We outline basic concepts of framed combinatorial topology and provide links and references for further reading.
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 (homotopy theory, higher operads, computads, geometric models thereof, etc.).
Basic ideas and definitions
Many useful ideas are explained in bite-sized chunks in the introductory material listed below.
Readers familiar with the intuition that relates stratified manifolds and higher category theory (in the sense of string diagrams, but generalized to higher dimensions), might want to see definitions straight away, and can do so in this expository paper.
Pictures of the basic building blocks
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  in Chapter 1 and Chapter 2 respectively.
From framed cells to stratifications
If we geometrically realize framed regular cells (and forget their PL structure) then we arrive at a framed topological notion called “mesh cells”. Mesh cells locally assemble into so-called meshes. The notion of meshes is flexible and can be dualized, leading to a distinction of “closed” meshes (which are gluings of mesh cells) and “open” meshes (which are gluings of Poincaré duals of mesh cells).
A “tame” stratification is a stratification that is meshable, that is, it can be refined by a mesh. Tame 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): in fact, these are very special tame stratifications… they are tangles (with defects)!
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.
There are two broad directions: we can study framed stratified spaces either “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 of stratifications that can be characterized as being “generic” with respect to the framing (and satisfy a conicality condition).
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 .
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 , 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 (see e.g. the categorical Thom-Pontryagin construction). We regard this as a potential starting point for the study of “geometric higher category theory”.
Towards computable manifold theory and higher-dimensional logic
The previous two research directions have two noteworthy applications.
Firstly, the above combinatorial approach to geometric topology may overcome certain foundational obstructions in the computability theory of manifolds. This is further discussed in this note.
Secondly, the geometric perspective on higher category theory is especially convenient for thinking about higher-categorical foundations, as it automatically generates the required coherences in the “$(\infty,\infty)$-topos of $(\infty,\infty)$-categories”. This is discussed in this note.
List of open problems
The following is an (incomplete, currently still very short) list of open problems.
- The conjecture that weak computads in the above sense are a model for higher categories (i.e. comparison to other existing models). Lukas Heidemann is currently working on this problem in his PhD thesis.
- 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).
Papers, books, theses
 “Associative \(n\)-categories”, 2018, Christoph Dorn
 “High-level methods for homotopy construction in associative \(n\)-categories”, 2019, David Reutter & Jamie Vicary
 “Framed combinatorial topology”, 2021, Christoph Dorn & Christopher Douglas
 “Zigzag normalisation for associative $n$-categories”, 2022, Lukas Heidemann, David Reutter & Jamie Vicary
 “Manifold diagrams and tame tangles”, 2022, Dorn + Douglas
 “A brief introduction to framed combinatorial topology”, Dorn + Douglas
 “Globular manifold diagrams”, Dorn + Douglas + Heidemann
 “Weak computadic shapes for higher category theory”, Dorn + Douglas + Heidemann
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)
- A note on elementary isotopies (i.e. perturbation-stable isotopies) and their heuristic relationship to the homotopy groups of spheres.
The following further material may be helpful to look at:
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