nLab n-truss



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higher category theory

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


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



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

  1. (T,)[n]=(01...n)(T,\preceq) \cong [n] = (0 \to 1 \to ... \to n)
  2. either i<i+1i \lt i+1 or i+1<ii + 1 \lt i for all ini \prec n
  3. dim\dim is conservative.


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

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

where \Rightarrow and \Leftarrow denote natural transformations of functors (T,)[1] op=(01)(T,\leq) \to [1]^{\mathrm{op}} = (0 \leftarrow 1).


Given a truss TT, denote by T (i)T_{(i)} the subset of objects xx with dim(x)=i\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’.


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


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


Given 1-trusses TT and SS, a 1-truss bordism R:TR : TSS is a Boolean profunctor TTSS satisfying the following:

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

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


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


A 1-truss bundle map F:q 1q 2F : q_1 \to q_2 of 1-truss bundles q i:T iP iq_i : T_i \to P_i is a map F:T 1T 2F : T_1 \to T_2 that factors through q iq_i by a ‘base’ map F 0:P 1P 2F_0 : P_1 \to P_2, such that FF is fiberwise a 1-truss map (cf. Def. ). We further say FF is regular resp. singular resp. dimension-preserving if it is fiberwise so.

n-Truss bundles and n-trusses


An nn-truss bundle TT over a poset PP is a tower of 1-truss bundles (see Def. )


An nn-truss bundle map F:TTF : T \to T' is a tower of 1-truss bundle maps F i:q iq iF_i : q_i \to q'_i where F i1F_{i-1} is the base map of F iF_i and F nF:T nT nF_n \equiv F : T_n \to T'_n. The adjectives ‘regular’ resp. ‘singular’ resp. ‘dimension-preserving’ apply to FF if they apply to all F iF_i. If TT and TT' have the same base PP, then FF is called base-preserving if F 0=id PF_0 = \mathrm{id}_P.


An nn-truss bundle over the terminal poset *\ast is called an nn-truss.

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

Labels and classifying categories

Truss bundles with labels


Given a category CC, a CC-labeled nn-truss bundle T=(T̲,lbl T)T = (\underline T, \mathsf{lbl}_T) over PP consists of an ‘underlying’ nn-truss bundle T̲=(T n...T 1P)\underline T = (T_n \to ... \to T_1 \to P) together with a ‘labeling’ functor lbl T:T nC\mathsf{lbl}_T : T_n \to C. In other words, TT is of the form


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


A labeled nn-truss bundle map F=(F̲,lbl F):TTF = (\underline F, \mathsf{lbl}_F) : T \to T' from a CC-labeled nn-truss bundle TT to a CC'-labeled nn-truss bundle TT' consists of an nn-truss bundle map F̲:T̲T̲\underline F : \underline T \to \underline T' and a functor lbl F:CC\mathsf{lbl}_F : C \to C' such that lbl TF̲lbl Flbl T\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 F̲\underline F. Further, we say FF is label-preserving if lbl F=id C\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.


Given 1-truss bordisms R:TR : TSS and Q:SQ : SUU, their composite profunctor RQR \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 𝔗 1\mathfrak{T}^1 of 1-trusses and their bordisms.


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


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.


Given a category CC, the category 𝔗 1(C)\mathfrak{T}^1(C) of CC-labeled 1-trusses and their bordisms is defined as follows: objects of 𝔗 1(C)\mathfrak{T}^1(C) are CC-labeled 1-truss bundles over [0][0]; morphisms are CC-labeled 1-truss bundles over [1][1] (with domain and codomain given by restricting to fibers over 00 resp. 11); two morphisms compose to a third iff there is a CC-labeled bundle over [2][2] that restricts over (01)(0 \to 1), (12)(1 \to 2), and (02)(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.


CC-labelled 11-truss bundles over a base poset PP up to dimension-preserving base-preserving label-preserving isomorphism bijectively correspond to functors P𝔗 1(C)P \to \mathfrak{T}^1(C) up to natural isomorphism.


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


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


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

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

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


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


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


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

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


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


(Labels in \infty-categories) The construction of 𝔗 1()\mathfrak{T}^1(-) generalizes to an endofunctor on \infty-categories Cat \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.


Given CC-labeled nn-truss bundles TT and TT' over PP, a normalizing map F:TTF : T \to T' is a labeled nn-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 lbl F=id\mathsf{lbl}_F = \mathrm{id} and lbl TF=lbl T\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.)


We sometimes write normalizing maps as F:TnormTF : T \underset{\mathsf{norm}}{\longrightarrow} T', and say TT' is a reduct of TT.


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


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

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 n\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.


nn-Meshes up to framed stratified homeomorphism bijectively correspond to nn-trusses.

Proof sketch. Given an n n -mesh MM

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

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


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

esh n(B,g)Truss n(ntr(B,g)) \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)(B,g) and the \infty-category of truss bundles over the entrance path \infty-category ntr(B,g)\mathcal{E}\mathit{ntr}(B,g) of gg (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’)


(Stratifying posets). Recall a characteristic map f:XEntr(f)f : X \to \mathsf{Entr}(f) of a stratification (X,f)(X,f) maps points in a stratum ss to the corresponding poset element sEntr(f)s \in \mathsf{Entr}(f). Considering posets PP 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:PEntr(f)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).


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


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


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


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


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


(Cone types). A stratified open nn-truss TT is said to be a combinatorial cone type if the underlying truss of TT is atomic with cone point xx, and {x}=lbl 1lbl(x)\{x\} = \mathsf{lbl}^{-1}\circ \mathsf{lbl}(x) (in words: ‘xx is its own stratum’).


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


(Products). More generally, one can similarly define labeled (k+m)(k+m)-trusses U×TU \times T as products between unlabeled kk-trusses UU and labeled mm-trusses TT.

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.


A combinatorial manifold nn-diagram TT is a stratifed open nn-truss such that for all xTx \in T we have

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

where C xC_x is a combinatorial cone type.


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

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

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


There is a covariant dualization functor

:Truss nTruss n \dagger : \mathsf{Truss}_n \to \mathsf{Truss}_n

from the category of nn-trusses and nn-truss maps to itself: the functor is defined by dualizing orders op\leq \mapsto \leq^{\mathrm{op}} and fiberwise replacing dimension maps dimdim op\dim \mapsto \dim^{\mathrm{op}} (using that [1] opop[1] op[1]^{\mathrm{op op}} \cong [1]^{\mathrm{op}} uniquely).

Dualization of nn-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:

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

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



Last revised on May 13, 2023 at 13:53:25. See the history of this page for a list of all contributions to it.