nLab n-mesh



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



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.


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.


A 1-mesh is a framed contractible kk-manifold MM, k1k \leq 1, together with a stratification ff on MM whose strata are open ll-disks, l1l \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 :𝒮tratCat \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).


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


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

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

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


A concrete way to think about 1-mesh bundles is explained in Dorn-Douglas 2021, Ch. 4: a 1-mesh bundle pp 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 bbb \to b' in the base, with p(x)=bp(x) = b for xx a point in a 00-stratum in the fiber over bb, have unique lifts to entrance paths xxx \to x'.


An nn-mesh bundle MM over a base stratification (B,g)(B,g) is a tower of 1-mesh bundles

(M n,f n)(M n1,f n1)...(M 1,f 1)(M 0,f 0)=(B,g) (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)=*(B,g) = \ast is the trivial stratification then the nn-mesh bundle is simply called an nn-mesh MM.


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


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


An nn-mesh map F:MNF : 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.)


Cell structures

Closed meshes are regular cell complexes (cf. Wikipedia entry). More generally, meshes MM are stratified 0 0 -types (meaning the fundamental \infty-categories (M i,f i)\mathcal{E}(M_i,f_i) are 00-truncated as \infty-categories, i.e. equivalent to posets).


The fundamental categories (M)=(f nf n1...f 0)\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, nn-meshes up to framed stratified homeomorphism are classified by nn-trusses (see at trusses for details), and nn-mesh bundles up to bundle isomorphism are classified by nn-truss bundles (and thus share the same classifying category).


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. ).


Last revised on May 12, 2023 at 10:01:53. See the history of this page for a list of all contributions to it.