From trusses to diagrammatic computads to combinatorial manifold diagrams

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Abstract. This note gives a fast introduction to the rich combinatorial theory of trusses. We also discuss how presheafs of truss blocks give rise to diagrammatic computads, and how this can be used to understand manifold diagrams in purely combinatorial terms.

This note is part of a series.

UPDATE (Mar 2023). The vague term ‘weak computads’ was replaced by the much better term ‘diagrammatic computads’. Also several new articles appeared which contain introductory material about trusses.

IntroductionPermalink

We give an overview of the basic combinatorial theory of trusses. The notion of “trusses” was introduced in [1], and extends the notion of “singular cubes” from [2] to a self-dual theory that combinatorially describes both “cell geometry” and “string geometry”. In particular, as discussed in [1], trusses provide a computable foundation for framed combinatorial topology which deals with spaces that are built from “framed” cells (the resulting notion of framed space can, for instance, be thought of as a geometric model for higher categorical structures). In this note, we will focus mainly on the combinatorics and less on the geometry. Our main aim, firstly, is to give the core combinatorial definitions in the theory of trusses. Secondly, we will sketch how to bridge the theory of trusses with classical ideas in higher categories by introducing a notion of “diagrammatic computad”, which will lead us to formally rediscover “computadic pasting diagrams” and, dually, “manifold diagrams”—both notions were discussed already in earlier notes.

Trusses are “towers of combinatorial constructible bundles”. Fibers of these bundles are “1-dimensional trusses”, usually referred to as “1-trusses”. Conversely, trusses are referred to as “nn-trusses” are given by a tower of nn bundles of 11-trusses (ending in the 00-truss, the “point”). There are many interesting aspects to the theory of trusses, and we shall introduce notions of “bundles”, “maps”, and “duals” for them; centrally. As we will see, similar to other combinatorial shapes (such as simplices, cubes, opetopes, etc.), maps include “faces” and “degeneracies”—importantly, we will also be able to combinatorially represent the dual notions of maps, namely notions of “embeddings” and “subdivisions”. The natural starting point for this discussion is dimension 11, which we will discuss first before introducing trusses in dimension nn.

1-TrussesPermalink

1-Trusses are the entrance path posets of (tangentially) framed stratified manifolds of dimension 00 or 11—these stratifications are called 1-meshes.

Definition (1-Meshes; see [1], Ch. 4). A 1-mesh (M,f,γ)(M,f,\gamma) is a connected kk-manifold, k{0,1}k \in \{0,1\}, together with a stratification ff of MM whose strata are open disks, as well as a tangential framing γ\gamma of MM. \blacklozenge

Note that a framing of a connected 00-dimensional manifold is trivial. A framing of a connected 11-dimensional manifold is equivalently an orientation (in particular, up to equivalence, there are exactly two framings).

Terminology (Linear and circular 1-meshes). A 11-mesh (M,f,γ)(M,f,\gamma) is called linear if MM is contractible, and circular otherwise. \blacklozenge

Example (1-Meshes) In Figure 1 we illustrate 1-meshes (both linear and circular ones). 0-disk strata are highlighted in red, 1-disk strata in blue. Framings are indicated by green arrows. \blacklozenge

Figure 1. 1-meshes of linear and circular type.

1-Trusses are the combinatorial counterparts of 1-meshes, obtained by passing to their entrance path posets together with combinatorial data representing disk dimensions and the framing. (Recall from a previous note that entrance path posets of stratifications are posets whose objects are strata and whose arrows indicate when one stratum borders on another.) Like 1-meshes, 1-trusses come both in “linear” and “circular” flavors. For simplicity we will only consider the linear case here—therefore, henceforth “1-mesh” will mean “linear 1-mesh” and similarly, the reader can think of “1-truss” as referring to “linear 1-truss” (as opposed to a notion of “circular 1-truss”, which we will not define here).

Definition (1-Trusses; see [1], Ch. 2). A 1-truss (T,,dim,)(T,\leq, \dim, \fleq) is a poset (T,)(T,\leq) together with a poset map dim:(T,)[1]op\dim : (T,\leq) \to [1]\op (where [1]op[1]\op is the poset 101 \to 0), as well as a total order (T,)(T,\fleq), satisfying the following: there exist a 1-mesh (T,f,γ)(\abs{T},f,\gamma) that admits

  • an isomorphism ϕ:(T,)Entr(f)\phi : (T,\leq) \iso \Entr(f),
  • with dim(s)=dim(ϕ(s))\dim(s) = \dim(\phi(s)) for all sTs \in T,
  • and sts \fleq t if and only if there is an oriented path starting in the stratum ϕ(s)\phi(s) and ending in ϕ(t)\phi(t). \blacklozenge

In the preceding definition, the 1-mesh (T,f,γ)(\abs{T},f,\gamma) is called a geometric realization of the 1-truss (T,,dim,)(T,\leq, \dim, \fleq). Note, for fixed geometric realization, the choice of isomorphisms ϕ\phi is necessarily unique in the previous definition (and the choice of geometric realization is itself unique up to contractible choice; we will come back to this point later).

Example (1-Trusses). In Figure 2 we illustrate 1-trusses (T,,dim,)(T,\leq,\dim,\fleq); the order \leq is indicated by arrows between objects; the map dim\dim is indicated by coloring the objects ss in red if dim(s)=0\dim(s) = 0 and in blue if dim(s)=1\dim(s) = 1; the linear order \fleq is indicated by arranging objects on a line together with a direction given by a (green) “coordinate axis” arrow. \blacklozenge

Figure 2. Examples of (linear) 1-trusses.

Terminology (Basic 1-truss terms). Given a 1-truss (T,,dim,)(T,\leq,\dim,\fleq) we call \leq the face order, dim\dim the dimension map, \fleq the frame order. We say sTs \in T is a singular element if dim(s)=1\dim(s) = 1, and a regular element otherwise. The subsets of singular resp. regular objects of TT will be written as T0T_0 resp. T1T_1 (note: in [1] these are written as sing(T)\mathrm{sing(T)} resp. reg(T)\mathrm{reg}(T)). A 1-truss is called closed if its geometric realization is compact (as a space). A 1-truss called open if its geometric realization is an open interval (as a space). \blacklozenge

In the next sections we will discuss three central aspects of 1-truss theory: (1), they behave well in families, (2), they admit several types of maps with concrete geometric interpretation (going beyond traditional “face” and “degeneracy” maps), and (3), they can be combinatorially dualized, which again has concrete geometric interpretation. We will discuss (1), (2), and (3) in order.

BundlesPermalink

Considering 1-truss in families gives rise to “bundles of 1-trusses”. Just as 1-trusses are combinatorial models of 1-meshes, “1-truss bundles” are combinatorial models of “1-mesh bundles”. To motivate our definitions, let us therefore start by illustrating the desired notion of bundles geometricbally. Recall that “stratified bundles” are fiber bundles of stratified spaces p:(M,f)(B,g)p : (M,f) \to (B,g) admitting local trivialization over each base stratum. We will consider stratified bundles in which each fiber p1(b)p\inv(b) has the structure of a 1-mesh. Crucially, in a 1-mesh bundle we want transition maps of fibers in these bundles to preserve the data of a 1-mesh: this means, firstly, that we need to preserve framings when passing between fibers (in other words, the orientations of fibers locally point in the “same direction”); secondly, we want to preserve dimensions of strata appropriately, in that 00-dimensional strata in the generic fiber must “transition to” 00-dimensional strata in the special fiber, while dually, 11-dimensional strata in the special fiber must “be transitioned into” from 11-dimensional strata in the generic fiber.

Example (1-Mesh bundles). The resulting types of bundles are illustrated in Figure 3: underneath each bundle, we illustrate the bundle generic and special fiber, together with an indication of functional relations (dotted arrows) that indicate how 00-strata “transition-to” other 00-strata (red/purple arrows), and how to 11-strata we “transition-from” other 11-strata (blue arrows). (Note in the last example, we omit blue “transition-from” arrows for simplicity). \blacklozenge

Figure 3. The idea of 1-mesh bundles (illustrated both for linear and circular 1-mesh fibers).

In general, the base of such 1-mesh bundles may be any stratified space (B,g)(B,g)—combinatorially, we will slightly simplify our life, and work with a base poset (B,)(B,\leq) in its place (note, while stratifications generally cannot be recovered from their entrance path posets, they can be recovered if strata are “regular cells”; and this is the case we are ultimately interested in).

Phrasing the two core properties of fiber transitions in 1-mesh bundles (namely, “frame” and “dimension” preservation) observed above in combinatorial terms, we are now led to the following definition of 1-truss bundles.

Definition (1-Truss bundles). A 1-truss bundle

p:(T,,dim,)(B,)p : (T,\leq,\dim,\fleq) \to (B,\leq)

consists of a poset map p:(T,)(B,)p : (T,\leq) \to (B,\leq), another poset map dim:(T,)[1]op\dim : (T,\leq) \to [1]\op, and a fiberwise total order (p1(x),)(p\inv(x),\fleq) (for each xBx \in B) such that the following holds.

  • For each object xx in BB, the datum (p1(x),,dim,)(p\inv(x),\leq,\dim,\fleq) (with \leq and dim\dim restricted to the fiber p1(x)p\inv(x)) is a 1-truss.
  • For each morphism xyx \to y in BB, p1(xy)p\inv(x \to y) is a valid fiber transition, meaning that the relation R(a,b)    abR(a,b) \iff a \to b (where ap1(x)a \in p\inv(x) and bp1(y)b \in p\inv(y)) satifies the following:
    • Frame preservation (called “bimonotocity” in [1]). If aaa \fles a', R(a,b)R(a,b) and R(a,b)R(a',b') then bbb \fleq b'. Dually, if bbb \fles b', R(a,b)R(a,b) and R(a,b)R(a',b') then aaa \fleq a'.
    • Dimension preservation (called “bifunctionality” in [1]). There exists a “transition-to” function R0:p1(x)0p1(y)0R_0 : p\inv(x)_0 \to p\inv(y)_0 such that, for ap1(x)0,bp1(x)0a \in p\inv(x)_0, b \in p\inv(x)_0 we have R(a,b)    R0(a)=bR(a,b) \iff R_0(a) = b. Dually, there exists a “transition-from” function R1:p1(y)1p1(x)1R_1 : p\inv(y)_1 \to p\inv(x)_1 such that, for ap1(x)1,bp1(x)1a \in p\inv(x)_1, b \in p\inv(x)_1 we have R(a,b)    a=R1(b)R(a,b) \iff a = R_1(b). \blacklozenge

If we were to spell out our definition of 1-mesh bundles further (which is done in [1]) we would find that 1-mesh bundles up to an appropriate notion of equivalence (and over sufficiently regular base) are exactly 1-truss bundles up to an appropriate notion of structure-preserving isomorphism. This provides, if you want, a geometric-semantic correctness result for the above definition. We will return to this comparison later in the context of nn-trusses (and their geometric realization as nn-meshes).

Note that 1-trusses (T,,dim,)(T,\leq,\dim,\fleq) are exactly 1-truss bundles over the terminal poset [0]={0}[0] = \{0\}.

Notation (Keeping truss [bundle] data implicit). We usually keep much of the data of a 1-truss bundle implicit and simply write nn-truss bundles as maps p:TBp : T \to B. The simplfication similarly applies to 1-trusses where we write TT in place of (T,,dim,)(T,\leq,\dim,\fleq); analogously, we usually write 1-meshes simply as MM and 1-mesh bundles as MBM \to B. \blacklozenge

Construction (Bordisms and classification of bundles). Given trusses TT and SS, a 1-truss bordism R:TSR : T \proto S is a 1-truss bundle p:R[1]p : R \to [1] over the interval poset [1]={01}[1] = \{0 \to 1\}, whose first fiber is p1(0)=Tp\inv(0) = T and whose second fiber is p1(1)=Sp\inv(1) = S. In [1] we give a more direct definition of 1-truss bordisms (and, in fact, define bundles in terms of bordisms and not the other way around) for the following reason: 1-truss bordisms are the arrows of a category of 1-truss bordisms T1\ttr 1. Each truss bundle p:TBp : T \to B has a classifying functor χp:BT1\fcl {} p : B \to \ttr 1, which maps objects xBx \in B to the fiber truss p1(x)p\inv(x), and arrows (xy)(x \to y) to the fiber bordism p1(xy)p\inv(x \to y) (or more precisely, to the pullback 1-truss bundle of pp along the map [1](xy)B[1] \iso (x \to y) \to B). 1-Truss bundles over BB up to equivalence (equivalences are defined in the next section) are in correspondence with functors BT1B \to \ttr 1 up to natural isomorphism. \blacklozenge

Remark (Truss induction). One aspect of the theory of 1-truss bundles which we do properly not touch upon is “truss induction” (thorougly discussed in [1], Section 2.2); this proves that “(n+1)(n+1)-simplices in 1-truss bundles over nn-simplices are linearly ordered”. Here, we will only provide a visual cue. Any truss bundle p:T[n]p : T \to [n] has a suspension Σp:ΣTΣ[n]\Sigma p : \Sigma T \to \Sigma [n] obtained by adjoining new initial and terminal elements to TT resp. [n][n] (require the initial element to have dimension 11 and the terminal element dimension 00). The restriction of Σp\Sigma p to the spine of Σ[n]\Sigma [n] gives a projection of graphs that can be visualized in a plane (see the example in Figure 4). There is a unique path such that: the path passes through all arrows in the fibers of the projection; in between any two such passages (and before/after the first/last such passage) it passes through exactly one other arrow; this arrow runs between a dim-1 and dim-0 object. We illustrate the path in Figure 4. This path encodes a linear order of (n+1)(n+1)-simplices in TT—inducting along this path is a useful technique when dealing with truss bundles, and we refer to this technique as truss induction. \blacklozenge

Figure 4. The "flow of truss induction" illustrated by a directed path.

MapsPermalink

We next discuss maps of 1-trusses. In the previous section, we saw that bordisms of 1-trusses are combinatorial models of fiber transitions in bundles of 1-meshes. In contrast, maps of 1-trusses are combinatorial models of “maps of 1-meshes”—the latter notion does the obvious: a map of 1-meshes is a stratified map which preserve the framings (in other words, they are orientation-preserving). Several such maps of 1-meshes are shown in Figure 5. Importantly, depending on how such maps interact with strata dimension they may be further distinguish (examples are given in Figure 5).

  • A 1-mesh map that maps 00-strata to 00-strata is called a singular map. Further, it is called
    • a face map (or F\mathbf{F}-map) if it is injective on strata,
    • a degeneracy (or D\mathbf{D}-map) map if is surjective on strata.
  • A 1-mesh map that maps 11-strata to 11-strata is called a regular map. Further, it is called
    • an embedding (or E\mathbf{E}-map) if it is injective on strata.
    • a coarsening (or C\mathbf{C}-map) if it is surjective on strata.
  • A map that is both regular and singular is called balanced. A balanced 1-mesh map that is bijective on strata is called a 1-mesh equivalence (it is, up to homotopy, a stratified homeomorphism).

Note coarsenings are sometimes called “subdivisions” or “refinements”.

Example (1-mesh maps). We illustrate 1-mesh maps in Figure 5; note that we illustrate all map types in the case of closed linear 1-meshes (the classification of course equally applies to other mesh types). \blacklozenge

Figure 5. The idea of 1-mesh maps.

Note that we may also refer to coarsenings as both subdivisions or refinements (often as a description of the “opposite” process of coarsening).

Let us now translate these notions of maps into the setting of 1-trusses.

Definition (1-Truss maps) Given 11-trusses TT and SS, a 1-truss map F:TSF : T \to S is a map that preserves both face orders F:(T,)(S,)F : (T,\leq) \to (S,\leq) and frame orders F:(T,)(S,)F : (T,\leq) \to (S,\leq). \blacklozenge

We saw that every 1-truss TT is geometrically realized by an 1-mesh T\abs{T} (uniquely so up to 1-mesh isomorphism) such that EntrT=T\Entr \abs{T} = T. Similarly, 1-truss maps have geometric realizations: we say that a 1-mesh map F:TS\abs{F} : \abs{T} \to \abs{S} geometrically realizes the 1-truss map F:TSF : T \to S if EntrF=F\Entr \abs{F} = F. Geometric realization of 1-truss maps are unique up to contractible choice in the space of realizations.

Terminology (Map types). A 1-truss map F:TSF : T \to S is called a C/D/E/F\bC/\bD/\bE/\bF-map if its realization is an C/D/E/F\bC/\bD/\bE/\bF-map of 1-meshes. Similarly one defines 1-truss maps that are singular, regular, balanced and equivalences. \blacklozenge

The notion of maps of 1-truss is easily generalized to the case of 1-truss bundles.

Definition (Truss bundle maps). Given 1-truss bundles p:TBp : T \to B and q:SCq : S \to C a bundle map F:pqF : p \to q is a poset map F:TSF : T \to S commuting with pp and qq by a square

TFSpqBC\begin{CD} T @>{F}>> S\\ @VpVV @VVqV \\ B @>>> C \end{CD}

such that FF restricts to a 1-truss map on each fiber. We say FF is a C/D/E/F\bC/\bD/\bE/\bF-map (resp. singular, regular, balanced or an equivalence) if it is so fiberwise. \blacklozenge

Notation (Categories of trusses and truss bundles). Let’s denote the category of 1-trusses and their maps by T ⁣rs1\truss 1, and the category of 1-truss bundles and their maps by TrsBun1\trussbun 1.

DualizationPermalink

Let us turn to a final important aspect of 1-truss theory: the combinatorial phrasing of (Poincaré) duality. Geometrically, dualization does the following: given a 1-mesh MM, its dual MM^\dagger is the 1-mesh in which 00-strata and 11-strata of MM are turned into 11-strata resp. 00-strata without affecting the direction of the framing—we illustrate this in Figure 6 (we also show how the idea applies to 1-mesh bundles).

Figure 6. The idea of 1-mesh duals.

Formally, and combinatorially, this is defined as follows.

Definition (Dualization of 1-trusses, bundles, bordisms and maps) Given a 1-truss T(T,,dim,)T \equiv (T,\leq,\dim,\fleq) its dual truss TT^\dagger is the 1-truss (T,op,dimop,)(T,\leq\op,\dim\op,\fleq) (where dimop:(T,op)([1]op)op[1]op\dim\op : (T,\leq\op) \to ([1]\op)\op \iso [1]\op).

Similarly, given a 1-truss bundle p:(T,,dim,)Bp : (T,\leq,\dim,\fleq) \to B, its dual bundle pp^\dagger is the bundle p:(T,op,dimop,)(B,op)p : (T,\leq\op,\dim\op,\fleq) \to (B,\leq\op). In particular, given a 1-truss bordism R:TSR : T \proto S, its dual bordism is the 1-truss bordism R:STR^\dagger : S \proto T such that R(a,b)    R(b,a)R(a,b) \iff R^\dagger(b,a).

Given a 1-truss bundle map F:TSF : T \to S, its dual map FF^\dagger is the map F:TSF : T^\dagger \to S^\dagger (i.e. identical to FF on objects). \blacklozenge

Earlier, we introduced the categories T1\ttr 1, T ⁣rs1\truss 1 and TrsBun1\trussbun 1; dualization acts as follows on these categories.

Observation (Dualization on categories). Dualization of 1-trusses and their bordisms yields an isomorphism of categories

:T1(T1)op:\dagger : \ttr 1 \iso (\ttr 1)\op : \dagger

Dualization of 1-truss (bundles) and their maps yields isomorphisms

:T ⁣rs1T ⁣rs1::TrsBun1TrsBun1:\begin{aligned} \dagger : \truss 1 &\iso \truss 1 : \dagger \\ \dagger : \trussbun 1 &\iso \trussbun 1 : \dagger \end{aligned}

The functor \dagger further maps

  • F\bF-maps to E\bE-maps and vice versa,
  • D\bD-maps to C\bC-maps and vice versa. \blacklozenge

Let us further distinguish two important subcategoeries in the above categories.

Remark (Open and closed 1-trusses). Note that \dagger sends closed 1-trusses to open 1-trusses and vice versa. \blacklozenge

Labels and stratificationsPermalink

LabelsPermalink

Given a topological space XX, and a categorical structure C\iC (e.g. a poset, category, or higher category) a “labeling” of XX is a functor ΠXC\Pi_\infty X \to \iC from the fundamental (\infty-)groupoid of XX into C\iC. In the case of stratified topological spaces, this generalizes to a functor from the fundamental \infty-category (whose morphisms are not mere paths but entrance paths) into C\iC. If we work with sufficiently simple stratifications (which we ultimately do), this fundamental \infty-category becomes equivalent to the stratification’s entrance path poset. As we’ve seen, in the case of 1-meshes (resp. their bundles) entrance path posets are modeled by the corresponding 1-trusses (resp. their bundles). This motivates the following definition.

Definition (Labeled 1-trusses and bundles). Let C\iC be a (ordinary) category. A C\iC-labeled 1-truss T(T,lblT)T \equiv (\und T, \lbl_T) consists of an ‘underlying’ 1-truss T(T,,dim,)\und T \equiv (\und T,\leq,\dim,\fleq) together with a ‘labeling’ functor lblT:(T,)C\lbl_T : (\und T,\leq) \to \iC. Similarly, a C\iC-labeled 1-truss bundle p(p,lblp)p \equiv (\und p, \lbl_p) consists of an ‘underlying’ 1-truss bundle p:(T,,dim,)B\und p : (T,\leq,\dim,\fleq) \to B together with a ‘labeling’ functor lblp:(T,)C\lbl_p : (T,\leq) \to \iC. \blacklozenge

Definition (Maps of labeled 1-trusses and bundles). Given two labeled 1-truss bundles p(p:TB,lblp:TC)p \equiv (\und p : T \to B, \lbl_p : T \to \iC) and q(q:SC,lblq:SC)q \equiv (\und q : S \to C, \lbl_q : S \to \iC), a labeled bundle map F:pqF : p \to q is a map F:pq\und F : \und p \to \und q of underlying bundles together with a functor lblF:CD\lbl_F : \iC \to \iD of labels such that the following commutes

ClblpTpBlblFFDlblqSqC\begin{CD} \iC @<{\lbl_p}<< T @>{\und p}>> B \\ @V{\lbl_F}VV @VVFV @VVV \\ \iD @<{\lbl_q}<< S @>{\und q}>> C \end{CD}

The definition specializes to a notion of labeled 1-truss maps if we set B=C=[0]B = C = [0]. (One may also weaken the definition, by allowing the left square in the above diagram to commute up to natural (iso)morphism). \blacklozenge

The central observation about C\iC-labeled 1-truss bundles is that they have a classifying category; this is the category of C\iC-labeled 1-truss and their bordisms.

Construction (Labeled bordisms and classification of labeled bundles) A C\iC-labeled 1-truss bordism R:TSR : T \proto S is a labeled 1-truss bundle (R:UB,lblR:UC)(\und R : U \to B, \lbl_R : U \to \iC) with base equal to the interval poset B=[1]B = [1]; its domain TT is the C\iC-labeled 1-truss obtain by restricting R\und R to {0}[1]\{0\} \into [1] and accordingly restricting lblR\lbl_R, its codomain SS is similarly obtain by restriction to {1}[1]\{1\} \into [1]. Importantly, C\iC-labeled 1-truss bordisms compose: given C\iC-labeled 1-truss bordisms R:TSR : T \proto S, and Q:SUQ : S \proto U, then there exists a unique labeled 1-truss bordism P:TUP : T \proto U such that P,Q,RP,Q,R are the restrictions of a single C\iC-labeled 1-truss bundle over the 2-simplex [2][2] (to the arrows (02)(0 \to 2), (12)(1 \to 2) resp. (01)(0 \to 1)). We set PRQP \equiv R \circ Q. The proof of this construction uses truss induction over the 2-simplex (as we will briefly revisit later on, there are also \infty-categorical generalizations, which then use truss induction in its full power over general nn-simplices).

As a result, we obtain the category of C\iC-labeled 1-truss bordisms T1(C)\lttr 1 \iC, whose objects are C\iC-labeled 1-trusses and whose morphisms are C\iC-labeled 1-truss bordisms. Each C\iC-labeled truss bundle p(p:TB,lblp:TC)p \equiv (\und p : T \to B, \lbl_p : T \to \iC) has again a classifying functor χp:BT1(C)\fcl {} p : B \to \lttr 1 \iC: this maps objects xBx \in B to the C\iC-labeled fiber truss (p1(x),lblp:p1(x)C)(\und p\inv(x), \lbl_p : \und p\inv(x) \to \iC), and arrows (xy)(x \to y) to the C\iC-labeled fiber bordism (p1(xy),lblp:p1(xy)C)(\und p\inv(x \to y),\lbl_p : \und p\inv(x \to y) \to \iC). This yields a correspondence of C\iC-labeled 1-truss bundles over BB with functors BT1(C)B \to \lttr 1 \iC (up to appropriate notions of equivalences). \blacklozenge

As an aside, for the category theorists among us it could be fun to think about the following observation.

Remark (Labeled truss bordisms as a vertical comma category). There is an embedding T1Prof\ttr 1 \into \mathrm{Prof} of the category of 1-truss bordisms into the bicategory of profunctors, which regards 1-truss bordisms as Bool\Bool-enriched profunctors and post-composes them with the inclusion BoolSet\Bool \into \SetCat (the fact that this works is non-trivial; for instance relations Rel\mathrm{Rel} do not (non-laxly) embed into profunctors Prof\mathrm{Prof}). Using this embedding, a C\iC-labeled 1-truss bordism is then exactly a square

TRS lblTlblRlblS CHomCC\begin{array}{ccc} \und T & \xrightarrow{\quad \raisebox{.3ex}{$\scriptstyle \und R$} \quad}\mkern{-4.5ex}{\raisebox{.3ex}{$\tiny\vert$}}\mkern{4.2ex} & \und S\\\ \kern-3ex\raisebox{.3ex}{\rlap{$\scriptstyle \lbl_T$}}\kern+3ex\bigg\downarrow & \quad \Darr {\scriptstyle \lbl_R} & \bigg\downarrow\raisebox{.5ex}{\rlap{$\scriptstyle \lbl_S$}} \\\ \iC & \xrightarrow{\kern{.4em} \raisebox{.3ex}{$\scriptstyle \Hom_\iC$} \kern{.4em}}\mkern{-4.5ex}{\raisebox{.3ex}{$\tiny\vert$}}\mkern{4.2ex} & \iC \end{array}

in the (pseudo) double category of profunctors Prof\mathbb{P}\mathrm{rof}. One can therefore think of T1(C)\lttr 1 \iC as a “vertical comma category” of the horizontal embedding T1ProfProf\ttr 1 \into \mathrm{Prof} \into \mathbb{P}\mathrm{rof} over C\iC. \blacklozenge

StratificationsPermalink

The “base case” of interest to us are labeling functors that are characteristic maps of stratifications. Recall that finite stratifications (X,f)(X,f) have (and are determined by their) continous characteristic maps f:XEntr(f)f : X \to \Entr(f). Topologizing posets using the downward closed topology, this similar applies to posets: a (finitely) stratified poset (P,f)(P,f) determines and is determined by its characteristic map f:PEntr(f)f : P \to \Entr(f)—the latter now is a poset map with the property that preimages f1(s)f\inv(s) are non-empty connected subposets of PP such that srs \to r in Entr(f)\Entr(f) if and only if the upwards closure of f1(s)f\inv(s) non-trivially intersects f1(r)f\inv(r) in PP (in [1], Appendix B, such maps are also called “connected-quotient” maps of posets).

Definition (Stratified 1-truss bundle). A stratified 1-truss bundle is a labeled 1-truss bundle in which the labeling functor is the characteristic map of a stratification. (The definition specializes to a notion of stratified 1-trusses if the base is trivial). \blacklozenge

Every poset labeled 1-truss bundle determines a stratified 1-truss bundle as the following construction shows.

Construction (Connected component splitting). Let PP be a poset, and consider a PP-labeled 1-truss bundle p(p:TB,lblp:TP)p \equiv (\und p : T \to B, \lbl_p : T \to P). There exists a unique factorization of the labeling by maps cc(p):TEntr(p)\mathrm{cc}(p) : T \to \Entr(p) and dd:Entr(p)P\mathrm{dd} : \Entr(p) \to P such that cc(p)\mathrm{cc}(p) is the characteristic map of a stratification, and dd(p)\mathrm{dd}(p) is a poset map with discrete preimages (this factorization has several universal properties, see [1], Appendix B). We can thus associate a stratified 1-truss bundle (p,cc(p))(\und p, \mathrm{cc}(p)) to pp, which is called the connected component splitting of pp. Usually, when referring to stratifications in the context of poset-labeled bundles we implicitly pass to their connected component splittings. \blacklozenge

The idea of “connected component splitting” bridges our variation of the notion of stratification with, for instance, the notion introduced in [3]: Lurie introduces “PP-stratifications” as spaces XX endowed with a continuous map f:XPf : X \to P to a poset PP. In the case of locally finite stratifications, this map splits uniquely by maps XEntr(f)PX \to \Entr(f) \to P in which the first map is a continuous characteristic map and the second map is a map with discrete preimages.

nn-TrussesPermalink

We are finally ready to generalize our discussion to higher dimension. The process is straight-forward and has the following analog. The interval I\bI is a 1-dimensional manifold. The total space of an I\bI-fiber bundle over I\bI is a 2-dimensional manifold—in fact, it homeomorphic to the square I2=I×I\bI^2 = \bI \times \bI. By iteratively considering I\bI-bundles over (the total space) of I\bI-bundles we can build the nn-cube In\bI^n. Replacing I\bI-bundles by 1-truss bundles, this idea can be directly applied to define nn-trusses. (Note also that the interval I\bI is an I\bI-bundle over the point \ast and, while this doesn’t add any information, we usually add this “terminal” bundle to all towers of bundles below.)

Definition (nn-Trusses). An nn-truss is a tower of 1-truss bundles pip_i

TnpnTn1pn1Tn2T2p2T1p1T0=[0]T_n \xto {p_n} T_{n-1} \xto {p_{n-1}} T_{n-2} \to \cdots \to T_2 \xto {p_2} T_1 \xto {p_1} T_0 = [0]

where the total poset (Ti,)(T_i,\leq) of pip_i is the base poset of pi+1p_{i+1}. \blacklozenge

BundlesPermalink

The above definition immediately generalizes to a notion of “nn-truss bundles” by omitting the condition T0=[0]T_0 = [0] and instead allowing T0T_0 to be an arbitrary (finite) poset. However, let us be more general and also allow these bundles to be labeled.

Definition (Labeled nn-truss bundles). For a category C\iC and a poset BB, a C\iC-labeled nn-truss bundle pp is a diagram of the form

ClblpTnpnTn1pn1p2T1p1T0=B\iC \xot{\lbl_p} T_n \xto {p_n} T_{n-1} \xto {p_{n-1}} \dots \xto {p_2} T_1 \xto {p_1} T_0 = B

where the left most ‘labeling functor’ lblp\lbl_p is a functor (Tn,C)(T_n,\leq \to \iC), while the remaining tower to its right consists of 1-truss bundles pip_i which together form the ‘underlying nn-truss bundle’ p\und p of pp. \blacklozenge

Construction (Classification of nn-truss bundles). A C\iC-labeled nn-truss bordism R:TSR : T \proto S is a C\iC-labeled nn-truss bundle over [1][1], which restricts to (the C\iC-labeled nn-truss) TT over {0}[1]\{0\} \into [1] and to SS over {1}[1]\{1\} \into [1]. As in the case n=1n = 1, such bordisms compose: given C\iC-labeled nn-truss bordisms R:TSR : T \proto S, and Q:SUQ : S \proto U, then there exists a unique labeled 1-truss bordism P:TUP : T \proto U such that P,Q,RP,Q,R are the restrictions of a single C\iC-labeled 1-truss bundle over the 2-simplex [2][2] (to the arrows (02)(0 \to 2), (12)(1 \to 2) resp. (01)(0 \to 1)). We set P=RQP = R \circ Q, and obtain the category of C\iC-labeled nn-truss bordisms Tn(C)\lttr n \iC.

This category can be constructed in a more explicit, inductive manner (which also shows that Tn(C)\lttr n \iC exactly classifies C\iC-labeled nn-truss bundles). Consider, in the following diagram, the upper row of 1-truss bundles pip_i together with the C\iC-labeling lblp\lbl_p: this defines a C\iC-labeled nn-truss bundles pp.

TnpnTn1pn1Tn1pn2p2T1p1T0lblp C T1(C) T1(T1(C)) Tn1(C) Tn(C)\scriptsize \begin{CD} T_n @>{p_{n}}>> T_{n-1} @>{p_{n-1}}>> T_{n-1} @>{p_{n-2}}>> {\quad \cdots \quad} @>{p_2}>> T_1 @>{p_1}>> T_0 \\ @V{\lbl_p}VV @VVV @VVV @. @VVV @VVV \\ \iC @. {\lttr 1 \iC} @. {\lttr 1 {\lttr 1 \iC}} @. {\cdots} @. {\lttr {n-1} \iC} @. {\lttr n \iC} \end{CD}

Just considering the top-bundle pnp_n together with the labeling lblp\lbl_p we obtain a C\iC-labeled 1-truss bundle (pn,lblp)(p_n,\lbl_p); as we’ve seen such labeled bundles are classified by functors from the base of pnp_n (namely Tn1T_{n-1}) to T1(C)\lttr 1 \iC. But this classifying functor Tn1T1(C)T_{n-1} \to \lttr 1 \iC now provides a labeling for the bundle pn1p_{n-1}; the resulting labeled 1-truss bundle is classified by a functor Tn2T1(T1(C))T_{n-2} \to \lttr 1 {\lttr 1 \iC}. Inductively continuing this process, and setting

Tk(C)=T1(Tk1(C))\lttr k \iC = \lttr 1 {\lttr {k-1} \iC}

we eventually construct the functor T0Tn(C)T_0 \to \lttr n \iC as shown in the diagram–this functor “classifies” the labeled nn-truss bundle pp. The seeming notational conflict (we’ve now doubly defined Tn(C)\lttr n \iC) is resolved by observing that both of our definitions are equivalent. Note further that the construction is functorial in C\iC (exercise!): we thus obtain functors Tn:CatCat\ttr n : \mathrm{Cat} \to \mathrm{Cat}, nNn \in \lN, on the category of categories, and this is the nn-fold iteration of the functor T1\ttr 1.

(A more direct view on the classifying functor T0Tn(C)T_0 \to \lttr n \iC is the following: objects xx in BB are mapped to the labeled fiber truss over xx in pp; morphisms (xy)(x \to y) in BB are mapped to the labeled fiber truss bordism over (xy)(x \to y) in pp.) \blacklozenge

Remark (The quasicategory of labeled nn-truss bordisms). Given a quasicategory C\mathcal{C} we can form a simplicial set Tn(C)\lttr n {\mathcal{C}} whose kk-simplices are diagrams

ClblpTnpnTn1pn1p2T1p1T0=[k]\mathcal{C} \xot{\lbl_p} T_n \xto {p_n} T_{n-1} \xto {p_{n-1}} \dots \xto {p_2} T_1 \xto {p_1} T_0 = [k]

where the pip_i’s form an nn-truss bundle over the kk-simplex [k][k], and lblp:(Tn,)C\lbl_p : (T_n,\leq) \to \mathcal{C} is a map of simplicial sets (we implicitly pass to the nerve of (Tn,)(T_n,\leq)). Faces and degeneracies are defined via pullback of nn-truss bundles (exercise!). The resulting simplicial set Tn(C)\lttr n {\mathcal{C}} is in fact a quasicategory itself, the quasicategory of C\mathcal{C}-labeled nn-truss bordisms—inner horn fillers can be constructed using truss induction. The construction is again functorial in C\mathcal{C} yielding an “nn-truss bordism” endofunctor

Tn:CatCat\ttr n : \infty\mathrm{Cat} \to \infty\mathrm{Cat}

on the quasicategory of quasicategories (Tn\ttr n is again equivalently obtained by nn-fold iteration of T1\ttr 1). \blacklozenge

We henceforth focus on the 1-categorical case; in fact, most of our use-cases (stratifications!) will see us set C\iC equal to a poset. Note also, in the special case that C=[0]\iC = [0] is terminal, labeling functors are trivial and we speak of unlabeled nn-trusses or nn-truss bundles (in which case we often omit the “(C)(\iC)” from our notation completely). Since the above constructions is functorial note that there is always a label forgetting functor

Tn(C)Tn.\lttr n \iC \to \ttr n .

Maps and dualizationPermalink

Notions of maps generalize from dimension 11 to dimension nn.

Definition (Maps of labeled nn-truss bundles). Bundle maps F:pqF : p \to q of labeled nn-truss bundles pp and qq are given by commutative diagrams

ClblpTnpnTn1pn1p1T0lblFFnFn1 F0DlblqSnqnSn1qn1q1S0\begin{CD} \iC @<{\lbl_p}<< T_n @>{p_n}>> T_{n-1} @>{p_{n-1}}>> \cdots @>{p_1}>> T_0\\ @V{\lbl_F}VV @VV{F_n}V @VV{F_{n-1}}V @. @VV{F_0}V \\ \iD @<<{\lbl_q}< S_n @>>{q_n}> S_{n-1} @>>{q_{n-1}}> \cdots @>>{q_1}> S_0 \end{CD}

where each Fi:piqiF_i : p_i \to q_i is a 1-truss bundle map. We say FF is an C/D/E/F\bC/\bD/\bE/\bF-map if each FiF_i is. If T0=S0=[0]T_0 = S_0 = [0] are trivial, these definitions specialize to the case of labeled nn-trusses. FF is said to be label (resp. base) preserving if lblF=id\lbl_F = \id (resp. F0=idF_0 = \id). \blacklozenge

As before, one may wish to weaken commutativity the left most square in the above diagram to hold up to natural (iso)morphism.

Observation (Closed and open nn-trusses). Our notions of open and closed 11-trusses generalize. An nn-truss (or nn-truss bundle) pp is said to be open (resp. closed) if all 1-truss fibers of each pip_i are open (resp. closed). Denote by Tˉ ⁣rs1\sctruss 1 the category of (unlabeled) closed nn-trusses, whose maps are singular. Then (D,F)(\bD,\bF) is an (epi,mono)-factorization system of this category into “degeneracies” and “faces”. Dually, denote by T˚ ⁣rs1\rotruss 1 the category of (unlabeled) open nn-trusses, whose maps are regular. Then (C,E)(\bC,\bE) is an (epi,mono)-factorization system of this category into “coarsenings” and “embeddings”. \blacklozenge

Construction (Dualization of nn-trusses). Our earlier dualization involution :T1(T1)op\dagger : \ttr 1 \iso (\ttr 1)\op generalizes to labeled 1-truss bundles by passing to opposite labeling functors, yielding an involution

:T1(C)(T1(Cop))op:\dagger : \lttr 1 \iC \iso (\lttr 1 {\iC\op})\op : \dagger

Applying this inductively, we find:

:Tn(C)(Tn(Cop))op:.\dagger : \lttr n \iC \iso (\lttr n {\iC\op})\op : \dagger.

Similarly, we may dualize nn-truss bundles and their maps, yielding isomorphisms

:LblT ⁣rsnLblT ⁣rsn:.\dagger : \trusslbl n \iso \trusslbl n : \dagger.

Dualization maps open nn-trusses to closed nn-truss and vice versa. It also maps C\bC-maps (resp. E\bE-maps) to D\bD (resp. F\bF-maps). \blacklozenge

nn-Trusses vs nn-meshesPermalink

Our discussion of 1-trusses was strongly geometrically motivated by the idea of 1-meshes (as framed stratified connected kk-manifolds, k1k \leq 1): this geometric interpretation carries over to higher dimension nn as well. An nn-mesh is a tower of 1-mesh bundles over the point. Maps of such nn-meshes are, fully analogous to our definition in the case of nn-trusses, maps of towers which fiberwise restrict to maps of 1-meshes—note that these maps have a topology making the category of nn-meshes an \infty-category. Passing to entrance path posets defines a functor from the category of nn-meshes to the category of nn-trusses. This functor is an equivalence: its inverse is geometric realization. Technically, to spell out this equivalence in full generality, we would have to take into account (,2)(\infty,2)-categorical structure (the category of posets is a 2-category). However, in the special case of closed trusses and singular maps (and, dually, open trusses and regular maps) 2-categorical structure trivializes, and we arrive at the following result.

Theorem ([1], Section 4.2). The \infty-category of closed nn-meshes and the 1-category of nn-trusses, both with singular maps, are equivalent. Dually, this holds in the open-regular case. \blacklozenge

We illustrate the equivalence of categories in Figure 7, in the case of an open 2-truss: the 2-truss geometrically realizes to (a contractible \infty-groupoid of) open 2-meshes; conversely, passing to the entrance path posets of these 2-meshes (and recording strata dimension and framing accordingly) recovers the 2-truss that we started with.

Figure 7. The equivalence of meshes and trusses.

Computads and diagramsPermalink

In what is effectively “part II” of this note, we now turn our attention away from “local” structures and towards “global” structures. Namely, we will discuss computads which are presheafs on trusses—or, more precisely, on the “atomic cells” that constitute trusses. We will refer to these cells as “blocks”.

Block setsPermalink

Blocks are the “atomic building blocks” of closed trusses: the reader may think of them as trusses with a single facet (that is, a single cell that is not in the boundary of another cell).

Definition (Blocks). An nn-truss kk-block BB is a closed nn-truss such that

  • the top poset BnB_n as a minimal element, called the facet of BB,
  • (nk)(n-k) of the 1-truss bundles pi:BiBi1p_i : B_i \to B_{i-1} are the identity.

The dimension vector of BB is the Boolean nn-vector (d1B,d2B,...,dnB)(d^B_1,d^B_2,...,d^B_n) where diB=0d^B_i = 0 if pi=idp_i = \id and d0B=1d^B_0 = 1 otherwise. A block is called vertical if its dimension vector is of the form (0,...,0,1,...,1)(0,...,0,1,...,1). \blacklozenge

Construction (Blocks in trusses). Given an nn-truss TT and an element xTnx \in T_n, then restricting the tower of bundles defining TT to the upper closure TnxTnT^{\geq x}_n \into T_n of xx defines an nn-truss kk-block TxTT^{\geq x} \into T. We call TxT^{\geq x} the face block of xx. \blacklozenge

Example (Blocks in trusses). In Figure 8 we depict (the geometric realizations of) a 2-truss 2-block resp. a 2-truss 1-block, both of which include as face blocks into their “parent” 22-truss (shown in the middle).

Figure 8. Including 2-mesh blocks into a 2-mesh.

Definition (Block category and block nerve). Write Blkn\blcat n for the category of nn-truss blocks and singular maps (which is a subcategory of Tˉ ⁣rsn\sctruss n, defined earlier). \blacklozenge

Note that (like Tˉ ⁣rsn\sctruss n) maps in Blkn\blcat n are generated by face and degeneracy maps. we will use this category in a very similar vein as other categories of “shapes” such as, for instance, the category Δ\Delta of simplices.

Definition (Block sets). A presheaf on the category of nn-truss blocks Blkn\blcat n is called a block nn-set (or simply, a “block set”). The category of such presheafs will be denoted by BlkSetn=PSh(Blkn)\blset n = \mathrm{PSh}(\blcat n). \blacklozenge

Note that, while morally similar, blocks behave differently from other categories of shapes in many ways. For instance, they do not form a Reedy category in the obvious way (when ordering blocks by dimension). They also fail to be an Eilenberg-Zilber category.

Observation (Non-EZ block sets). Given a block in a block set XX, i.e. a map b:BXb : B \to X out of a representable, we say bb is degenerate if it factors by a degeneracy BBB \to B' through another block b:BXb' : B' \to X. It need not be the case that bb is a degeneracy of a unique non-degenerate block. \blacklozenge

In light of the previous observation, it makes sense to require the following sheafiness condition, which guarantuees that block sets preserve “stratified gluings” of blocks; concretely, “stratified gluings” refers to colimits that are computed on entrance path posets in the following sense.

Definition (Geometric block sets). A geometrically absolute colimit in Blkn\blcat n is a colimit that is preserved under the functor BlknPosn\blcat n \to \Pos_n (here, Posn=Fun([n],Pos)\Pos_n = \Fun([n],\Pos) denotes nn-simplices in Pos\Pos, and the functor BlknPosn\blcat n \to \Pos_n takes 1-truss bundles to underlying poset maps). A block set called geometric if it maps geometrically absolute colimits in Blkn\blcat n to limits in Set\SetCat. \blacklozenge

Blocks in geometric block sets are degeneracies of unique non-degenerate blocks.

ComputadsPermalink

We can now define computads. Note that we use the term “(geometric) computad” not in the traditional (already-defined!) sense, but to mean a notion of “free (associative) weak higher category”.

Definition (Computads). An nn-computad is an nn-truss geometric block set whose non-degenerate blocks are vertical. \blacklozenge

Dropping the condition of verticality, the definition generalizes to a notion of “nn-fold computad” (note also that the term “vertical” alludes to nn-fold category lingo; relatedly, see also cat-nn-groups and crossed nn-cubes)). We will come back to nn-foldness in a moment, after having discussed morphisms in computads.

A non-degenerate nn-truss kk-block in an nn-computad can be, in traditional category theory terminology, be understood as a generating kk-morphisms. The process of adding generating kk-morphisms can be understood inductively: generating (k+1)(k+1)-morphisms are the “relations” satisfied by composites of kk-morphisms (in particular, to describe a traditional nn-category you would want to consider (n+1)(n+1)-computad, in order to capture the compositional relations satisfied by nn-morphisms). Every nn-computad is trivially an (n+1)(n+1)-computad by “categorical extension” as follows.

Construction (Embeddings). There’s an inclusion of categories ι:T ⁣rsnT ⁣rsn+1\iota : \truss n \into \truss {n+1} which takes an nn-truss TT and augments it by a trivial bundle p0:T0=T0p_{0} : T_0 = T_0 to obtain an (n+1)(n+1)-truss. This restricts to an inclusion ι:BlknBlkn+1\iota : \blcat n \into \blcat {n+1}. \blacklozenge

Definition (Categorical truncation and extensions). Precomposition with ι\iota defines a functor ι:BlkSetn+1BlkSetn\iota^\ast: \blset {n+1} \to \blset n called categorical truncation. Its left adjoint ι:BlkSetnBlkSetn+1\iota_\ast : \blset n \to \blset {n+1} is called categorical extension. \blacklozenge

Let us now define what “morphism” in a computad are (the definition equally applies, more generally, to nn-fold computads). The idea is, of course, that “shapes” are given by blocks and “diagrams of shapes” are given by trusses: therefore, we first note that trusses themselves are block sets (in fact, geometric block sets).

Construction (Block nerves). The restricted Yoneda embedding of the inclusion ι:BlknTˉ ⁣rsn\iota : \blcat n \into \sctruss n induces a functor N=Tˉ ⁣rsn(ι,):Tˉ ⁣rsnBlknN = \sctruss n (\iota-,-) : \sctruss n \to \blcat n which we call the block nerve. \blacklozenge

Remark (Blocks build trusses). Given a truss TT we have

T=colim(Blkn/TTˉ ⁣rsn)T = \mathrm{colim}(\blcat n / T \to \sctruss n)

where Blkn/TTˉ ⁣rsn\blcat n / T \to \sctruss n is the forgetful functor form the comma category Blkn/T\blcat n / T. The reader familiar with the general notion of nerves will know that there are equivalent ways to state this remark: namely, it is equivalent to the observation that the functor BlknTˉ ⁣rsn\blcat n \into \sctruss n is dense; it is also equivalent to the observation that the block nerve functor NN is fully faithful. \blacklozenge

We say a truss TT is kk-vertical if kk is the smallest index such that TT lies in the image of T ⁣rskT ⁣rsn\truss k \into \truss n (equivalently, its bundles pip_i satisfy pi=idp_i = \id exactly for inki \leq n-k).

Definition (Morphisms in computads). The kk-morphisms in an nn-computad XX are maps NTXNT \to X for a kk-vertical nn-truss TT. \blacklozenge

Remark (nn-fold computads). Dropping the notion of kk-verticality makes sense if the target XX is an nn-fold computad (and thus itself may contains non-vertical generating blocks): in this case, we get 2n2^n “types of morphisms” indexed by Boolean nn-vectors that record the positions of trivial bundles pi=idp_i = \id in the truss TT.

Problem (Computads). An open problem is the comparison of the category of computads (of which, here, we only defined its objects) to other models of higher categories (from our discussion in an earlier note about the different higher categorical paradigms in use, one may expect this to be not as “straight-forward” as other comparison results). Note that we forego a more in-depth discussion of functors of computads, but that further discussion of functors can be found in this note. \blacklozenge

Addendum (Definitional disclaimer). The definition of computads given here is tentative, and certainly other definitions are possible (e.g., together with Lukas Heidemann, we are thinking about representing computads as cofibrant objects in a model structure on block/truss sets). \blacklozenge

Pasting Diagrams and combinatorial manifold diagramsPermalink

Finally, let us formally answer two questions that have been posed in previous notes but so far only addressed “informally” (or “geometrically”); namely:

  • What is a computadic pasting nn-diagram?
  • What is a (combinatorial) manifold nn-diagram?

The answer to these questions has a further refinement: we will distinguish diagrams that are “unscoped” and “scoped” (in a computad XX). Let us first observe the following. Just as cell complexes have face posets (describing the incidence relation of their faces), computads have posets describing incidence relations of blocks: of course this is a variation of the idea of entrance path posets.

Construction (Entrance and exit path posets of computads). Given a computad XX, its entrance path poset Entr(X)\Entr(X) is the poset of non-degenerate blocks in XX with and arrow bcb \to c whenever some degeneracy of cc is a face of bb. Its dual, is the exit path poset Exit(X)=Entr(X)op\Exit(X) = \Entr(X)\op of XX.

Given a kk-morphisms f:NTXf : NT \to X, we obtain the Exit(X)\Exit(X)-labeling Exit(f):TExit(X)\Exit(f) : T \to \Exit(X) which maps xTx \in T to the unique nondegenerate block to which the block NTxNTfXNT^{\geq x} \into NT \xto f X degenerates. The opposite of Exit(f)\Exit(f) will be denoted by Entr(f):TEntr(X)\Entr(f) : T \to \Entr(X). \blacklozenge

Definition (Scoped pasting diagrams). Given an nn-computad XX, an XX-scoped pasting diagram is a Exit(X)\Exit(X)-labeled nn-truss TT where lblT\lbl_T is of the form Exit(f)\Exit(f) for some (necessarily unique) higher morphism f:NTXf : NT \to X in XX. \blacklozenge

Definition (Pasting diagrams). An unscoped pasting nn-diagram is a stratified truss TT for which there exists some (non-unique) morphism ff in some nn-computad YY such that lblT\lbl_T is the connected component splitting of Exit(f)\Exit(f). \blacklozenge

We often refer to “unscoped pasting diagrams” simply as “pasting diagrams”. Note that, given an unscoped pasting nn-diagram TT, there is always a canonical choice of ff and YY; namely, the “free nn-computad” generated by YY. This is characterized by the property that Exit(f):TExit(Y)\Exit(f) : T \to \Exit(Y) is a bijection, in which case we usually write Exit(Y)Exit(T)\Exit(Y) \equiv \Exit(T).

Construction (Source and targets). Let’s construct some familiar ideas: namely, “sources” and “targets” of pasting diagrams (we discuss the unscoped case, the scoped case works similarly). Let T=(T,lblT)T = (\und T, \lbl_T) be a pasting diagram. Let ±T1T1\partial_\pm T_1 \into T_1 be the endpoints of the 1-truss T1T_1 (with frame order (T1,)(T_1,\fleq)). Restricting the tower of (n1)(n-1) bundles (pn:TnTn1,...,p2:T2T1)(p_n : T_n \to T_{n-1},...,p_2 :T_2 \to T_1) over these objects ±T1\partial_\pm T_1, augmenting the resulting towers with another trivial bundle, and restricting the labeling Exit(f)\Exit(f) accordingly, we obtain the nn-trusses ±TT\partial_\pm T \into T; these are called the source and target of TT. \blacklozenge

Remark (Degeneracy-free diagrams). A labeled closed nn-truss TT is called normalized if for any degeneracy F:TSF : T \to S (which preserves labels, i.e. lblF=id\lbl_F = \id) we have F=idF = \id. (Dually, this applies to open trusses and their label-preserving coarsenings). Every labeled closed nn-truss admits a unique degeneracy TST \to S such that SS is normalized (this shown e.g. in [2]). As a consequence, each pasting diagram degenerates to a unique pasting diagram in which degenerate blocks (or, categorically, “identities”) have been maximally removed. \blacklozenge

Before finally, giving some example, let us now dualize the notion of pasting diagrams to obtain a notion of combinatorial manifold diagrams.

Definition (Scoped combinatorial manifold diagrams). Given a computad XX, an XX-scoped combinatorial manifold diagram MM is a stratified open nn-truss whose dual M=(M,lblmop)M^\dagger = (\und M^\dagger, \lbl_m\op) is an XX-scoped pasting diagram. \blacklozenge

Definition (Combinatorial manifold diagrams). An (unscoped) combinatorial manifold diagram MM is a stratified open nn-truss whose dual MM^\dagger is a (unscoped) pasting diagram. \blacklozenge

What is the connection of such “combinatorial manifold diagrams” with the “geometric” manifold diagrams defined in an earlier note? This is where the geometric realization of nn-trusses as nn-meshes comes into play.

Construction (Geometric realization). Given a combinatorial manifold diagram M=(M,lblM:MnEntr(M))M = (\und M,\lbl_M : \und M_n \to \Entr(M)), its geometric realization M\abs{M} is the stratification with characteristic map:

MMlblMEntr(M)\abs{\und M} \to \und M \xto {\lbl_M} \Entr(M)

where the first map is the characteristic map of the open nn-mesh M\abs{\und M} realizing the open nn-truss M\und M (in particular, the underlying space of M\abs{M} is the same as that of M\abs{\und M}: the open nn-cube). Note that in the XX-scoped case, we would replace lblM\lbl_M in the above with the connected component splitting of the labeling lblM:MEntr(X)\lbl_M : \und M \to \Entr(X).

Geometric realization produces a correspondence between the following combinatorial and geometric notions:

{combinatorialmanifold diagramsthat are normalized}{geometric manifolddiagrams up to framedstratified homeomorphism}.\footnotesize \left\{ \begin{matrix} \text{combinatorial} \\ \text{manifold diagrams} \\ \text{that are normalized} \end{matrix} \right\} \iso \left\{ \begin{matrix} \text{geometric manifold} \\ \text{diagrams up to framed} \\ \text{stratified homeomorphism} \end{matrix} \right\} .

(The result, in the more general case of framed stratifications, is discussed in [1], Chapter 5).

Note that the construction verbatim applies to define geometric realizations of pasting diagrams as well (in this case, underlying spaces are those of closed nn-meshes). \blacklozenge

Finally, examples!

Examples (Pasting diagrams and their dual manifold diagrams). In Figure 9 we illustrate (geometric realizations of) pastings diagrams and their dual manifold diagrams. In each case we illustrate the geometric realization T\abs{T} of a given labeled nn-truss (whether pasting or manifold diagram): to distingsuish strata in these stratifications we either leave ample space between them or use different shades. A recipe to combinatorially interpret the pictures is as follows: (1) find the coarsest open mesh M\abs{\und M} refining one of the given manifold diagrams M\abs{M} (note we use arrows here and there to indicate framing directions), (2) passing to entrance path posets, build an open truss M\und M, (3) derive a labeling lblM=Entr(MM)\lbl_M = \Entr(\abs{\und M} \to \abs{M}) from the action of the coarsening MM\abs{\und M} \to \abs{M} on entrance path posets; as a result, obtain a combinatorial manifold diagram M=(M,lblM)M = (\und M,\lbl_M), (4) dualize to obtain a pasting diagram MM^\dagger (of course, one may also start with building closed meshes for the given geometric pasting diagrams). We leave details to the reader as a fruitful exercise. \blacklozenge

Figure 9. Pasting diagrams (realized as geometric pasting diagrams), and their dual combinatorial manifold diagrams (realized as geometric manifold diagrams)

That’s it for now—as always, comments are most welcome!

ReferencesPermalink

[1] “Framed Combinatorial Topology”, Dorn + Douglas

[2] “Associative nn-categories”, Dorn

[3] “Higher Algebra”, Lurie