallow. Let us explain this with an example.

Consider a surface diagram in the cube (really a string diagram: there are no 2-strata) which represents the before-shot of a Reidemeister I move, as seen from the perspective of looking in the direction of the xx-axis.

(For those who like explicit formulae, a piece of it looks like locus of y=x 3xy = x^3 - x, z=x 2z = x^2. The obvious stratification to use to enforce progressivity is by taking X 0={(0,0,0)}X_0 = \{(0, 0, 0)\} and X 1=X 2=X_1 = X_2 = the locus. This may be reparametrized and straightened near the top so that it satisfies globularity in the standard cube.)

insertpicturehereinsert picture here

Now apply a Reidemeister I move, so as to remove the crossing as seen from the perspective of projection onto the yzy z-plane. In terms of an explicit formula, we may use the deformation whose slice at time tt is the surface diagram given by the locus LL of y=x 3(12t)xy = x^3 - (1 - 2t)x, z=x 2z = x^2 (with each slice stratified by taking X 0={(0,0,0)})X_0 = \{(0, 0, 0)\}) and X 1=X 2=LX_1 = X_2 = L).

After making some straightening adjustments near the boundary, we obtain an isotopy which satisfies the five deformation conditions above. However, the 3-cell modeled by the before-shot differs from the 3-cell modeled by the after-shot. Namely, the before-shot models a composite γ 2ϕ\gamma \circ_2 \phi (across a 2-morphism) of a Gray-interchange isomorphism γ\gamma with a 3-cell ϕ\phi from the terminal computad (the interchange can be thought of as the crossing of the knot projection). The after-shot models just a different 3-cell ϕ\phi' from the terminal computad. Since γ 2ϕϕ\gamma \circ_2 \phi \neq \phi', it is clear that algebraic modelings are not preserved under the naive notion of deformation specified by the five conditions above: this notion of deformation is not strong enough.

To make clear exactly what phenomena we need to disallow more generally, we need to track where coincidences occur in string diagram slices z=cz = c. Recall that a “coincidence” in a string diagram is where two or more 0-strata have the same yy-coordinate. Let us call…

(To be continued…)

Surface diagrams in n dimensions

Up to now, we have restricted attention to surface diagrams in dimensions 2 and 3, and even there we imposed a heavy “progressivity” restriction, which, while it permits a nice classification of diagrams describable by structures freely generated from computads, eliminates many surface diagrams which are of theoretical interest. Ultimately we would like to get rid of the progressivity assumptions and study surface diagrams more freely, relating them especially to the study of duals in algebraic nn-categories. This study would involve a fascinating marriage between topics in pure higher category theory (nn-categories with duals) and topics in pure differential topology (stratified Morse theory).

However, in order to get good classification results, we must guard against an ever-present threat of pathology: for example, our manifolds should not wiggle around too wildly near points in their closures. We could just work within the language of stratified Morse functions throughout, but sometimes the hypotheses underlying that language are hard to check; what we really want is a class of spaces known to be stable under all the constructions we need to perform and which meets the “local niceness” criteria whose technical expression is in stratified Morse theory. In other words, we want our stratified spaces to emerge naturally from a class of “tame spaces” which are readily describable and intuitive.

Technically, the solution we propose invokes the theory of o-minimal structures, a certain branch of model theory with applications to and extensions of real algebraic and analytic geometry. An outline of some of the major points of this theory (at least the points we need) is given in Appendix B, but for now we can just say: there is a class Σ\Sigma of subsets of Euclidean spaces n\mathbb{R}^n which

The smallest class Σ\Sigma which meets these conditions is the class of subexponential sets (as explained in appendix B), but any class Σ\Sigma with these specified properties will do. Let us fix such a class Σ\Sigma. We remark that the topological closure of a member of Σ\Sigma also belongs to Σ\Sigma, because the closure is definable by a first-order property.

Definition

A Σ\Sigma-surface diagram in I n=[0,1] nI^n = [0, 1]^n is a filtration by closed subsets belonging to Σ\Sigma,

=X 1X 0X n=I n,\emptyset = X_{-1} \subseteq X_0 \subseteq \ldots \subseteq X_n = I^n,

such that each X kX k1X_k - X_{k-1} is a kk-dimensional smooth manifold with corners (whose corner sets are embedded in I n\partial I^n), such that

  • The connected components of the X kX k1X_k - X_{k-1} are strata of a Whitney stratification of [0,1] n[0, 1]^n;

  • (Globularity) Let S kS_k be the span of the last kk standard basis vectors e nk+1,,e n1,e ne_{n-k+1}, \ldots, e_{n-1}, e_n. For ε>0\varepsilon \gt 0 and i{0,1}i \in \{0, 1\}, let (I n) k,i,ε(\partial I^n)_{k, i, \varepsilon} denote an ε\varepsilon-neighborhood of I nk×{i}×I k1I^{n-k} \times \{i\} \times I^{k-1} in I nI^n. Then, for each (k,i)(k, i), there are finitely many interior points p jp_j (i.e., p jI np_j \notin \partial I^n) such that

    X k(I n) k,i,ε=( jp j+S k)(I n) k,i,ε,X k1(I n) k,i,ε=X_k \cap (\partial I^n)_{k, i, \varepsilon} = (\bigcup_j p_j + S_k) \cap (\partial I^n)_{k, i, \varepsilon}, \qquad X_{k-1} \cap (\partial I^n)_{k, i, \varepsilon} = \emptyset

    for all sufficiently small ε>0\varepsilon \gt 0.

Such surface diagrams include all the higher-dimensional diagrams we need to consider in formulating the Generalized Tangle Hypothesis and the Generalized Cobordism Hypothesis (and much more).

As an important special case, we have

Definition

An nn-dimensional surface diagram XX in [0,1] n[0, 1]^n is progressive if the restriction of the projection

[0,1] n=[0,1]×[0,1] n1[0,1] n1[0, 1]^n = [0, 1] \times [0, 1]^{n-1} \to [0, 1]^{n-1}

to each stratum is a regular function.

The following geometric results are fundamental:

Proposition

Let XX be a progressive nn-dimensional Σ\Sigma-surface diagram, and let π n:[0,1] n[0,1]\pi_n: [0, 1]^n \to [0, 1] denote the last projection. Then every slice X c=π n 1(c)XX_c = \pi_{n}^{-1}(c) \cap X is a progressive (n1)(n-1)-dimensional Σ\Sigma-surface diagram.

Theorem (Isotopy)

Let XX be a progressive nn-dimensional Σ\Sigma-surface diagram with no 0-dimensional strata. Then XX is (Σ\Sigma-definably) isomorphic to a graph of an isotopy, i.e., there is a (definable) homeomorphism

h:XX 0×Ih: X \to X_0 \times I

that restricts to a diffeomorphism on each stratum, and such that π n=πh\pi_n = \pi \circ h where π\pi denotes projection to the interval II.

Theorem (Coning)

Let XX be an nn-dimensional Σ\Sigma-surface diagram, and let pp be a 0-stratum. Then there is a small closed nn-cube NN of pp such that the stratified set NXN \cap X is isomorphic to a cone on the stratified set NX\partial N \cap X.

For these results to be applicable to general surface diagrams, we want to following conjecture to be true:

Appendix A: Concepts from differential topology

Manifolds with corners

The manifolds of interest in surface diagrams are manifolds with corners. The pattern of definition is the usual one; a general context is as follows.

A pseudogroup on a topological space XX is a groupoid GG each of whose objects is an open set of XX, and whose morphisms are homeomorphisms between such open sets, satisfying the following conditions:

Let GG be a pseudogroup on XX. A GG-chart on a topological space MM is an open subset UU of MM together with an embedding

ϕ:UX.\phi: U \to X.

Two charts ϕ:UX\phi: U \to X and ψ:VX\psi: V \to X are compatible if

ψϕ 1:ϕ(UV)ψ(UV)\psi \circ \phi^{-1}: \phi(U \cap V) \to \psi(U \cap V)

belongs to GG. A GG-atlas on MM is a family of compatible charts (ϕ α:U αX) α(\phi_\alpha: U_\alpha \to X)_\alpha such that (U α) α) α(U_\alpha)_\alpha)_\alpha covers MM. The (restricted) maps ϕ αβ=ϕ βϕ α 1\phi_{\alpha \beta} = \phi_\beta \circ \phi_{\alpha}^{-1} are called transition functions between the charts of the atlas.

Finally, a GG-manifold is a topological space equipped with a GG-atlas. We can think of a GG-manifold as a space which is locally modeled on XX according to the geometry GG.

For nn-dimensional manifolds with corners, we take X=[0,1] nX = [0, 1]^n, and we take the morphisms of GG to be diffeomorphisms between open sets of XX, i.e., C C^{\infty} maps that have a C C^{\infty} inverse. (If A mA \subseteq \mathbb{R}^m and B nB \subseteq \mathbb{R}^n are arbitrary subsets, a smooth map f:ABf: A \to B is by definition a function such that the composite

AfB nA \stackrel{f}{\to} B \hookrightarrow \mathbb{R}^n

has an extension to a smooth function F:U nF: U \to \mathbb{R}^n defined on an open set UU containing AA.)

Remark: An atlas is not considered an essential part of the structure of a manifold: two different atlases may yield the same manifold structure. Here are the relevant definitions:

An isomorphism of GG-manifolds f:MNf: M \to N (defined by chosen atlas structures) is a homeomorphism ff such that

ϕ(Uf 1(V))ϕ 1Uf 1(V)ff(U)Vψψ(f(U)V)\phi(U \cap f^{-1}(V)) \overset{\phi^{-1}}{\to} U \cap f^{-1}(V) \overset{f}{\to} f(U) \cap V \overset{\psi}{\to} \psi(f(U) \cap V)

is in GG whenever (U,ϕ)(U, \phi) is a coordinate chart of xMx \in M, and (V,ψ)(V, \psi) is a coordinate chart of f(x)Nf(x) \in N. If M 1M_1 and M 2M_2 are two GG-manifold structures on the same topological space MM, then M 1M_1 and M 2M_2 are considered equal as GG-manifolds if id:MMid: M \to M is an isomorphism from M 1M_1 to M 2M_2 (and hence also from M 2M_2 to M 1M_1).

Embedded submanifolds

Many standard notions from the differential geometry of ordinary manifolds carries over to manifolds with corners. In particular, each point xMx \in M has an algebra O xO_x of germs of smooth (real-valued) functions defined on neighborhoods of xx, and a tangent vector at xx is just a derivation

v:O xv: O_x \to \mathbb{R}

The vector space of tangent vectors T x(M)T_x(M) is nn-dimensional, no matter if xx is a boundary point or not. A smooth map f:MNf: M \to N induces a linear map f ,x:T x(M)T f(x)(N)f_{\star, x}: T_x(M) \to T_{f(x)}(N).

As usual, we say a smooth map f:MNf: M \to N is a smooth embedding if ff maps MM homeomorphically onto its image and each map f ,xf_{\star, x} is a linear embedding.

Proposition

Given a manifold with corners NN and a topological embedding f:MNf: M \to N, there is at most one smooth structure on MM that renders ff a smooth embedding.

In particular, if i:SNi: S \subseteq N is a locally closed subset of a manifold with corners, it makes sense to ask whether there is a smooth structure on SS which makes the inclusion a smooth embedding. If so, there is only one, and thus an embedded submanifold (submanifold for short) refers to a property of a subset, not extra structure.

Tubular neighborhoods

Let i:SXi: S \subseteq X be an (embedded) submanifold. The normal bundle over SS in XX is the cokernel N X(S)N_X(S) in the exact sequence

0TSTX| SN X(S)00 \to T S \to T X|_S \to N_X(S) \to 0

in the abelian category of vector bundles over SS. If N X,0(S)N X(S)N_{X, 0}(S) \subseteq N_X(S) denotes the subset of zero vectors, there is an evident diffeomorphism ϕ 0:SN X,0(S)\phi_0: S \to N_{X, 0}(S).

A tubular neighborhood of SS in XX is an open neighborhood UU of SS which admits a diffeomorphism ϕ:UN X(S)\phi: U \to N_X(S) that extends ϕ 0\phi_0.

Theorem (Tubular neighborhood theorem)

Every closed submanifold SXS \subseteq X has a tubular neighborhood.

A proof may be found here for example.

Whitney conditions

Let XX and YY be embedded submanifolds of a manifold ZZ (typically n\mathbb{R}^n), of dimensions ii and jj respectively, with YY contained in the closure of XX. Then

John Mather, in his unpublished (but widely distributed) notes on topological stability, observed that condition A follows from condition B.

The significance of condition B is not particularly easy to grasp, but one of the more significant and accessible consequences is that the intersection of XX with some tubular neighborhood of YY in ZZ has the structure of a locally trivial fiber bundle.

A Whitney stratification of a set XX in n\mathbb{R}^n is a filtration by closed subsets,

\\emptyset = X_{-1} \subseteq X_0 \subseteq X_1 \subseteq \ldots \subseteq X_n = X

such that

We remark that for any pair of strata (c,d)(c, d) whose closures have nonempty intersection, either cd¯c \subseteq \bar{d} or dc¯d \subseteq \bar{c}.

Much of the interest in Whitney stratified spaces is in having sufficient hypotheses in which to extend Morse theory to singular spaces. This extension of Morse theory, called stratified Morse theory, is of crucial importance in intersection homology theory. The book by Goresky-MacPherson gives a very good account of the subject.

First isotopy lemma

A result of fundamental importance for Whitney stratified spaces is:

Theorem (Thom’s first isotopy lemma)

Suppose XX is a Whitney stratified set in n\mathbb{R}^n and π:X k\pi: X \to \mathbb{R}^k is a proper submersion when restricted to each stratum. Then there is isomorphism of Whitney-stratified sets, i.e., a stratum-preserving homeomorphism

h:X k×(π 1(0)X)h: X \to \mathbb{R}^k \times (\pi^{-1}(0) \cap X)

which restricts to a diffeomorphism on each stratum and such that π=π 1h\pi = \pi_1 \circ h. In particular, every fiber of π\pi is isomorphic to the fiber π 1(0)\pi^{-1}(0) as a Whitney-stratified set.

Perhaps needless to say, the main application is to the case k=1k = 1, where the slices π 1(t)\pi^{-1}(t) define an isotopy through Whitney stratified sets.

Appendix B: Tame Topology

In his Esquisse d’un Programme (section 5), Grothendieck enunciated the desire for an axiomatic “tame topology”, flexible and general enough to permit many constructions with ease, but also free from pathologies which are largely spurious for the topological study of geometric shapes. A quick summary of Grothendieck’s deep ideas here is next to impossible, but here are some quotes which should give a taste of what he has in mind (from the translation by Leila Schneps in Geometric Galois Actions; the original in French is included in that volume).

I would like to say a few words now on some topological considerations which have made me understand the necessity of new foundations for “geometric” topology… The problem I started from… was that of defining a theory of devissage for stratified structures…

The simplest non-trivial example of a stratified structure is obtained by considering a pair (X,Y)(X, Y) of a space XX and a closed subspace YY, and assuming… that both strata YY and X\YX \backslash Y are topological manifolds. The naive idea, in such a situation, is to consider “the” tubular neighborhood TT of YY in XX, whose boundary T\partial T should also be a smooth manifold, fibred with compact smooth fibers over YY, whereas TT itself can be associated with the conical fibration associated to the above one…

This naive vision immediately encounters various difficulties… This triggered a renewal of the reflection on the foundations of such a topology, whose necessity appears more and more clearly to me.

After some ten years, I would now say, with hindsight, that “general topology” was developed (during the thirties and forties) by analysts and in order to meet the needs of analysis, not for topology per se, i.e., the study of the topological properties of the various geometric shapes. That the foundations of topology are inadequate is manifest from the very beginning, in the form of “flase problems” (at least from the point of view of the topological intuition of shapes) such as the “invariance of domains”, even if the solution to this problem by Brouwer led him to introduce new geometrical ideas. Even now, just as in the hroic times when one anxiously witnessed for the first time curves cheerfully filling squares and cubes, when one tries to do topological geometry in the technical context of topological spaces, one is confronted at each step with spurious difficulties related to wild phenomena… Topologists elude the difficulty, without tackling it, moving to contexts which are close to the topological one and less subject to wildness, such as differentiable manifolds, PL spaces (piecewise linear) etc., of which it is clear that none is “good”, i.e., stable under the most obvious topological operations, such as contraction-glueing operations (not to mention operations like XAut(X)X \to Aut(X) which oblige one to leave the paradise of finite dimensional “spaces”). This is a way of beating about the bush! This situation, like so often already in the history of our science, simply reveals the almost insurmountable inertia of the mind, burdened by a heavy weight of conditioning, which makes it difficult to take a real look at a foundational question, thus at the context in which we live, breathe, work – accepting it, rather, as immutable data… It is this again which explains why the rigid framework of general topology is patiently dragged along by generation after generation of topologists for whom “wildness” is a fatal necessity, rooted in the nature of things.

My approach toward possible foundations for a tame topology has been an axiomatic one. Rather than declaring (which would be a perfectly acceptable thing to do) that the desired “tame spaces” are no other than (say) Hironaka’s semianalytic spaces, and then developing in this context the toolbox of constructions and notions which are familiar from topology, supplemented with those which had not been developed up to now, for that very reason, I preferred to work on extracting which exactly, among the geometrical properties of the semianalytic sets in a space n\mathbb{R}^n, make it possible to use these as local “models” for a notion of “tame space” (here semianalytic) and what (hopefully!) makes this notion flexible enough to use it effectively as the fundamental notion for a “tame topology” which would express with ease the topological intuition of shapes.

Our own interest in “tame topology” is to be able to easily implement standard constructions from the smooth (C C^\infty) category that we would like for surface diagrams, without being beset with pathologies. The types of shapes we want for surface diagrams should look essentially like semi-algebraic sets, but we would like a little more freedom than that in order to regularize various constructions (for example, to achieve globularity) by use of standard techniques like bump functions. On the other hand, we don’t want too much freedom: if we work with the full generality of (Whitney stratified sets) in the C C^{\infty} category, one has do deal with annoying pathologies that elude classification and miss the whole point of our enterprise; for example, the Cantor set is the zero set of a smooth function. We want something in between the semi-algebraic category and the piecewise smooth category, along the lines of what Grothendieck is suggesting.

In this appendix, we give an outline of one very satisfying answer to Grothendieck’s plea that is based on model-theoretic considerations, namely the theory of o-minimal structures.

Definition of o-minimal structure

In model theory, a structure on \mathbb{R} consists of collections of subsets of n\mathbb{R}^n, Σ nP( n)\Sigma_n \subseteq P(\mathbb{R}^n) which are closed under the operations of first order logic with equality:

A structure on \mathbb{R} is o-minimal if it contains the order relation, <Σ 2\lt \in \Sigma_2, and satisfies the following order-minimality condition:

It is the first several axioms, closure under first-order definability, that account for the power of o-minimal structures, while it is the last axiom which accounts for the “tameness”. Let us call an element belonging to Σ= n0Σ n\Sigma = \sum_{n \geq 0} \Sigma_n a (Σ\Sigma)-definable set. Then the o-minimality axiom guarantees that N\subseteq{N} is not definable, so that all the pathological behavior of \mathbb{N} from a logical perspective (related to Gödel incompleteness phenomena, the results of Davis-Matiyasevich-Putnam-Robinson, etc.) is unavailable, i.e., cannot be exploited to construct “wild sets” in Σ\Sigma, or space-filling curves, etc.

Perhaps the archetypal example of an o-minimal structure is the class of real semi-algebraic sets, where SS is definable if and only if it is the union of finitely many loci of joint equalities and inequalities of polynomials (in finitely many variables x=(x 1,,x k)x = (x_1, \ldots, x_k)) with coefficients in \mathbb{R}:

p 1(x)=p 2(x)==p m(x)=0;q 1(x)<q 2(x)<<q n(x)<0p_1(x) = p_2(x) = \ldots = p_m(x) = 0; \qquad q_1(x) \lt q_2(x) \lt \ldots \lt q_n(x) \lt 0

It is easy to check that the Σ n\Sigma_n are Boolean algebras and that the definability is preserved by inverse images along the f\mathbb{R}^f; the substantial part is the following “quantifier elimination” theorem:

Theorem (Tarski-Seidenberg)

The direct image of a semi-algebraic set under a map f\mathbb{R}^f is semi-algebraic.

This essentially means that if R(x 1,,x m,y 1,lots,y n)R(x_1, \ldots, x_m, y_1, \lots, y_n) is any formula in propositional language of ordered fields plus all real numbers as constants, then x 1,,x mR(x 1,,x m,y 1,,y n\exists_{x_1, \ldots, x_m} R(x_1, \ldots, x_m, y_1, \ldots, y_n is equivalent to such a formula (hence we eliminate the quantifier \exists).

The Tarski-Seidenberg theorem has many nice consequences, among them decidability of the first-order theory of the model \mathbb{R} as ordered field (which includes for example the theory of “Euclidean geometry”, and much more). There are numerous consequences of the Tarski-Seidenberg theorem for the structure of semi-algebraic sets (for example, the fact that they admit Whitney stratifications), and in fact this structure theory carries over to general sets that are definable in an o-minimal structure, as we will explain below.

Before turning to examples and applications, a few more basic concepts. If XX and YY are definable sets, then we may speak of a definable relation RX×YR \subseteq X \times Y. There is a bicategory of definable relations. A definable map f:XYf: X \to Y is a definable relation that is the graph of a function from XX to YY. There is an evident category of definable sets and definable maps.

Examples of o-minimal structures

There are many known examples of o-minimal structures:

Here are the relevant definitions. A exponential set is a finite Boolean combination of loci of the form

f(x 1,,x n,e x 1,,e x n)=0f(x_1, \ldots, x_n, e^{x_1}, \ldots, e^{x_n}) = 0

and a subexponential set is the direct image of an exponential set under a map f: n m\mathbb{R}^f: \mathbb{R}^n \to \mathbb{R}^m. Here are two critical facts:

Theorem (Khovanskii)

Each subexponential set has only finitely many connected components.

Theorem (Wilkie)

The complement of a subexponential set S nS \subseteq \mathbb{R}^n is also subexponential.

Given these facts, it is not hard to verify that subexponential sets are the definable sets of an o-minimal structure.

Subexponential sets are technically necessary in our development because they include graphs of standard smooth bump functions which we need to smoothly straighten various constructions, as in globularization. For 0<t<10 \lt t \lt 1 let

ϕ(t)=exp(1/t)exp(1/1t)\phi(t) = \exp(-1/t)\exp(1/1-t)
ψ(t)=ϕ(t)1+ϕ(t)\psi(t) = \frac{\phi(t)}{1 + \phi(t)}

For a<b<c<da \lt b \lt c \lt d define

= 0 ift0 = ψ(taba) ifa<t<b ψ abcd(t) = 1 ifbtc = ψ(tdcd) ifc<t<d = 0 ifdt\array{ & = & 0 & if t \leq 0 \\ & = & \psi(\frac{t - a}{b - a}) & if a \lt t \lt b \\ \psi_{a b c d}(t) & = & 1 & if b \leq t \leq c \\ & = & \psi(\frac{t - d}{c - d}) & if c \lt t \lt d \\ & = & 0 & if d \leq t }

This function is supported on [a,d][a, d], valued in [0,1][0, 1], and identically 11 on the interval [b,c][b, c].

In some sense subexponential sets should also be sufficient for any surface diagram purpose we have in mind. More technically, surface diagrams in nn-dimensional space (relative to a given o-minimal structure Σ\Sigma containing the subexponential sets), together with isotopies between them, isotopies between isotopies, etc., should form an (,n)(\infty, n)-category nn-Surf ΣSurf_\Sigma, and the conjecture (based on a passage from the Esquisse!) is that for any inclusion ΣΣ\Sigma \hookrightarrow \Sigma' of such o-minimal structures, the induced inclusion

nSurf ΣnSurf Σn-Surf_\Sigma \hookrightarrow n-Surf_{\Sigma'}

should be an equivalence of (,n)(\infty, n)-categories.

For the record, however, it is useful to have some examples of o-minimal structures which are likely to cover any surface diagram needs we may have in the future. The following example might just fit the bill:

This result is due to Patrick Speissegger (see references).

Whitney stratifications

Every definable set in an o-minimal structure admits a Whitney stratification whose strata are definable. A proof may be found in this paper by Ta Lê Loi.

Crucial for our purposes is that there is an o-minimal version of Thom’s first isotopy lemma:

Theorem (J. Escribano)

Suppose given an o-minimal structure, and a definable set XX which is Whitney stratified into definable strata. If π:X k\pi: X \to \mathbb{R}^k is a definable proper submersion when restricted to each stratum, then there is a definable isomorphism of Whitney-stratified sets

h:X k×(Xπ 1(0))h: X \to \mathbb{R}^k \times (X \cap \pi^{-1}(0))

which is smooth on each stratum and commutes with the projections, as in Thom’s first isotopy lemma.

There is a lacuna in that the smoothness can be assumed up to C kC^k diffeomorphism for each finite kk, but in the full generality of o-minimal structures there is currently no proof which assures C C^\infty diffeomorphisms on the strata. This phenomenon of a gap between C C^\infty and C kC^k is well-known in the theory of o-minimal structures, but fortunately for all known examples of o-minimal structures, we can actually achieve C C^\infty smoothness. For the semi-algebraic case, the reader may consult the reference by Coste-Shiota below.

References

The next reference sketches a proof of the o-minimal definable version of the first isotopy lemma.

Revised on October 2, 2010 13:39:38 by Todd Trimble