nLab n-mesh

Idea

Meshes are towers of constructible stratified bundles whose fibers are points or framed stratified 1-manifolds. Meshes play a central role in the local description of directed (namely, framed) regular cell and dual-cell complexes.

Definition

Meshes are certain stratified spaces with additional structure (namely, structure that records spatial directions via a notion of framings). In the following definition, framings can be understood in the ordinary sense.

Definition

A 1-mesh is a framed contractible $k$-manifold $M$, $k \leq 1$, together with a stratification $f$ on $M$ whose strata are open $l$-disks, $l \leq 1$.

Contractability implies that any 1-mesh framed embeds in standard framed $\mathbb{R}$: we call such embeddings mesh-trivializations.

To prepare for the definition of a bundle of 1-meshes, recall, the fundamental $\infty$-category functor $\mathcal{E} : \mathcal{S}\mathit{trat} \to \mathbf{Cat}_\infty$ maps conical stratifications to their fundamental ∞-categories: for defining the latter, we work with entrance paths (which is the opposite convention to working with exit paths).

Remark

For simplicity, below one can replace the $\infty$-functor $\mathcal{E}$ with the ordinary fundamental poset functor $\mathsf{E}$. This yields an equivalent definition of $n$-meshes (even though the definition of $n$-mesh bundles will differ).

The following definition abstractly characterizes bundles of 1-meshes (see Rmk. for an elaboration). We assume all stratifications to be conical.

Definition

A 1-mesh bundle $p : (M,f) \to (B,g)$ is a stratified bundle with 1-mesh fibers $(M_x, f_x)$, for $x \in B$, and a fiberwise mesh-trivializing bundle embedding $p \hookrightarrow (\pi_B : B \times \mathbb{R} \to B)$ into the trivial $\mathbb{R}$-bundle over $B$, such that

1. $\mathcal{E}(p)$ is an exponentiable fibration,
2. $\mathcal{E}(p_{(0)})$ is an opfibration,
3. $\mathcal{E}(p_{(1)})$ is a fibration,

where $p_{(i)}$ denotes the restriction of $p$ to the union of strata in its domain whose non-empty intersections with fibers are all of dimension $i$.

Remark

A concrete way to think about 1-mesh bundles is explained in Dorn-Douglas 2021, Ch. 4: a 1-mesh bundle $p$ is a stratified bundle of 1-meshes that can be trivialized in the trivial framed line bundle, such that the image of the trivialization has continuous upper and lower fiberwise bounds, and such that entrance paths $b \to b'$ in the base, with $p(x) = b$ for $x$ a point in a $0$-stratum in the fiber over $b$, have unique lifts to entrance paths $x \to x'$.

Definition

An $n$-mesh bundle $M$ over a base stratification $(B,g)$ is a tower of 1-mesh bundles

$$(M_n, f_n) \to (M_{n-1}, f_{n-1}) \to ... \to (M_1, f_1) \to (M_0,f_0) = (B,g)$$

If $(B,g) = \ast$ is the trivial stratification then the $n$-mesh bundle is simply called an $n$-mesh $M$.

Remark

There are several equivalent phrasings and variations of the definitions of 1- and $n$-mesh bundles. For instance, the assumption of contractability in the definition of 1-meshes can be weakened.

Terminology

An $n$-mesh $M$ is call closed if $M_n$ is compact, and open if $M_n \cong \mathbb{R}^n$.

Definition

An $n$-mesh map $F : M \to N$ is a stratified map of towers that fiberwise preserves the framing. (In fact, this makes mesh maps framed maps in the sense of framed directed spaces.)

Properties

Cell structures

Closed meshes are regular cell complexes (cf. Wikipedia entry). More generally, meshes $M$ are stratified $0$-types (meaning the fundamental $\infty$-categories $\mathcal{E}(M_i,f_i)$ are $0$-truncated as $\infty$-categories, i.e. equivalent to posets).

Classification

The fundamental categories $\mathcal{E}(M) = \mathcal{E}(f_n \to f_{n-1} \to ... \to f_0)$ of meshes are certain towers of poset maps ‘with structure’: this gives rise to a combinatorial notion called trusses. In fact, $n$-meshes up to framed stratified homeomorphism are classified by $n$-trusses (see at trusses for details), and $n$-mesh bundles up to bundle isomorphism are classified by $n$-truss bundles (and thus share the same classifying category).

Duality

As a consequence of truss duality and of the classification of meshes by trusses, we obtain a involutive dualization endofunctor on the category of meshes. This functor can be understood as a geometric dualization operation in the sense of Poincare duality. It maps open to closed meshes and vice versa (see Trm. ).

Related concepts

• n-trusses

• manifold diagrams

• framed combinatorial topology

References

• Christoph Dorn and Christopher Douglas, Manifold diagrams and tame tangles, 2022 (arXiv, latest)

• Christoph Dorn and Christopher Douglas, Framed combinatorial topology, 2021 (arXiv, latest)