# Framed combinatorial topology

We outline basic concepts of framed combinatorial topology and provide links and references for further reading.

## Program overview

Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology and higher algebra. The theory synthesizes notions from classical combinatorial topology with a new combinatorial approach to framings. Its aim is to exhibit elements of a unifying language relating, in particular, the following:

- the theory of stratified manifolds (singularity theory, higher Morse theory, classification of smooth structures, etc.),
- quantum algebra (TQFTs, invariants, knots, knotted surfaces and higher knots, etc.),
- and higher category theory (computable foundations, homotopy coherence, operads, computads, etc.).

### From framed cells to trusses

The basic building blocks of framed combinatorial spaces are “framed regular cells”. These are classical regular cells endowed with a “framing” structure. In Figure 1 we depict several such cells: in each case, we indicate the framing of the cell (as well as of some of its boundary cells) by a frame of vectors (indicated in green). Next to each cell we depict another structure: a “tower of posets” (with additional dimension and framing structure, in indicated by coloring object in red and blue respectively endowing fibers of maps with a direction indicated by the green arrows). These towers of posets are called trusses. Trusses are highly structured combinatorial objects (namely, each map in the tower is a combinatorial constructible bundle), and they serve to classify all framed regular cell shapes.

Framed regular cells and trusses are discussed in [3] in Chapter 1 and Chapter 2 respectively.

### From framed cells to framed stratifications

Technically, framed regular cells carry piecewise linear structure (they are “combinatorial” topological objects). If we forget this structure, we arrive at a topological notion, which we call a “mesh cell”. Mesh cells (locally) assemble into so-called meshes. The notion of meshes is flexible and can be dualized (the “geometric duality” is not easily accessible in the world of framed regular cells), leading to a distinction of “closed” meshes (those that are gluings of mesh cells) and “open” meshes (those that are gluings of dual mesh cells). The topologically enriched category of (closed resp. open) meshes is weakly equivalent to that of (closed resp. open) trusses.

A flat framed stratification is a stratification that is “meshable”, that is, it can be refined by a mesh. Flat framed stratifications are local models of framed stratified space (*locally* framings are *flat*). In Figure 2 we depict several examples of flat framed stratification of the open \(3\)-cube (all examples can be refined by an open mesh). Importantly (and in contrast to the classical fact that triangulations of a space may have neither mutual refinements nor coarsenings), there is always a unique coarsest refining mesh which is coarser than all other meshes refining a given flat framed stratification. This allows to established a faithful combinatorialization of the theory of flat framed stratifications.

Meshes and flat framed stratifications are discussed in [3] in Chapter 4 and Chapter 5 respectively.

## Program outlook

There are two broad directions: study framed (stratified) space “locally” or “globally”.

### Towards combinatorial geometric topology

Let us start with the study of local models for framed stratified spaces, that is, of flat framed stratifications. We mention two interesting subclasses.

Firstly, flat framed stratifications specialize to manifold diagrams (and, dually, to pasting diagrams of framed computadic cells). Manifold diagrams are framed conical stratifications. The class of manifold diagrams is particularly interesting as it contains a wealth of higher categorical coherences (also called isotopies) such as the braid, the Yang-Baxterator (a.k.a. the Reidemeister III move) and many others. One can ask which of these isotopies are “elementary”: elementary isotopies cannot be perturbed to a composite of simpler isotopies. Classifying and better understanding (elementary) isotopies is an interesting open problem, promising insights into the structure of manifold diagrams (and thus of pasting diagrams) in higher dimensions, and by extension into the systematics of higher algebraic coherences.

Another interesting subclass of flat framed stratifications is that of tame tangles. Tame tangles are manifolds (tamely) embedded in the unit cube. Tame tangles may be refined by a (unique coarsest) manifold diagram, and this allows us to translate tame tangles into categorical pasting diagrams in the spirit of the tangle hypothesis. Framed homeomorphism of tame tangles provides an interesting equivalence relation: around each point of a tangle it allows to distinguish tangle “singularities”, such as the saddle, the cusp, the swallowtails singularities. Again, one can ask which singularities are “elementary” i.e. stable under perturbation. The classification of elementary singularities is another open problem. It has close ties to classical (differential) singularity theory, but the combinatorial approach evades certain problems that the differential approach encounters with increasing numbers of parameters. There are further interesting relations to notions from higher Morse theory.

Both of the above directions are further discussed in [3].

### Towards geometric higher category theory

We may consider framed structures “globally” as well. For this, note that the category of framed regular cells and cellular maps is generated by face and degeneracy maps (just as other categories of shapes such as simplices or cubes). To define a notion of *framed spaces* we may consider presheafs on this category (technically, it makes sense to consider “sheafs”, meaning that we require certain colimits of cells to be preserved). Elements of a presheaf are the “cells” of the framed space that it defines. Face and degeneracy maps allow us to speak about “cell faces” and “degenerate cells” with the usual geometric interpretation.

Forgetting framings, one can similarly consider presheafs on the category of *non-framed* (i.e. ordinary) regular cells (N.B: this category can be defined in purely combinatorial terms; see [3], Section 1.3.1). Such presheafs model (non-framed) spaces build from regular cells. Algebraically, we regard these presheafs as playing the role of free higher *groupoids*: each non-degenerate cell freely attaches a new invertible non-identity morphism (the same interpretation of course applies for other combinatorial models of space, such as simplicial or cubical sets). The framed case works analogously. Namely, presheafs on the category of *framed* regular cells can be regarded as free higher structures as follows: without additional conditions such presheafs model a notion of free higher-fold categories; if we impose each cell in a framed space to be globular) in an appropriate sense, then we obtain a model for a notion of free higher categories, also called *weak computads*.

Finally, the equivalence of trusses and meshes induces an equivalence between the category of framed regular cells and the topologically enriched category of mesh cells. The above approach is thus simultaneously combinatorial and topological in nature: it both provides a traditional combinatorial approach on higher categories (via presheafs on a category of framed combinatorial shapes), while also allowing for a geometric interpretation of these structures (as spaces equipped with framing structure). As a result several interesting constructions linking combinatorics and geometry can be easily formalized. We regard this as a potential starting point for the study of “geometric higher category theory”.

### List of open problems

The following is an (incomplete, currently still very short) list of open problems.

- The classification of elementary singularities.
- The classification of elementary isotopies.
- The conjecture that all smooth structures can be presented by tame tangles (i.e. comparison to classical differential model of manifolds).
- The conjecture that weak computads in the above sense are a model for higher categories (i.e. comparison to other existing models)

## References

### Papers, books, theses

[1] “Associative \(n\)-categories”, 2018, Christoph Dorn, Oxford University Research Archive

This thesis defines a combinatorial notion of manifold diagrams by introducing the idea of *singular \(n\)-cubes labeled in \(\mathsf{C}\)* (these are now, in more general form, called “\(n\)-trusses [labeled] in \(\mathsf{C}\)”—see here for a terminological comparison). Based on a combinatorial definition of *manifold diagrams*, the thesis goes on to define combinatorial models of both free and non-free higher categories. It also sketches the path towards the categorical study of singularities and the PT construction (as outlined above). However, it certainly stops short of giving a full account of these ideas (in particular, it does not compare either notion to other models of higher categories)—the thesis is superseded, and shortcomings hopefully partially fixed, by [4],[5] and [7] below.

[2] “High-level methods for homotopy construction in associative \(n\)-categories”, 2019, David Reutter & Jamie Vicary, arXiv

This paper takes first steps towards a computer implementation of the idea of associative \(n\)-categories respectively manifold diagrams from [1], focusing in particular on the algorithmic construction of isotopies. Note that “singular \(n\)-cubes in \(\mathsf{C}\)” are called “\(n\)-iterated zigzags in \(\mathsf{C}\)” and the definition is slightly reworked from that in [1] (the resulting structures are the same; see here for a terminological comparison).

[3] “Framed combinatorial topology”, 2021, Christoph Dorn & Christopher Douglas, arXiv most recent version

This book aims to lay comprehensive foundations for the subject. It in particular introduces the notion of \(n\)-trusses (labeled in a category \(\mathsf{C}\)) which generalizes the notion of singular \(n\)-cubes from [1]. This generalization achieves that the theory of trusses becomes self-dual (with both the initial and terminal truss both living in the same category), which allows to uniformly treat the theory of “framed cells” and their duals, “framed (manifold diagram) singularities”. The theory of framed cells can then be introduced from first principles in a similar fashion to classical combinatorial topology, by endowing classical combinatorial structures with framings, and many interesting question (such as the comparison \(\mathrm{TOP}/\mathrm{PL}/\mathrm{DIFF}\)) carry over to the framed setting.

[4] “Efficient algorithm for normalization in associative \(n\)-categories”?, 2022, Lukas Heidemann, David Reutter & Jamie Vicary, arXiv

This paper describes an explicit (and computer-implemented) algorithms for normalization of labeled trusses with degeneracies in associative \(n\)-categories.

[5] “Manifold diagrams, tame tangles and singularities”, Dorn + Douglas, pre-preprint

The notion of “manifold diagrams” generalizes that of ordinary string diagrams to higher dimensions, and can be interpreted in familiar higher categorical terms by a process of geometric dualization. The closely related notion of “tame tangles” describes a well-behaved class of embedded manifolds with corners that can likewise be interpreted categorically. In this paper we formally introduce both of these notions in topological terms, and show that they admit a combinatorial classification, by using results from the toolbox of framed combinatorial topology. We then study bundles of tame tangles, and their stability under perturbation; this results in a combinatorial approach to differential singularity theory. We describe the resulting perturbation-stable singularities of tame tangles in low dimensions. We show that all smooth 4-manifolds can be presented as tame tangles, and conjecture that the same is true for smooth manifolds of any dimension.

[6] “An intrinsic geometric definition of manifold diagrams”, Dorn + Douglas + Heidemann, *In preparation*

[7] “Weak computadic shapes for higher category theory”, 2022, Dorn + Douglas + Heidemann, *In preparation*

### Introductory material

The following notes present various ideas in the context of FCT; they are written sequentially (with later notes often referring to earlier ones). The focus is mainly on intuition and brevity, and less on mathematical formality; however, some mathematical substance can nonetheless be found: to indicate this notes are rated with a “formality level” (**FL**) below.

- A note on framed computadic and regular cells, explaining the classical problem of defining universal classes of shapes and how to resolve it. (
**FL**1/5) - A note on manifold diagrams, the dual notion of pasting diagrams of framed computadic cells, introduced via a geometric approach. (
**FL**2/5) - A note on two paradigms of higher category theory, explaining syntactically powerful sources of categorical coherences. (
**FL**1/5) - A note on tangles, elementary singularities and higher Morse theory, explaining the close connections of pasting diagrams and the geometry of smooth manifolds. (
**FL**2/5) - A note on the basic combinatorial theory of trusses, leading up to formal defintion of weak computads, as well as combinatorial defintions of (combinatorial) manifold and pasting diagrams. (
**FL**4/5) - A note on the the formalization of the categorical Pontryagin-Thom construction, relating functors of weak computads to stratifications on them, and discussing how this should lead to a “combinatorialization” of smooth manifolds and cobordisms. (
**FL**3/5)

Planned note:

- A note on elementary isotopy (i.e. perturbation-stable isotopies) and their heuristic relationship to the homotopy groups of spheres.

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