# nLab n-truss

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

$n$-Trusses are the fundamental categories of $n$-meshes. They are central to the combinatorial classification of manifold diagrams and play an important role in framed combinatorial topology.

## Definition

Trusses in $n$-dimensions, or $n$-trusses for short, are defined inductively in the dimension $n$.

### 1-Trusses

###### Definition

A 1-truss $T \equiv (T,\leq, \preceq, \mathrm{dim})$ is a set $T$ with two partial orders (the ‘face’ order $\leq$ and the ‘frame’ order $\preceq$) as well as a ‘dimension’ map $\dim : (T,\leq) \to ^{\mathrm{op}}$ such that

1. $(T,\preceq) \cong [n] = (0 \to 1 \to ... \to n)$
2. either $i \lt i+1$ or $i + 1 \lt i$ for all $i \prec n$
3. $\dim$ is conservative.

###### Definition

A 1-truss map $F : T \to S$ is a function of sets that preserves both face and frame order. Further,

• $F$ is called regular if $\dim \circ F \Rightarrow \dim$,
• $F$ is called singular if $\dim \circ F \Leftarrow \dim$,
• $F$ is called dimension-preserving if $\dim \circ F = \dim$,

where $\Rightarrow$ and $\Leftarrow$ denote natural transformations of functors $(T,\leq) \to ^{\mathrm{op}} = (0 \leftarrow 1)$.

###### Notation

Given a truss $T$, denote by $T_{(i)}$ the subset of objects $x$ with $\dim(x) = i$.

### 1-Truss bundles

To define bundles of 1-trusses, we first define what are the valid fiber transitions. We dub these ‘1-truss bordisms’.

###### Remark

Below, a Boolean profunctor is an ordinary profunctor $H : C$$D$ whose values are either the initial set $\emptyset \equiv \bot$ or the terminal set $\ast \equiv \top$. If $C$ and $D$ are discrete, then such a profunctor $H$ is simply a relation of sets. In this case, we call the profunctor $H$ a function if it is a functional relation or a cofunction if the dual profunctor $H^{\mathrm{op}}$ is a function.

###### Remark

For any map of posets $F : P \to Q$, the fiber $F^{-1}(x \to y)$ over an arrow $x \to y$ of $Q$ defines a Boolean profunctor $F^{-1}(x)$$F^{-1}(y)$ by mapping $(a,b)$ to $\top$ iff $a \to b$ is an arrow in $P$.

###### Definition

Given 1-trusses $T$ and $S$, a 1-truss bordism $R : T$$S$ is a Boolean profunctor $T$$S$ satisfying the following:

1. $R$ restricts to a function $R_{(0)} : T_{(0)}$$S_{(0)}$ and a cofunction $R_{(1)} : T_{(1)}$$S_{(1)}$.
2. Whenever $R(t,s) = \top = R(t',s')$, then either $t \prec t'$ or $s' \prec s$ but not both.

Importantly, 1-truss bordisms are morphisms of a category $\mathfrak{T}^1$ that embeds into the category of profunctors $\mathbf{Prof}$ (unlike general Boolean profunctors). See the discussion of ‘labels’ below for details.

###### Definition

A 1-truss bundle over a ‘base’ poset $(P,\leq)$ is a poset map $q : (T,\leq) \to (P,\leq)$ in which, for each $x \in P$, the fiber $(T^x,\leq) = q^{-1}(x)$ is equipped with the additional structures $(\preceq,\dim)$ of a 1-truss, and, for each arrow $x \to y$ in $P$, the fiber $q^{-1}(x \to y)$ is a 1-truss bordisms $T^x$$T^y$ (cf. Rmk. ).

###### Definition

A 1-truss bundle map $F : q_1 \to q_2$ of 1-truss bundles $q_i : T_i \to P_i$ is a map $F : T_1 \to T_2$ that factors through $q_i$ by a ‘base’ map $F_0 : P_1 \to P_2$, such that $F$ is fiberwise a 1-truss map (cf. Def. ). We further say $F$ is regular resp. singular resp. dimension-preserving if it is fiberwise so.

### n-Truss bundles and n-trusses

###### Definition

An $n$-truss bundle $T$ over a poset $P$ is a tower of 1-truss bundles (see Def. )

###### Definition

An $n$-truss bundle map $F : T \to T'$ is a tower of 1-truss bundle maps $F_i : q_i \to q'_i$ where $F_{i-1}$ is the base map of $F_i$ and $F_n \equiv F : T_n \to T'_n$. The adjectives ‘regular’ resp. ‘singular’ resp. ‘dimension-preserving’ apply to $F$ if they apply to all $F_i$. If $T$ and $T'$ have the same base $P$, then $F$ is called base-preserving if $F_0 = \mathrm{id}_P$.

###### Terminology

An $n$-truss bundle over the terminal poset $\ast$ is called an $n$-truss.

We can think of $n$-trusses as the ‘fibers’ of $n$-truss bundles: indeed, given an $n$-truss bundle $p$ over $P$, then any point $x \in P$ determines an $n$-truss by (iteratively) restricting $p$ over $\{x\}$.

## Labels and classifying categories

### Truss bundles with labels

###### Definition

Given a category $C$, a $C$-labeled $n$-truss bundle $T = (\underline T, \mathsf{lbl}_T)$ over $P$ consists of an ‘underlying’ $n$-truss bundle $\underline T = (T_n \to ... \to T_1 \to P)$ together with a ‘labeling’ functor $\mathsf{lbl}_T : T_n \to C$. In other words, $T$ is of the form

###### Remark

If $C = \ast$ is the terminal category in the previous definition, then we recover ordinary $n$-truss bundles.

###### Definition

A labeled $n$-truss bundle map $F = (\underline F, \mathsf{lbl}_F) : T \to T'$ from a $C$-labeled $n$-truss bundle $T$ to a $C'$-labeled $n$-truss bundle $T'$ consists of an $n$-truss bundle map $\underline F : \underline T \to \underline T'$ and a functor $\mathsf{lbl}_F : C \to C'$ such that $\mathsf{lbl}_{T'} \circ \underline F \cong \mathsf{lbl}_F \circ \mathsf{lbl}_{T}$ commutes up to natural isomorphism. Adjectives ‘regular’, ‘singular’, ‘dimension-preserving’, ‘base-preserving’ apply if they apply to $\underline F$. Further, we say $F$ is label-preserving if $\mathsf{lbl}_F = \mathrm{id}_C$.

Labeled truss bundles are a central ingredient in truss theory as the next section will demonstrate.

### Truss bundle classifications

Since fiber transitions in 1-truss bundles are 1-truss bordisms, it comes as no surprise that the category of 1-truss bordisms classifies 1-truss bundles.

###### Definition

Given 1-truss bordisms $R : T$$S$ and $Q : S$$U$, their composite profunctor $R \circ Q$ (composed as ordinary profunctors) is again a 1-truss bordism. (In contrast, composites of general Boolean profunctors (composed as ordinary profunctors) in general need not themselves be Boolean.) This defines the category $\mathfrak{T}^1$ of 1-trusses and their bordisms.

###### Theorem

$1$-truss bundles over a base poset $P$ up to dimension-preserving base-preserving isomorphism bijectively correspond to functors $P \to \mathfrak{T}^1$ up to natural isomorphism.

###### Proof

Follows from the definition of 1-truss bundles.

(There are also more categorical phrasings of this theorem, as well as the subsequent theorems below, which replace bijection with categorical equivalence; see e.g. Dorn-Douglas 2021, Ch. 2.)

The theorem now generalizes to labelled 1-truss bundles as follows.

###### Definition

Given a category $C$, the category $\mathfrak{T}^1(C)$ of $C$-labeled 1-trusses and their bordisms is defined as follows: objects of $\mathfrak{T}^1(C)$ are $C$-labeled 1-truss bundles over $$; morphisms are $C$-labeled 1-truss bundles over $$ (with domain and codomain given by restricting to fibers over $0$ resp. $1$); two morphisms compose to a third iff there is a $C$-labeled bundle over $$ that restricts over $(0 \to 1)$, $(1 \to 2)$, and $(0 \to 2)$ to the first, second, resp. third morphism. The fact that this defines a category is shown in Dorn-Douglas 2021, Sec. 2.3.1.

###### Theorem

$C$-labelled $1$-truss bundles over a base poset $P$ up to dimension-preserving base-preserving label-preserving isomorphism bijectively correspond to functors $P \to \mathfrak{T}^1(C)$ up to natural isomorphism.

###### Proof

Follows from the definition of $\mathfrak{T}^1(C)$.

###### Remark

(Recovering the unlabeled case). In particular, the preceding two definitions coincide $\mathfrak{T}^1(\ast) \cong \mathfrak{T}^1$ up to (essentially unique!) isomorphism of categories when $C = \ast$ is terminal.

###### Remark

(Functoriality of labeled bordism). Importantly, the construction of $\mathfrak{T}(C)$ is functorial in $C$. Indeed, given a functor $F : C \to D$, then $\mathfrak{T}^1(F) : \mathfrak{T}^1(C) \to \mathfrak{T}^1(D)$ acts on objects and morphisms of $\mathfrak{T}^1(C)$ by post-composing their labelings with $F$. This yields the labeled bordism functor

$\mathfrak{T}^1 : \mathbf{Cat} \to \mathbf{Cat}$

For a given category $C \in \mathbf{Cat}$ we can thus apply the labeled bordism functor $n$ times to it: the resulting category $\mathfrak{T}^n(C)$ classifies $C$-labeled $n$-truss bundless as follows.

###### Theorem

$C$-labelled $n$-truss bundles over a base poset $P$ up to dimension-preserving base-preserving label-preserving isomorphism bijectively correspond to functors $P \to \mathfrak{T}^n(C)$ up to natural isomorphism.

###### Proof

Inductively apply the previous theorem, starting with the highest 1-truss bundle and working your way downwards.

###### Remark

($n$-Truss bundles over categories). The theorem makes it evident that nothing would have stopped us from defining $n$-truss bundles over categories $B$ (in place of just posets): indeed, such bundles may be thought of as functors $B \to \mathfrak{T}^n(\ast)$ (or $B \to \mathfrak{T}^n(C)$ in the labeled case).

Lukas Heidemann points out the following nice perspective on the labeled bordism functor.

###### Remark

(Universal construction of the labeled bordism functor) Applying the profunctorial Grothendieck construction to the (frame-order-forgetting) functor $\mathfrak{T}^1 \to \mathbf{Prof}$, yields an exponentiable fibration $E\mathfrak{T}^1 \to \mathfrak{T}^1$. By general nonsense, the composition of the pullback $\mathbf{Cat}_{/ \mathfrak{T}^1} \to \mathbf{Cat}_{/ E\mathfrak{T}^1}$ and forgetful functor $\mathbf{Cat}_{/ E\mathfrak{T}^1} \to \mathbf{Cat}$ has a right adjoint $\mathbf{Cat} \to \mathbf{Cat}_{/ \mathfrak{T}^1}$; this adjoint is exactly the functor $C \mapsto \mathfrak{T}^1(C \to \ast)$. (Note: more generally, this construction applies to all normal pseudofunctors $H : D \to \mathbf{Prof}$, where it characterizes the constructions of ‘vertical comma categories’ $H_{/\!/ C}$ for such functors $H$ (see Dorn-Douglas 20, Term. 2.3.18) as right adjoints.)

###### Remark

(Labels in $\infty$-categories) The construction of $\mathfrak{T}^1(-)$ generalizes to an endofunctor on $\infty$-categories $\mathbf{Cat}_\infty$, which immediately leads to a notion of truss bundles labeled in $\infty$-categories.

Thus there is a ‘spectrum’ of base/label structures on which we can reasonably consider truss bundles, ranging from posets over to categories to $\infty$-categories. Most of the theory works the same across the spectrum. In this article, we work with the simplest possible choice, i.e. with posets (initially as base structure, but later even as label structures for the purpose of defining ‘stratifications’).

### Normalization theorem

Another important part of truss theory is normalization.

###### Definition

Given $C$-labeled $n$-truss bundles $T$ and $T'$ over $P$, a reduction $F : T \to T'$ is a labeled $n$-truss bundle map which is:

1. regular;
2. endpoint-dimension-preserving, meaning it is dimension-preserving on the endpoints of all 1-truss fibers;
3. base-preserving;
4. label-preserving-on-the-nose, meaning that $\mathsf{lbl}_F = \mathrm{id}$ and $\mathsf{lbl}_{T'} \circ F = \mathsf{lbl}_{T}$ commutes strictly.

(There’s a dual version of the definition that replaces ‘regular’ by ‘singular’; cf. the section on ‘duality’ below.)

###### Terminology

We sometimes write reductions as $F : T \underset{\mathsf{red}}{\longrightarrow} T'$, and say $T'$ is a reduct of $T$.

###### Terminology

A labeled truss whose only reduct is itself is called normalized (or, said to be in normal form).

###### Theorem

(Reduction ends in normal forms). The category $\mathsf{Norm}(T)$ of reducts of $T$ and reduction between them has a unique terminal object $[[T]]$ (called the normal form of $T$).

Various proofs have been given, the first in Dorn 2018, 5.2.2 (in the case of open trusses, see Def. , in which case condition 2. is implied by condition 1. above; the proof generalizes to the case of general trusses, however.), and another (shorter) proof in Heidemann-Reutter-Vicary 2022 (also for open trusses).

A geometric derivation and interpretation of the normalization theorem can be given in terms of ‘the existence of coarsest subdividing framed regular cell complexes of stratification in framed $\mathbb{R}^n$’, see Dorn-Douglas 2021, Ch. 5.

## Combinatorialization theorems

Trusses are the combinatorial analogues (namely, the fundamental categorical structures) of certain framed stratified topological structures. We describe how this works in two examples.

### Combinatorial meshes

Bare (unlabeled) trusses are combinatorial analogues of meshes. The relation is given by the following theorem.

###### Theorem

$n$-Meshes up to framed stratified homeomorphism bijectively correspond to $n$-trusses.

Proof sketch. Given an $n$-mesh $M$

apply to it the entrance path poset functor $\mathsf{Entr}(-)$ to obtain the corresponding $n$-truss

(framings and strata dimensions of the mesh canonically induce the required truss structures $\preceq$ and $\dim$). $\square$

###### Remark

The theorem has various (more categorical) generalizations, which essentially capture versions of the equivalence

$\mathcal{M}\mathit{esh}_n(B,g) \simeq \mathsf{Truss}_n(\mathcal{E}\mathit{ntr}(B,g))$

of the $\infty$-category of mesh bundles over a base stratification $(B,g)$ and the $\infty$-category of truss bundles over the entrance path $\infty$-category $\mathcal{E}\mathit{ntr}(B,g)$ of $g$ (recall, entrance path $\infty$-categories are the duals of exit path $\infty$-categories). See Dorn-Douglas 2021, Ch. 4 for details.

### Combinatorial manifold diagrams

Certain labeled trusses are the ‘combinatorial’ analogues of manifold diagrams. We first define the class of labeled trusses we are interested in. We need four ingredients:

1. Truss stratifications (special case of labelings)
2. Open trusses (special type of trusses)
3. Open truss neighborhoods and stratifed neighborhoods
4. Truss products (or, more specifically, ‘cylinders’)

###### Remark

(Stratifying posets). Recall a characteristic map $f : X \to \mathsf{Entr}(f)$ of a stratification $(X,f)$ maps points in a stratum $s$ to the corresponding poset element $s \in \mathsf{Entr}(f)$. Considering posets $P$ as spaces (by their specialization topology, with the convention that downward closed subposets are open), we may equally consider stratified posets by characteristic maps $f : P \to \mathsf{Entr}(f)$. Such characteristic maps are exactly poset maps which are poset quotients with connected preimages (see Dorn-Douglas 2021, App. B.1).

###### Definition

(Stratified truss). A stratified $n$-truss $T$ is a labeled $n$-truss $T$ whose labeling $\mathsf{lbl}_T$ is a characteristic map.

###### Definition

(Open truss). A 1-truss is called open if its endpoint have dimension 1. A (labeled) $n$-truss $T$ is called open if all 1-truss fibers in all 1-truss bundles that comprise $T$ are open.

###### Definition

(Open neigborhood). Given an open $n$-truss $T$ and an element $x \in T_n$, define the neighborhood $T^{\leq x}$ of $x$ to be the unique open truss that comes with a dimension-preserving map $F : T^{\leq x} \hookrightarrow T$ such that $F : T^{\leq x}_n \hookrightarrow T_n$ is an inclusion of the downward closure of $x$ into the poset $(T_n,\leq)$.

###### Terminology

(Atomic truss). Given an open $n$-truss $T$ with $x \in T_n$ such that $T^{\leq x} = T$, we say $T$ is an atomic open $n$-truss with cone point $x$ (in Dorn-Douglas 2021, Ch. 2, atomic trusses are called ‘truss braces’).

###### Definition

(Stratified open neighborhood). Given a stratified open $n$-truss $T$ and an element $x \in T_n$, define the stratified neighborhood $T^{\leq x}$ to be the unique stratified trusses that stratified embeds in $T$ with underlying truss map being the (non-stratified) neighborhood inclusion of $x$.

###### Terminology

(Cone types). A stratified open $n$-truss $T$ is said to be a combinatorial cone type if the underlying truss of $T$ is atomic with cone point $x$, and $\{x\} = \mathsf{lbl}^{-1}\circ \mathsf{lbl}(x)$ (in words: ‘$x$ is its own stratum’).

###### Definition

(Cylinders). Given a labeled $m$-truss $T$ defined the $k$-cylinder $\mathbb{I}^k \times T$ of $T$ to be the labeled $(m+k)$ obtained from $T$ be adding $k$ trivial truss bundles $\ast \to \ast$ to its underlying truss.

###### Remark

(Products). More generally, one can similarly define labeled $(k+m)$-trusses $U \times T$ as products between unlabeled $k$-trusses $U$ and labeled $m$-trusses $T$.

Recall the definition of manifold diagrams: these are framed conical (compactly triangulable) stratifications. Putting the preceding notions together, we obtain a combinatorial version of framed conicality as follows.

###### Definition

A combinatorial manifold $n$-diagram $T$ is a stratifed open $n$-truss such that for all $x \in T$ we have

$[[T^{\leq x}]] = \mathbb{I}^k \times C_x$

where $C_x$ is a combinatorial cone type.

###### Theorem

Manifold diagrams, up to framed stratified homeomorphism, bijectively correspond to normalized combinatorial manifold $n$-diagrams.

Proof sketch. Given a manifold $n$-diagram $(\mathbb{R}^n,f)$ its corresponding normalized combinatorial manifold $n$-diagram can be constructed by first refining $f$ by the unique coarsest $n$-mesh $M$, and then labeling the $n$-truss $\mathsf{Entr}(M)$ with the labeling $\mathsf{Entr}(M \to f)$. $\square$

(See Dorn-Douglas 2022, Sec. 2 for details.)

## Duality

There is a covariant dualization functor

$\dagger : \mathsf{Truss}_n \to \mathsf{Truss}_n$

from the category of $n$-trusses and $n$-truss maps to itself: the functor is defined by dualizing orders $\leq \mapsto \leq^{\mathrm{op}}$ and fiberwise replacing dimension maps $\dim \mapsto \dim^{\mathrm{op}}$ (using that $^{\mathrm{op op}} \cong ^{\mathrm{op}}$ uniquely).

Dualization of $n$-trusses is an involution. It maps:

1. Singular maps to regular maps and vice-versa
2. Open trusses to closed trusses and vice-versa

Geometrically, dualization dualizes stratifications in the sense of Poincare duality, which can for instance be used to translate between manifold diagrams and cellular pasting diagrams. This is discussed in e.g. in Dorn-Douglas 2022.

Dualization can also be applied to truss bordisms, where it yields a contravariant dualization functor:

$\dagger : \mathfrak{T}^n \to (\mathfrak{T}^n)^{\mathrm{op}}$

(or, in the labeled case: $\dagger : \mathfrak{T}^n(C) \to (\mathfrak{T}^n(C^{\mathrm{op}}))^{\mathrm{op}}$.)