Diagrammatic mathematics: an overview

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Abstract. Diagrammatic mathematics shines new light on many old problems, ranging from mathematical foundations to specific open problems in geometric topology, and structures from quantum physics.

What diagrams?

Manifold diagrams generalize string and surface diagrams to higher dimensions. They have seen substantial development over the last 5 years and their theory, owing to the natural integration of geometric, topological and algebraic ideas, continues to produce a lot of elegant and interesting mathematics.

Their theory goes far beyond of that of “categorical pasting diagrams”: they combine stratified geometry, combinatorial topology, knot and singularity theory into one cohesive framework. Nonetheless, a ‘killer’ application of this framework to other areas of contemporary mathematics is still missing. Personally, I still really liked working on manifold diagrams—to me, it felt like catching two birds with one stone.

• Firstly, the theory addresses interesting foundational questions spanning across geometry, topology and higher algebra, including questions such as: do we really need topological machinery to easily talk about braids and their higher-dimensional analogs? Do we really need differential machinery to easily talk about manifold singularities?

• Secondly, the theory studies structures that often appear to be relevant to mathematical physics, via the many interactions of Algebraic Topology and Quantum Field Theory or String theory. Importantly, it does so in (canonically) combinatorial, and often computable, terms. This reduction of geometry in physics to discrete structure is vaguely reminiscent of ideas going by names like ‘it from bit’, ‘computational universe’, and ‘digital physics’, or more peculiar approaches such as ‘causal set theory’ (here, ‘peculiar’ is not meant in any negative way … I would call manifold diagrams peculiar themselves at this point in time).

These two vague points aside, in this note, I wanted to briefly list my aspirations for the ‘manifold-diagrammatic’ line of research at the intersection of geometry and higher algebra. Many of the mentioned projects are still actively worked on by others in the field: in particular, Christopher Douglas, David Reutter, Jamie Vicary, … . If you are interested, you should not hesitate to contact these researchers.

Branches of diagrammatic mathematics

Living at the intersection of fields, diagrammatic mathematics provide new perspective on many ideas. It also unfiy a lot of ideas in this vein.

Diagrammatic category theory and type theory

Manifold diagrams can be used as a basis for defining what it means to compose in a higher-dimensional categorical setting; in other words, they can be used as pasting diagrams of morphisms. In some sense they are the canonical “most general” general of such pastings.

What is really intriguing about this approach is that it address categorical coherences in a purely geometric fashion, conceiving them as isotopies of stratified manifold; for example, the fundamental naturality law of natural transformations in category theory is simply the braid isotopy. This not only thinks along the lines of a (stratified) cobordism hypothesis, it build geometry into the fundament of higher categories.

This approach pays off in an unexpected way. Inductive types (a.k.a. computads) become elegant mathematical objects: in the diagrammatic setting we also speak of diagrammatic computads. Indeed, by conceiving coherences as isotopies, we save our “large computations” to work with “small example”. The ability to work with “small examples” is a deep property of this approach. See also the nine short stories about diagrammatic higher categories for more. (edit: and this later article).

Diagrammatic knot theory, singularity theory, and geometry

Diagrammatic geometry is my favorite diagrammatic mathematics: it studies knots, manifolds, and their singularities through the lens of manifold diagrams. There are also some far-reaching conjectures about diagrammatic representations of manifolds being faithful invariants for their smooth structure: this would be extremely cool, and a first “truly combinatorial” representation of smooth manifolds.

And let me add one more thing. I proclaim (it’s a bit much, but hey, nothing to lose!) that the diagrammatic approach to embedded manifolds could play a key role in understanding the remaining case of the smooth Poincaré conjecture: by capturing and classifying deformations as manifold diagrams. (In fact, I know that some are thinking along these lines already!) This can be based on the key insight that exotic $S^4$ must embed in $\mathbf{R}^5$.

Diagrammatic combinatorial topology

There are some cool things to be said also to the classical combinatorial topologists among us. Dual to any manifold diagrams there is a cell diagrams. These cells form cell complexes, which give rise to a sort of “framed cell complex” notion. There are many intriguing ideas waiting to be explored here (algorithms, higher analogs of DAGs, i.e. directed acyclic graphs, etc.), some of which you’ll quickly spot if you have a look at the Framed combinatorial topology book.

Things I’ve been working on

I recently posted the following research summary on the Category Theory Zulip server. I’ll be lazy and just re-post it here.

For a while I’ve been thinking about the “geometry of composition”, or in more technical terms, the higher-dimensional analogs of string diagrams (see manifold diagram). There is a bigger story to be told here, about “geometric”, “topological” and “algebraic” models of higher structures, and I tried to sketch this story in a recent \(n\)-Category Café post. There are two distinct types of interactions in the geometry of composition, and one of my long-term goals is to understand these two parts individually and how they interplay.

• distant interactions: isotopies (also called homotopies, or just coherences) are higher-dimensional analogs of braids. Examples include, for instance, the braid, the Yang-Baxterator, the Zamolodchikov eversion, etc.
• local interactions: singularities are local neighborhoods of the critical points of strata in diagrams (more precisely, in tangle diagrams). Examples include, for instance, the cup/cap, the birth/death of a circle, the saddle, the monkey saddle, the swallowtail, etc. …

In combination, isotopies and singularities make up all of homotopical behaviour (such as the homotopy groups of spheres; e.g. the hopf fibration is a capped-off braid, and the quaterniotic hopf fibration is a ‘capped-off’ Zamolodchikov eversion). In general, homotopical behaviour is very complicated… I’d say much of Algebraic Topology is dedicated to finding patterns in it. However, individually (to the extend that they can be separated) isotopies and singularities don’t seem that complicated! Well, still complicated, but somehow tractably so. Indeed, both can be formalized in a few words in the framework of manifold diagrams, and due to the combinatorial classification of such diagrams, they can be worked with easily ‘by hand’ up to the higher low dimension range. Both appear to have connections to loads of existing math, but the challenge is to understand those connections.

While the above provides my motivation for studying manifold diagrams, here are some more concrete short-term projects I’ve been thinking about.

1. Genericity properties of manifold diagrams: show that, while by default manifold diagrams are weakly globular, generically they are strictly globular. (Project with Lukas Heidemann and Christopher Douglas)

2. Theory of diagrammatic computads: diagrammatic computads and their functors are easily defined, but describing the higher category of diagrammatic computads and its properties is work in progress. (Project with Lukas Heidemann)

3. Complexification: there are a more or less straight-forward complex analogs of meshes (the ‘regular cell structures’ of diagrams). The role of resulting notions of \(\mathbb{C}\)-tangle diagrams should be similar to the role of complex singularities in Arnold’s singularity theory. (Project with Christopher Douglas)

Key problems: classifying higher knots and singularities

We can keep this short: really, there are only two key problems which address the respective classification of the two types of ‘behaviors’ pointed out above. We briefly discuss them below.

1. Classification of (perturbation-stable) isotopies. Using the geometry of manifold diagrams to define higher categories naturally yields ‘unbiased’ notions of composition since gluings of manifold diagrams themselves are unbiased in an appropriate sense (you could also say ‘associative’ in place of ‘unbiased’). This similarly affects the description of higher categorical coherences via isotopies of manifold diagrams. A priori there are infinitely many isotopies starting in dimension 3. However, the conjecture (alluded to, for instance, in the manifold diagram paper) is that there is a sub-class of so-called perturbation-stable isotopies. An example note that the ‘triple’ braid can be perturbed into three simple ‘binary’ braids; indeed, the latter is an elementary isotopy, which, as a categorical coherence, appears for instance in the definition of Gray categories.

2. Classification of (perturbation-stable) singularities. As explained in this blog article, at the root of the connection between geometry and higher algebra is the Thom-Pontryagin construction. In fact, classically, in place of relating higher morphisms and manifold diagrams, this relates invertible morphisms (namely, paths in spaces) and stratified tangles. (Up to putting these tangles into generic position) stratified tangles are closely related to manifold diagrams: in fact, any tangle can be uniquely refined by a manifold diagram by its ‘higher critical points’, also called ‘singularities’, see here and here for details. In order to classify singularities, we must once more restrict our attention to the perturbation-stable case. Similar to the case of isotopies, there is a wealth structure to be explored here: in the linked paper, we made the ‘ADE pattern conjecture’ that perturbation-stable singularities found in smooth tangles (embedded in codimension 1) are governed by the classical ADE classification of Arnold. (The ADE pattern is, in the words of Terry Gannon, a far-reaching ‘meta-pattern’.)

As mentioned in the the previous section, an incredible amount of behaviour ‘emerges’ when combining elementary isotopies and singularities; namely, all homotopical behaviour! (Hopf fibrations, homotopy groups of spheres, Bott periodicity, … you name it). A fun simple example to consider here is the computation of $\pi_3 S^2$ in manifold diagrams. But, given homotopical behaviour is complicated, the following word of warning will not come as a surprise: both points (1.) and (2.) above are not easy. In fact, Chris Douglas and I discussed that we expect (2.) to be closely related to the smooth Poincaré conjecture in dimension 4… one of the big(gest) open problems in geometric topology!

Final thoughts

Who knows where this line of research will go. For example, it would be great if, say, in 20 years from now, we wouldn’t think of Morse and singularity theory as something inherent to differential geometry any more, but understand that it’s simply another facet of a deeper higher-categorical idea. This and other unifications do seem within reach. And, yet more speculative, I’d even hope that this line of research will then ultimately affect our understanding of ‘grand unified physics’, and maybe we are already witnessing the beginning of that.

So long, and thanks for all the fish!