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 $n$-dimensional geometric structures whose isotopy classes represent $n$-cells in $n$-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 $n = 3$, 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 $n$-dimensional diagrams, whose isotopy classes were intended to represent $n$-cells in a “semi-strictified” $n$-category freely generated from an $n$-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 $n$-category. The conjecture however is that these progressive $n$-diagrams naturally continue the series starting at $n=1$ and up to $n=3$, where one does have appropriate notions of semi-strict $n$-category and where the conjecture is indeed a theorem. For example, in the case $n=2$, 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 $n=3$, 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 $n$-dimensional (progressive) surface diagrams give a clue as to how analogous semi-strictifications of algebraic $n$-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.
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 $[a_1, b_1] \times [a_2, b_2]$, 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.
A progressive planar string diagram in the rectangle $R = [0, 1] \times [0, 1]$ is a filtration
where each $X_k$ is compact and each $X_k - X_{k-1}$ is a smoothly embedded submanifold of dimension $k$, satisfying the following conditions:
The connected components of the manifolds $X_k - X_{k-1}$ are the strata of a Whitney stratification.
Globularity: $X_1$ does not intersect the vertical edges of $R$, $X_0$ does not intersect the boundary $\partial R$, and $X_1$ meets some neighborhood of $\partial R$ in straight vertical lines.
Progressivity: The second projection $(x, y) \mapsto y$ is a regular map when restricted to any 1-stratum (any edge) $e$.
Some notes on this definition:
The notion of a Whitney stratified set is familiar to differential topologists: there are some technical conditions called Whitney conditions which ensure that the strata fit together well, as one moves within one stratum toward another stratum lying in its closure. In such low dimension $n=2$, we don’t have to worry about this, but as we move to higher dimensions, the Whitney conditions assume much greater importance, as they are crucial as hypotheses for technical results such as Thom’s first isotopy lemma. For now, think of this as foreshadowing a certain preoccupation with tameness issues which we will need to deal with later on.
When we say the union of 1-strata $X_1 - X_0$ is a 1-manifold, we actually mean a 1-dimensional manifold with boundary, whose boundary endpoints meet $\partial R$ along the horizontal edges. In higher dimension, we need to deal with manifolds with corners.
Progressivity means we disallowing “cups” and “caps”: edges move up the page without turning around, and there are no critical points along edges with respect to the vertical projection function.
The domain of the string diagram is defined to be the stratification of $[0, 1]$ isomorphic to the stratification
(via projection $\pi_1: [0, 1] \times \{0\} \cong [0, 1]$), and the codomain of the string diagram is defined to the stratification of $[0, 1]$ isomorphic to the stratification
(again via projection). When the codomain of one string diagram $X$ matches the domain of another $Y$, the string diagrams can be glued together by placing one on top of the other and taking their union (squashing down the resulting rectangle to a unit square in a standard way). This will be denoted $Y \circ_1 X$. The straight line condition under “globularity” guarantees that this gluing is smooth, so that $Y \circ_1 X$ is a string diagram.
Similarly, any two string diagrams $X$, $Y$ can be horizontally composed by juxtaposition, by pasting the right edge of $X$ to the left edge of $Y$ and taking the union. This yields a string diagram $Y \circ_0 X$.
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 $X$ and $Y$ be two (progressive) string diagrams in a rectangle $R$ that have the same domain and have the same codomain.
Let $X$ and $Y$ be progressive string diagrams in a rectangle $R$. A deformation from $X$ to $Y$ is an isotopy
satisfying the following conditions:
$h$ is constant along some neighborhood of $\partial R$ (i.e., $h(0, x) = h(s, x)$ for all $0 \leq s \leq 1$ and all $x$ in some neighborhood of $\partial R$);
$h(0, -): R \to R$ is the identity;
Each $h(s, -): R \to R$ is a homeomorphism that maps each manifold $X_k - X_{k-1}$ diffeomorphically onto its image, for all $0 \leq s \leq 1$, and the filtration defined by the sets $h(s, X_k)$ is a progressive string diagram;
$h(1, -): R \to R$ maps $X_k - X_{k-1}$ onto $Y_k - Y_{k-1}$.
There is an obvious notion of deformation-equivalence of string diagrams, denoted $X \sim X'$. The following statements have straightforward proofs:
If $X \sim X'$ and $Y \sim Y'$, then $Y \circ_1 X \sim Y' \circ_1 X'$ whenever either (hence each) of the composites is defined.
If $X \sim X'$ and $Y \sim Y'$, then $Y \circ_0 X \sim Y' \circ_0 X'$ whenever either (hence each) of the composites is defined.
The operations $\circ_1$ and $\circ_0$ on string diagrams induce a strict 2-category structure on the globular set where
(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 $n \geq 0$.)
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
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:
The 2-category of progressive string diagrams is 2-equivalent to the 2-category $Free(\mathbf{1})$ freely generated from the terminal 2-computad $\mathbf{1}$.
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 $n \geq 0$, and the unique 2-cell $m \to n$ in the terminal 2-computad corresponds to a deformation class of string diagrams of shape
with exactly one point in the 0-stratum, lying in the closure of every edge. This diagram is called a cone (of type $(m, n)$).
The Joyal-Street theorem asserts in part that every progressive string diagram $X$ is deformation-equivalent to another, $Y$, that is built up as a vertical ($\circ_1$) composite of string diagram layers, each layer being a whiskering of a cone with isomorphisms on either side (whiskerings being $\circ_0$-composites):
This is accomplished by first finding a mesh $0 = y_0 \lt y_1 \lt \ldots \lt y_n = 1$ so fine that
$\{(x, y) \in X_0: \exists_i y = y_i\} = \emptyset$
If $(x, y) \in X_0$, $(x', y') \in X_0$, and $y_{i-1} \lt y, y' \lt y_i$, then $y = y'$
For any $i$, if $C$ and $C'$ are connected components of $X_1 \cap ([0, 1] \times [y_{i-1}, y_i]$, and if there exist $(x, y) \in C$ and $(x', y') \in C'$ such that $x \lt x'$, then for all $(x, y) \in C$ and $(x', y') \in C'$ we have $x \lt x'$.
In that case, inside each box $[0, 1] \times [y_{i-1}, y_i]$, there is a mesh $0 = x_0 \lt x_1 \lt \ldots \lt x_m = 1$ so fine that each connected component of $X_1 \cap ([0, 1] \times [y_{i-1}, y_i]$ is contained in the interior of some sub-box $[x_{j-1}, x_j] \times [y_{i-1}, y_i]$.
Now what we do is deform the string diagram $X$ so as to vertically straighten where the edges in $X_1$ meet the sub-box boundaries. To this end, let $\psi: [0, 1] \to [0, 1]$ be a smooth function such that
$\psi$ is increasing, $\psi(0) = 0$, $\psi(1) = 1$;
$\psi$ is constant on a neighborhood of each $y_i$, and $\psi(y_i) = y_i$;
$\psi'(y) \gt 0$ whenever there exists $(x, y) \in X_0$.
Now define a progressive string diagram $Y$ by
Then the edges of $Y_1$ are nice and vertical where they meet the sub-box boundaries. Moreover, one can define a deformation from $X$ to $Y$,
For more general 2-computads $C$, the 2-category $Free(C)$ may be constructed in terms of $Free(\mathbf{1})$ and $C$ by a kind of wreath product, one which has a nice intuitive description if we replace $Free(\mathbf{1})$ up to 2-equivalence by the 2-category of progressive string diagrams. The key point is that each progressive string diagram $X$ possesses an underlying computad $U(X)$:
The 2-cells of $U(X)$ are 0-strata (vertices) of $X_0$;
The 1-cells of $U(X)$ are 1-strata (edges) of $X_1 - X_0$;
The 0-cells of $U(X)$ are 2-strata (planar regions) of $X_2 - X_1$.
The source and target of an edge $e$ are the planar regions to the immediate left and right of $e$, and the source and target of a vertex $v$ are obtained by viewing $v$ as the vertex of a local cone diagram surrounding $v$, and reading off the domain and codomain of the local cone. The underlying 2-computad is invariant with respect to deformation-equivalence classes $[X]$.
In that case, $Free(C)$ may be described up to 2-equivalence as a 2-category whose 2-cells are pairs $([X], \phi: U(X) \to C)$, where $\phi$ is a 2-computad map.
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.
A progressive string diagram is generic if no two 0-strata have the same second coordinate. The sum
is called the degree of coincidence $deg(X)$ of a progressive string diagram $X$ (so $X$ is generic if $deg(X) = 0$).
Observations:
Generic diagrams can be interpreted in sesquicategories. There is a sesquicategory whose 2-cells are equivalence classes of generic diagrams with respect to deformations valued in the space of generic diagrams.
This sesquicategory is equivalent to the free sesquicategory on the terminal 2-computad.
The free sesquicategory on $F_{ses}(C)$ on a 2-computad $C$ is equivalent to a sesquicategory whose 2-cells are of the form $([X], \phi: C \to U(X))$, where $[X]$ refers to equivalence up to deformations valued in generic diagrams.
Now we move up a dimension, with a view toward a geometry for $Gray$-categories. We define a notion of progressive surface diagram in the 3-cube $[0, 1]^3$, and show that suitable deformation classes of such surface diagrams are the 3-cells of a $Gray$-category freely generated from the terminal “$Gray$-computad”. Again, this implies a more general result that the free $Gray$-category generated from a Gray-computad $C$ is equivalent to a $Gray$-category whose 3-cells are deformation classes of $C$-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 $Gray$-computad? What does the free $Gray$-category on a $Gray$-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 $n$-dimensional surface diagrams.
For the most part we assume known the notion of a $Gray$-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 $[C, D]$ consists of strict 2-functors $C \to D$, pseudonatural transformations, and modifications. The tensor product is called the “Gray tensor product”. The resulting monoidal category is denoted $Gray$, and a $Gray$-category is simply a category enriched in $Gray$. Such a $Gray$-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 $Gray$-category.
Next, let us consider what we should mean by a $Gray$-computad. The rough idea is that we have sets of $k$-cells $C_k$ ($k = 0, 1, 2, 3$) together with source and target maps
where $M_{k-1}$ is the monad with respect to an monadic underlying functor that, roughly speaking, takes the $(k-1)$-skeleton of a Gray-category to its underlying $(k-1)$-computad, and $C^{(k-1)}$ denotes a $(k-1)$-dimensional computad skeleton of $C$. The source and target maps must satisfy certain globularity conditions.
Let’s now parse this. The underlying 0-skeleton of a $Gray$-category is of course just a set of 0-cells, as is a 0-computad, and $M_0: Set \to Set$ is just the identity monad. Thus the first level of a $Gray$-computad is a function
and this gives us a directed graph. This directed graph is the 1-computad skeleton $C^{(1)}$ of $C$. Next, the underlying 1-skeleton of a $Gray$-category bears a category structure, so the appropriate monadic underlying functor is,
with corresponding monad the free category construction $M_1$. The second level of a $Gray$-computad thus consists of source-target maps
and this gives a 2-computad. This 2-computad is the 2-skeleton $C^{(2)}$ of $C$.
Finally, the underlying 2-skeleton of a $Gray$-category is… what? It is not a 2-category, because in a $Gray$-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
with corresponding monad the free sesquicategory construction, $M_2$. The third level of a $Gray$-computad thus consists of source-target maps
and this completes the data of a $Gray$-computad.
Each $Gray$-category $G$ has an underlying $Gray$-computad $U(G)$, in a straightforward way. For the 0-cells, define $U(G)_0$ to be $G_0$. For the 1-cells, let $U(G)_1$ be $G_1$, equipped with source-target data data
For the 2-cells, using the fact that $G^{(1)}$ is a 1-category, define the 2-cells $U(G)_2$ as the pullback
where $\varepsilon: M_1(G^{(1)}) \to G^{(1)}$ is the canonical counit or evaluation from the free category on $G^{(1)}$ to $G^{(1)}$. The top horizontal arrow of the pullback gives the source and target data on 2-cells of $U(G)$.
For the 3-cells, using the facts that $G^{(2)}$ is a sesquicategory, define the 3-cells $U(G)_3$ as the pullback
where $\mu: M_2(G^{(2)}) \to G^{(2)}$ 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 $U(G)$.
The underlying functor
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 $Gray$-category $F(C)$ generated from a $Gray$-computad $C$. The easy part is what the $Gray$-category skeleta $F(C)^{(k)}$ look like for $k \lt 3$: they are just $M_k(C^{(k)}$, where $M_2(C^{(2)})$ 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 $C^{(2)}$, at least up to equivalence.
The harder part is getting a clear picture of the 3-cells of $F(C)$. 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:
The 3-cells of the $Gray$-computad $C$.
3-cells which arise from $Gray$-interchange isomorphisms.
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 $Gray$-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 $Gray$-category on the $Gray$-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 $x$-, $y$-, and $z$-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.
We now focus attention on the free $Gray$-category $F(\mathbf{1})$ generated from the terminal $Gray$-computad $\mathbf{1}$, and consider surface diagrams for modeling the primitive 3-cells used to generate $F(\mathbf{1})$.
First, the primitive 3-cells coming from the 3-cells of $\mathbf{1}$. There is exactly one 3-cell between any two given 2-cells of $\mathbf{1}$ that have the same source and have the same target. Now 2-cells of $\mathbf{1}$ 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 $X$, $X'$ 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 $X \to X'$. (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 $v = (\frac1{2}, \frac1{2}, \frac1{2})$ be the barycenter of the cube $[0, 1]^3$. Embed $X$ at the bottom of the cube:
and $X'$ at the top $[0, 1]^2 \times \{1\}$. Further, extend the line segments where $X$ meets the edges $y = 0$ and $y = 1$ to rays extending out to infinity (in the plane $z = 0$), and similarly extend the line segments where $X'$ meets the edges $y = 0$ and $y = 1$ 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 $X_1 \cup X_{1}'$ to $v$. The closure of
defines a (at most) 2-dimensional set $C_2 \subseteq [0, 1]^3$. Similarly,
defines a (at most) 1-dimensional set $C_1 \subseteq [0, 1]^3$. The filtration
defines a surface diagram, called the coning $C(X, X')$. This models a primitive 3-cell coming from the $Gray$-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 ($z = 0$ and $z = 1$) orthogonally, by applying a suitable deformation. Otherwise, this construction gives the correct picture.)
Next, suppose given generic string diagrams $X: s \to t$, $X': s' \to t'$ (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)
which “performs a $Gray$-interchange”, i.e., deforms the diagram $D$ given by
to the diagram $D'$ given by
and in fact it would be easy to write an actual standard formula for $h$. The following picture should give the idea:
The graph of this isotopy is the filtered set $Y$ whereby
There are no 0-strata in $Y$: $Y_0 = \emptyset$. This surface diagram $Y$ models a Gray 3-cell interchange isomorphism (and once again, it needs to be straightened a bit so that $Y$ meets the top and bottom faces orthogonally).
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.
A progressive surface diagram in $K = [0, 1]^3$ is a filtration $X$ by compact subspaces,
such that each $X_{k} - X_{k-1}$ is a smoothly embedded $k$-dimensional manifold with corners (with corner sets in $\partial K$), satisfying the following conditions:
The connected components of the manifolds $X_k - X_{k-1}$ are strata of a Whitney stratification of $K$;
Globularity: $X_0$ does not meet the planes $z = 0$ and $z = 1$, $X_1$ does not meet the planes $y = 0$ and $y = 1$, and $X_2$ does not meet the planes $x = 0$, $x = 1$. For all sufficiently small $\varepsilon \gt 0$, the 1-strata and 2-strata meet the planes $z = \varepsilon$ and $z = 1 - \varepsilon$ orthogonally, and the 2-strata meet the planes $y = \varepsilon$ and $y = 1 - \varepsilon$ in lines parallel to the $z$-axis.
Progressivity: The projection $(x, y, z) \mapsto (y, z)$, 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 $z = c$ 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 $\varepsilon$ seconds, and that within an $\varepsilon$-neighborhood of the boundary of the square, the picture of the string diagram $z = c$ in the unit square is always the same throughout the movie ($0 \leq c \leq 1$). This rigidity near the boundary still leaves plenty of room for action inside the square.
Each slice of a progressive surface diagram $X$ is a progressive string diagram.
This is straightforward; we touch on the essential points. The progressivity condition on $X$ implies that the third projection $\pi_3: (x, y, z) \mapto z$ is regular on the submanifold $X_k - X_{k-1}$, so that for any $c \in [0, 1]$,
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 $X$ also implies that the projection $(x, y) \mapsto y$, applied to any stratum within the slice $z = c$, 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:
By regularity of the map $\pi_3: (x, y, z) \mapsto z$ when applied to any stratum, this map is a submersion when applied to any $k$-stratum for $k \gt 0$. It follows from stratified Morse theory that the topology of the stratified sets $z = t$ undergoes no change between $c \leq t \leq d$, provided that there are no 0-strata with $z$-coordinates in this region. More precisely still, the region $c \leq z \leq d$ is the graph of an isotopy (a deformation) from the string diagram $z = c$ to $z = d$, by Thom’s first isotopy lemma.
Therefore the topology of the slices changes only through passage through a 0-stratum. In some small (closed) box or rectilinear neighborhood $N$ of a 0-stratum, the intersection $N \cap X$ has the topology of a coning of two string diagrams.
Regarding the second point, we are not yet claiming that $N \cap X$ is an honest-to-god coning: first we would need that the top and bottom slices of $N$ describe generic string diagrams, and moreover there is the somewhat irksome globularity condition we would additionally need for this box $N \cap X$, which practically never occurs. However, we do claim that it is possible to deform $X$ to achieve such a thing.
Thus, the ultimate claim is that any progressive surface diagram $X$ is deformation-equivalent to a progressive surface diagram $Y$ which can be partitioned into finitely many boxes $N$, where each $N \cap Y$ either is the graph of an isotopy between generic string diagrams or is a coning of generic string diagrams. This $Y$ thus models a 3-cell in the free Gray-computad $F(\mathbf{1})$. Moreover, any choice of such $Y$ in the deformation class of $X$ models the same 3-cell. Therefore, we may assign this 3-cell to the (class of) $X$ 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.
At a first pass, we might try defining deformations by analogy with the string diagram case. Suppose $X$ and $Y$ are progressive surface diagrams having strictly the same generic string diagram as domain and strictly the same generic diagram as codomain. A deformation from $X$ to $Y$ should be a map $h: I \times K \to K$ (remember $K = [0, 1]^3$) such that
$h$ is constant along some neighborhood of $\partial K$ (i.e., $h(0, x) = h(s, x)$ for all $0 \leq s \leq 1$ and all $x$ in some neighborhood of $\partial K$);
$h(0, -): K \to K$ is the identity;
Each $h(s, -): K \to K$ is a homeomorphism that maps each manifold $X_k - X_{k-1}$ diffeomorphically onto its image, for all $0 \leq s \leq 1$;
$h(1, -): R \to R$ maps $X_k - X_{k-1}$ onto $Y_k - Y_{k-1}$.
$h(0, x) = h(t, x)$ and $h(1, x) = h(1-t, x)$ for all sufficiently small $t$.
(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 $x$-axis.
(For those who like explicit formulae, a piece of it looks like locus of $y = x^3 - x$, $z = x^2$. The obvious stratification to use to enforce progressivity is by taking $X_0 = \{(0, 0, 0)\}$ and $X_1 = X_2 =$ the locus. This may be reparametrized and straightened near the top so that it satisfies globularity in the standard cube.)
Now apply a Reidemeister I move, so as to remove the crossing as seen from the perspective of projection onto the $y z$-plane. In terms of an explicit formula, we may use the deformation whose slice at time $t$ is the surface diagram given by the locus $L$ of $y = x^3 - (1 - 2t)x$, $z = x^2$ (with each slice stratified by taking $X_0 = \{(0, 0, 0)\})$ and $X_1 = X_2 = L$).
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 $\gamma \circ_2 \phi$ (across a 2-morphism) of a Gray-interchange isomorphism $\gamma$ with a 3-cell $\phi$ 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 $\phi'$ from the terminal computad. Since $\gamma \circ_2 \phi \neq \phi'$, 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 $z = c$. Recall that a “coincidence” in a string diagram is where two or more 0-strata have the same $y$-coordinate. Let us call…
(To be continued…)
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 $n$-categories. This study would involve a fascinating marriage between topics in pure higher category theory ($n$-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 $\Sigma$ of subsets of Euclidean spaces $\mathbb{R}^n$ which
Is closed under all first-order logical constructions: it contains the equality relation on $\mathbb{R}$, and is closed under finite intersections, finite unions, complements, finite products, and direct images under coordinate projection maps from one Euclidean space to another,
Contains some basic relations and functions to get off the ground: every constant $a \in \mathbb{R}$, the relation $\lt$, and the graphs of the addition function, the multiplication function, and the exponential function,
Is guaranteed to be “free of pathology”; for example, all sets in $\Sigma$ have only finitely many connected components, and all admit Whitney stratifications (by a canonical process).
The smallest class $\Sigma$ which meets these conditions is the class of subexponential sets (as explained in appendix B), but any class $\Sigma$ with these specified properties will do. Let us fix such a class $\Sigma$. We remark that the topological closure of a member of $\Sigma$ also belongs to $\Sigma$, because the closure is definable by a first-order property.
A $\Sigma$-surface diagram in $I^n = [0, 1]^n$ is a filtration by closed subsets belonging to $\Sigma$,
such that each $X_k - X_{k-1}$ is a $k$-dimensional smooth manifold with corners (whose corner sets are embedded in $\partial I^n$), such that
The connected components of the $X_k - X_{k-1}$ are strata of a Whitney stratification of $[0, 1]^n$;
(Globularity) Let $S_k$ be the span of the last $k$ standard basis vectors $e_{n-k+1}, \ldots, e_{n-1}, e_n$. For $\varepsilon \gt 0$ and $i \in \{0, 1\}$, let $(\partial I^n)_{k, i, \varepsilon}$ denote an $\varepsilon$-neighborhood of $I^{n-k} \times \{i\} \times I^{k-1}$ in $I^n$. Then, for each $(k, i)$, there are finitely many interior points $p_j$ (i.e., $p_j \notin \partial I^n$) such that
for all sufficiently small $\varepsilon \gt 0$.
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
An $n$-dimensional surface diagram $X$ in $[0, 1]^n$ is progressive if the restriction of the projection
to each stratum is a regular function.
The following geometric results are fundamental:
Let $X$ be a progressive $n$-dimensional $\Sigma$-surface diagram, and let $\pi_n: [0, 1]^n \to [0, 1]$ denote the last projection. Then every slice $X_c = \pi_{n}^{-1}(c) \cap X$ is a progressive $(n-1)$-dimensional $\Sigma$-surface diagram.
Let $X$ be a progressive $n$-dimensional $\Sigma$-surface diagram with no 0-dimensional strata. Then $X$ is ($\Sigma$-definably) isomorphic to a graph of an isotopy, i.e., there is a (definable) homeomorphism
that restricts to a diffeomorphism on each stratum, and such that $\pi_n = \pi \circ h$ where $\pi$ denotes projection to the interval $I$.
Let $X$ be an $n$-dimensional $\Sigma$-surface diagram, and let $p$ be a 0-stratum. Then there is a small closed $n$-cube $N$ of $p$ such that the stratified set $N \cap X$ is isomorphic to a cone on the stratified set $\partial N \cap X$.
For these results to be applicable to general surface diagrams, we want to following conjecture to be true:
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 $X$ is a groupoid $G$ each of whose objects is an open set of $X$, and whose morphisms are homeomorphisms between such open sets, satisfying the following conditions:
Let $G$ be a pseudogroup on $X$. A $G$-chart on a topological space $M$ is an open subset $U$ of $M$ together with an embedding
Two charts $\phi: U \to X$ and $\psi: V \to X$ are compatible if
belongs to $G$. A $G$-atlas on $M$ is a family of compatible charts $(\phi_\alpha: U_\alpha \to X)_\alpha$ such that $(U_\alpha)_\alpha)_\alpha$ covers $M$. The (restricted) maps $\phi_{\alpha \beta} = \phi_\beta \circ \phi_{\alpha}^{-1}$ are called transition functions between the charts of the atlas.
Finally, a $G$-manifold is a topological space equipped with a $G$-atlas. We can think of a $G$-manifold as a space which is locally modeled on $X$ according to the geometry $G$.
For $n$-dimensional manifolds with corners, we take $X = [0, 1]^n$, and we take the morphisms of $G$ to be diffeomorphisms between open sets of $X$, i.e., $C^{\infty}$ maps that have a $C^{\infty}$ inverse. (If $A \subseteq \mathbb{R}^m$ and $B \subseteq \mathbb{R}^n$ are arbitrary subsets, a smooth map $f: A \to B$ is by definition a function such that the composite
has an extension to a smooth function $F: U \to \mathbb{R}^n$ defined on an open set $U$ containing $A$.)
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 $G$-manifolds $f: M \to N$ (defined by chosen atlas structures) is a homeomorphism $f$ such that
is in $G$ whenever $(U, \phi)$ is a coordinate chart of $x \in M$, and $(V, \psi)$ is a coordinate chart of $f(x) \in N$. If $M_1$ and $M_2$ are two $G$-manifold structures on the same topological space $M$, then $M_1$ and $M_2$ are considered equal as $G$-manifolds if $id: M \to M$ is an isomorphism from $M_1$ to $M_2$ (and hence also from $M_2$ to $M_1$).
Many standard notions from the differential geometry of ordinary manifolds carries over to manifolds with corners. In particular, each point $x \in M$ has an algebra $O_x$ of germs of smooth (real-valued) functions defined on neighborhoods of $x$, and a tangent vector at $x$ is just a derivation
The vector space of tangent vectors $T_x(M)$ is $n$-dimensional, no matter if $x$ is a boundary point or not. A smooth map $f: M \to N$ induces a linear map $f_{\star, x}: T_x(M) \to T_{f(x)}(N)$.
As usual, we say a smooth map $f: M \to N$ is a smooth embedding if $f$ maps $M$ homeomorphically onto its image and each map $f_{\star, x}$ is a linear embedding.
Given a manifold with corners $N$ and a topological embedding $f: M \to N$, there is at most one smooth structure on $M$ that renders $f$ a smooth embedding.
In particular, if $i: S \subseteq N$ is a locally closed subset of a manifold with corners, it makes sense to ask whether there is a smooth structure on $S$ 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.
Let $i: S \subseteq X$ be an (embedded) submanifold. The normal bundle over $S$ in $X$ is the cokernel $N_X(S)$ in the exact sequence
in the abelian category of vector bundles over $S$. If $N_{X, 0}(S) \subseteq N_X(S)$ denotes the subset of zero vectors, there is an evident diffeomorphism $\phi_0: S \to N_{X, 0}(S)$.
A tubular neighborhood of $S$ in $X$ is an open neighborhood $U$ of $S$ which admits a diffeomorphism $\phi: U \to N_X(S)$ that extends $\phi_0$.
Every closed submanifold $S \subseteq X$ has a tubular neighborhood.
A proof may be found here for example.
Let $X$ and $Y$ be embedded submanifolds of a manifold $Z$ (typically $\mathbb{R}^n$), of dimensions $i$ and $j$ respectively, with $Y$ contained in the closure of $X$. Then
The pair $(X, Y)$ satisfies Whitney’s condition A if whenever a sequence of points $x_1, x_2, \ldots$ in $X$ converges to a point $y$ in $Y$, and the sequence of tangent $i$-planes $T_{x_m} X$ converges to an $i$-plane $T$ as $m$ tends to infinity, then $T$ contains the tangent $j$-plane $T_y Y$.
The pair $(X, Y)$ satisfies Whitney’s condition B if for each sequence $x_1, x_2, \ldots$ of points in $X$ and each sequence $y_1, y_2, \ldots$ of points in $Y$, both converging to the same point $y$ in $Y$, such that the sequence of secant lines $L_m$ between $x_m$ and $y_m$ converges to a line $L$ as $m$ tends to infinity, and the sequence of tangent $i$-planes $T_{x_m} X$ converges to an $i$-plane $T$ as $m$ tends to infinity, then $L$ is contained in $T$.
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 $X$ with some tubular neighborhood of $Y$ in $Z$ has the structure of a locally trivial fiber bundle.
A Whitney stratification of a set $X$ in $\mathbb{R}^n$ is a filtration by closed subsets,
\\emptyset = X_{-1} \subseteq X_0 \subseteq X_1 \subseteq \ldots \subseteq X_n = X
such that
Each difference $\\X_i - X_{i-1}$ is an $i$-dimensional manifold (whose connected components are called strata), and
For each pair of strata $(c, d)$ with $d$ contained in the closure of $c$, the pair $(c, d)$ satisfies the Whitney conditions.
We remark that for any pair of strata $(c, d)$ whose closures have nonempty intersection, either $c \subseteq \bar{d}$ or $d \subseteq \bar{c}$.
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.
A result of fundamental importance for Whitney stratified spaces is:
Suppose $X$ is a Whitney stratified set in $\mathbb{R}^n$ and $\pi: X \to \mathbb{R}^k$ is a proper submersion when restricted to each stratum. Then there is isomorphism of Whitney-stratified sets, i.e., a stratum-preserving homeomorphism
which restricts to a diffeomorphism on each stratum and such that $\pi = \pi_1 \circ h$. In particular, every fiber of $\pi$ is isomorphic to the fiber $\pi^{-1}(0)$ as a Whitney-stratified set.
Perhaps needless to say, the main application is to the case $k = 1$, where the slices $\pi^{-1}(t)$ define an isotopy through Whitney stratified sets.
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 $(X, Y)$ of a space $X$ and a closed subspace $Y$, and assuming… that both strata $Y$ and $X \backslash Y$ are topological manifolds. The naive idea, in such a situation, is to consider “the” tubular neighborhood $T$ of $Y$ in $X$, whose boundary $\partial T$ should also be a smooth manifold, fibred with compact smooth fibers over $Y$, whereas $T$ 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 $X \to Aut(X)$ 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 $\mathbb{R}^n$, 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 ($C^\infty$) 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 $C^{\infty}$ 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.
In model theory, a structure on $\mathbb{R}$ consists of collections of subsets of $\mathbb{R}^n$, $\Sigma_n \subseteq P(\mathbb{R}^n)$ which are closed under the operations of first order logic with equality:
Each $\Sigma_n$ is a Boolean subalgebra of $P(\mathbb{R}^n)$,
For each function $f: m \to n$ between finite sets, the inverse image function
restricts to a map of Boolean algebras $\Sigma_m \to \Sigma_n$.
For each function $f: m \to n$ between finite sets, the direct image function
restricts to a function $\Sigma_n \to \Sigma_m$ which is left adjoint to $(\mathbb{R}^f)^{-1}: \Sigma_m \to \Sigma_n$.
A structure on $\mathbb{R}$ is o-minimal if it contains the order relation, $\lt \in \Sigma_2$, 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 $\Sigma = \sum_{n \geq 0} \Sigma_n$ a ($\Sigma$)-definable set. Then the o-minimality axiom guarantees that $\subseteq{N}$ 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 $\Sigma$, or space-filling curves, etc.
Perhaps the archetypal example of an o-minimal structure is the class of real semi-algebraic sets, where $S$ 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 $x = (x_1, \ldots, x_k)$) with coefficients in $\mathbb{R}$:
It is easy to check that the $\Sigma_n$ are Boolean algebras and that the definability is preserved by inverse images along the $\mathbb{R}^f$; the substantial part is the following “quantifier elimination” theorem:
The direct image of a semi-algebraic set under a map $\mathbb{R}^f$ is semi-algebraic.
This essentially means that if $R(x_1, \ldots, x_m, y_1, \lots, y_n)$ is any formula in propositional language of ordered fields plus all real numbers as constants, then $\exists_{x_1, \ldots, x_m} R(x_1, \ldots, x_m, y_1, \ldots, y_n$ 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 $X$ and $Y$ are definable sets, then we may speak of a definable relation $R \subseteq X \times Y$. There is a bicategory of definable relations. A definable map $f: X \to Y$ is a definable relation that is the graph of a function from $X$ to $Y$. There is an evident category of definable sets and definable maps.
There are many known examples of o-minimal structures:
Semilinear sets: the Boolean combinations of sets defined by linear inequalities $L(x) \leq 0$, where $L: \mathbb{R}^n \to \mathbb{R}$ is a linear functional, are the definable sets of an o-minimal structure.
Subexponential sets: there is an o-minimal structure which contains the graphs of addition, multiplication, and the exponential function, namely the o-minimal structure of subexponential sets.
Here are the relevant definitions. A exponential set is a finite Boolean combination of loci of the form
and a subexponential set is the direct image of an exponential set under a map $\mathbb{R}^f: \mathbb{R}^n \to \mathbb{R}^m$. Here are two critical facts:
Each subexponential set has only finitely many connected components.
The complement of a subexponential set $S \subseteq \mathbb{R}^n$ 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 $0 \lt t \lt 1$ let
For $a \lt b \lt c \lt d$ define
This function is supported on $[a, d]$, valued in $[0, 1]$, and identically $1$ on the interval $[b, c]$.
In some sense subexponential sets should also be sufficient for any surface diagram purpose we have in mind. More technically, surface diagrams in $n$-dimensional space (relative to a given o-minimal structure $\Sigma$ containing the subexponential sets), together with isotopies between them, isotopies between isotopies, etc., should form an $(\infty, n)$-category $n$-$Surf_\Sigma$, and the conjecture (based on a passage from the Esquisse!) is that for any inclusion $\Sigma \hookrightarrow \Sigma'$ of such o-minimal structures, the induced inclusion
should be an equivalence of $(\infty, n)$-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).
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:
Suppose given an o-minimal structure, and a definable set $X$ which is Whitney stratified into definable strata. If $\pi: X \to \mathbb{R}^k$ is a definable proper submersion when restricted to each stratum, then there is a definable isomorphism of Whitney-stratified sets
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 $C^k$ diffeomorphism for each finite $k$, but in the full generality of o-minimal structures there is currently no proof which assures $C^\infty$ diffeomorphisms on the strata. This phenomenon of a gap between $C^\infty$ and $C^k$ is well-known in the theory of o-minimal structures, but fortunately for all known examples of o-minimal structures, we can actually achieve $C^\infty$ smoothness. For the semi-algebraic case, the reader may consult the reference by Coste-Shiota below.
Michael Batanin, Computads for finitary monads on globular sets, Contemporary Math. 230, Amer. Math. Soc. (1998), 37-57.
Lou van den Dries, Tame Topology and O-minimal Structures, LMS Lecture Note Series, Cambridge University Press, 1998.
The next reference sketches a proof of the o-minimal definable version of the first isotopy lemma.
Jesús Escribano, Definable families of definable stratified sets, Talk given at the International Workshop on Real Geometry, Madrid, 2001. See also
M. Goresky and R. MacPherson, Stratified Morse Theory, Ergibnesse der Mathematik und ihrer Grenzgebiete, 3 Folge Band 14, Springer-Verlag (1988).
Ta Lê Loi, Verdier and strict Thom stratifications in o-minimal structures, Ill. J. Math. 42, No. 2 (1998), 347-356.
John Mather, Notes on topological stability, unpublished notes written at Harvard, 1970.
Leila Schneps and Pierre Lochak (eds.), Geometric Galois Actions 1: Around Grothendieck’s Esquisse d’un Programme, LMS Lecture Note Series, Cambridge University Press, 1997.
Patrick Speissegger, The Pfaffian closure of an o-minimal structure, Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 1999, Issue 508, 189–211.