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### Context

#### Higher category theory

+– {: .hide} [[!include higher category theory - contents]] =– =– =–

#Contents#

## Idea

Trusses are the [[fundamental categories]] (or, rather, fundamental posets) of [[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 dimension $n$.

### 1-Trusses

\begin{defn} A 1-truss $T \equiv (T,\leq, \preceq, \mathrm{dim})$ is a set $T$ with two [[partial orders]] (the ‘entrance’ order $\leq$ and the ‘frame’ order $\succeq$) 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]].

\end{defn}

\begin{defn} A 1-truss map $F : T \to S$ is a function of sets that preserves both entrance 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 $(T,\leq) \to ^{\mathrm{op}} = (0 \leftarrow 1)$. \end{defn}

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

### 1-Truss bundles

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

\begin{rmk} 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. \end{rmk}

\begin{rmk} 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 $P$. \end{rmk}

\begin{defn}[1-truss bordisms] \label{defn:1-truss-bordisms} 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.

\end{defn}

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.

\begin{defn} 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 $T^x$ ⇸ $T^y$. \end{defn}

\begin{defn} 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. We further say $F$ is regular resp. singular resp. dimension-preserving if it is fiberwise so. \end{defn}

### n-Truss bundles and $n$-trusses

\begin{defn} An $n$-truss bundle $T$ over a poset $P$ is a tower of 1-truss bundles $$T_n \xrightarrow{p_n} T_{n-1} \xrightarrow{p_{n-1}} ... \xrightarrow{p_2} T_1 \xrightarrow{p_1} T_0 = P$$ \end{defn}

\begin{defn} 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 $F_0 = \mathrm{id}_P$. \end{defn}

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

We can think of $n$-trusses as the ‘fibers’ of $n$-truss bundles.

## Labels and classifying categories

### Truss bundles with labels

\begin{defn} 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 $$C \xleftarrow {\mathsf{lbl}_T} T_n \xrightarrow{p_n} T_{n-1} \xrightarrow{p_{n-1}} ... \xrightarrow{p_2} T_1 \xrightarrow{p_1} T_0 = P$$

\end{defn}

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

\begin{defn} A labeled $n$-truss bundle map $F = (\underline F, \mathsf{lbl}F) : T \to T’$ of $C$- resp. $C’$-labeled $n$-trusses $T$ resp. $T’$ 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 $\mathsf{lbl}_F = \mathrm{id}_C$. \end{defn}

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.

\begin{defn} 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. \end{defn}

\begin{thm} $1$-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. \end{thm}

\begin{proof} Follows from the definition of 1-truss bundles. \end{proof}

(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.

\begin{defn} 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, 2.3.1. \end{defn}

\begin{thm} $C$-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. \end{thm}

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

\begin{rmk} (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. \end{rmk}

\begin{rmk} (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 $F$. \end{rmk}

The bordism functor is an endofunctor $$\mathfrak{T}^1 : \mathbf{Cat} \to \mathbf{Cat}$$ For $C \in \mathbf{Cat}$ we can thus apply the functor $n$ times: the resulting category $\mathfrak{T}^n(C)$ classifies $C$-labeled $n$-truss bundless as follows.

\begin{thm} $C$-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. \end{thm}

\begin{proof} Inductively apply the previous theorem, starting with the highest 1-truss bundle and working your way downwards. \end{proof}

\begin{rmk} ($n$-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, these may be thought of as functors $B \to \mathfrak{T}^n(\ast)$ (or $\mathfrak{T}^n(C)$ in the labeled case). \end{rmk}

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

\begin{rmk} (Universal construction of the labeled bordism functor) Applying the profunctorial Grothendieck construction to the forgetful 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 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 underlying bordism functor $C \mapsto \mathfrak{T}^1(C \to \ast)$. (Note: more generally, this observation applies to all [[normal]] [[pseudofunctors]] $H : D \to \mathbf{Prof}$, and characterizes the construction of ‘vertical comma categories’ $H_{/!/ C}$ for such functors $H$ (see Dorn-Douglas 20, Term. 2.3.18) as a right adjoint.) \end{rmk}

\begin{rmk} (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. \end{rmk}

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.

\begin{defn} Given $C$-labeled $n$-truss bundles $T$ and $T’$ over $P$, a normalizing map $F : T \to T’$ 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.

\end{defn}

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

\begin{terminology} We sometimes write normalizing maps as $F : T \xrightarrow{\mathsf{norm}} T’$, and say $T’$ is a reduct of $T$. \end{terminology}

\begin{terminology} A labeled truss whose only reduct is itself is called normalized. \end{terminology}

\begin{thm}(Normalization ends in normal forms). The category $\mathsf{Norm}(T)$ of reducts of $T$ and normalizing maps between them has a unique terminal object $[[T]]$ (called the normal form of $T$). \end{thm}

Various proofs have been given, the first in Dorn 2018, 5.2.2 (in the case of ‘open’ trusses, i.e. which dim-1 endpoints, in which case condition 2. follows from condition 1. above; the proof generalize to 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 category 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.

\begin{thm} $n$-Meshes up to framed stratified homeomorphism bijectively correspond to $n$-trusses. \end{thm}

Proof sketch. Given a mesh $$M_n \xrightarrow{p_n} M_{n-1} \xrightarrow{p_{n-1}} ... \xrightarrow{p_1} M_0 = \ast$$ apply to it the entrance path poset functor $\mathsf{Entr}(-)$ to obtain the corresponding $n$-truss $$\mathsf{Entr}(M_n) \xrightarrow{\mathsf{Entr}(p_n)} \mathsf{Entr}(M_{n-1}) \xrightarrow{\mathsf{Entr}(p_{n-1})} ... \xrightarrow{\mathsf{Entr}(p_1)} \mathsf{Entr}(M_0) = \ast$$ (framings and strata dimensions of the mesh canonically induce the required truss structures $\preceq$ and $\dim$). $\square$

\begin{rmk} The theorem has various (more categorical) generalizations, which essentially capture versions of the equivalence $$\mathcal{M}\mathit{esh}_n(X,f) \simeq \mathsf{Truss}_n(\mathcal{E}\mathit{ntr}(f))$$ of the $\infty$-category of mesh bundles over a base stratification $(X,f)$ and the $\infty$-category of truss bundles over the entrance path $\infty$-category $\mathcal{E}\mathit{ntr}(f)$ of $f$ (recall, entrance path categories are the duals of [[exit path categories]]). See Dorn-Douglas 2021, Ch. 4 for details. \end{rmk}

{#comb_mdiag}

### 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’)

\begin{rmk} 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). \end{rmk}

\begin{defn} A stratified $n$-truss $T$ is a labeled $n$-truss $T$ whose labeling $\mathsf{lbl}_T$ is a characteristic map. \end{defn}

\begin{defn} 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. \end{defn}

\begin{defn} 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 $(T_n,\leq)$. \end{defn}

\begin{terminology} 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 $T$ are called ‘truss braces’). \end{terminology}

\begin{defn} 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 $x$. \end{defn}

\begin{terminology} 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’). \end{terminology}

\begin{defn} 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. \end{defn}

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

Recall the definition of [[manifold diagrams]]: these are framed conical (compactly triangulable) stratifications. Putting the above construction together, we obtain a combinatorial version of framed conicality for the following definition.

\begin{defn} 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. \end{defn}

\begin{thm} Manifold diagrams, up to framed stratified homeomorphism, bijectively correspond to combinatorial manifold $n$-diagrams in normal form. \end{thm}

\begin{proof} Given a manifold $n$-diagram $(\mathbb{R}^n,f)$ its corresponding combinatorial manifold $n$-diagram in normal form 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 $\mathsf{Entr}(M \to f)$. \end{proof}

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

{#duality}

## Duality

There is a 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 the structure of a [[dagger category]]:

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

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

•  [[n-mesh n-mesh]]
•  [[manifold diagram manifold diagram]]
• [[manifold-diagrammatic n-category]]
• [[framed combinatorial topology]]
• {#Dorn18} [[Christoph Dorn]], Associative $n$-categories, PhD thesis (arXiv:1812.10586).

• {#DornDouglas21} Christoph Dorn and Christopher Douglas, Framed combinatorial topology, 2021 (pdfs)

• {#HRV22} [[Lukas Heidemann]], [[David Reutter]], [[Jamie Vicary]], Zigzag normalisation for associative $n$-categories, Proceedings of the Thirty-Seventh Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2022) &lbrack;arXiv:2205.08952, doi:10.1145/3531130.3533352&rbrack;

• {#DornDouglas22} Christoph Dorn and Christopher Douglas, Manifold diagrams and tame tangles, 2022 (pdfs)

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