# Combinatorial Cobordism Theory

Maxime Savoy

###### Abstract

We introduce a formalism based on a combinatorial notion of cell complex
subject to an inclusion-reversing duality operation. Our main goal is to open
the way for a functorial definition of field theories in a context where no
manifold or topological structure is assumed. This is achieved via a discrete
notion of cobordism for which a composition operation is defined. Our main
theorem enables the composition of cobordisms by showing that certain
sequences of maps between cell complexes are in bijective correspondence with
a cell complex of dimension one higher. As a result we obtain a category whose
morphisms are cobordisms having a causal structure generalizing that of Causal
Dynamical Triangulations as well as dualities inherited from the duality map
defined on cell complexes.

##### Key words

Discrete Geometry, Geometric Duality, Combinatorial cell complexes, Quantum
Gravity, Quantum Field Theory.

###### Contents

  1. Introduction
  2. 1 Laying down the formalism
    1. 1.1 Notation
    2. 1.2 Cell Complexes (cc)
    3. 1.3 Duality map for closed cc
    4. 1.4 Midsections and pinches
    5. 1.5 Subdividing and collapsing closed cc
      1. 1.5.1 Barycentric subdivision and dual cells
      2. 1.5.2 Reductions and collapses
  3. 2 Structure near the boundary and Correspondence Theorem
    1. 2.1 Relative cc
    2. 2.2 Transition and uniformity
    3. 2.3 Geometric sequences
    4. 2.4 Slices and Correspondence Theorem
  4. 3 Cobordisms
    1. 3.1 Definition and duality
    2. 3.2 Composition
    3. 3.3 The category of causal cobordisms
  5. A Lemmas A.1 to A.8
  6. B Proof of Proposition 1.24
  7. C Proof of Proposition 2.10
  8. D Proof of Proposition 2.13
  9. E Proof of Theorem 3.5

## Introduction

Motivated by the physics of space-time at low scales, this paper proposes a
novel framework of discrete geometry based on a combinatorial notion of cell
complex. By combinatorial we mean that a cell of a cell complex simply
corresponds to a subset of a given finite set of vertices, and each cell is
only assigned a rank (or dimension). The structure of cell complexes is then
purely characterized using conditions on how cells are included in one another
according to their rank and does not require to specify the topology
associated to each cell.

Our main goal is to identify a minimal set of assumptions on cell complexes
leading to a notion of Quantum Field Theory (QFT) taking values on the
resulting discrete structure and defined via a functor between categories.
Resting on ideas of Witten, similar functorial field theories were originally
defined in the work of Atiyah [Ati88] for the case of Topological Quantum
Field Theory, and Segal [Seg04] for the case of Conformal Field Theory and
were also used in various other contexts. Such field theories usually
correspond to a functor from a category of cobordisms (or bordisms, from which
we make no distinction here) to the category of vector spaces (and this is as
far as categorical notions will be used in the present work).

Figure 1:  A two-dimensional cobordism $M$ between an ingoing component
$\Sigma_{1}^{*}$ and outgoing components $\Sigma_{2}$ and $\Sigma_{3}.$

Cobordisms were initially studied by René Thom ([Tho54]) as defining an
equivalence relation on closed oriented manifolds of dimension $n,$ where
$\Sigma\cong\Sigma^{\prime}$ whenever there exists a smooth oriented compact
manifold $M$ of dimension $n+1,$ a cobordism, such that $\partial
M=\Sigma\sqcup\Sigma^{\prime},$ as for example illustrated in Figure 1. In our
context a cobordism more generally stands for a geometry of a given dimension
having a definite notion of boundary split in two types of components: ingoing
and outgoing. A cobordism can then be interpreted as interpolating between an
initial ”state of geometry” associated with the ingoing boundary components to
a finial state of geometry associated with the outgoing ones, and can be seen
as a possible transition between the two states. The definition of field
theory we aim for hence essentially requires a notion of cobordism that, seen
as a mapping between states of geometry, satisfies the composition law of
morphisms in a category. This in turn requires an appropriate notion of state
of geometry such that if a given state is both the final state of a cobordism
$C_{1}$ and the initial state of a cobordism $C_{2}$ then the composition
$C_{2}\circ C_{1}$ (i.e. the union $C_{2}\cup C_{1}$) is a cobordism from the
initial state of $C_{1}$ to the final state of $C_{2}.$

In the final section of this paper we introduce a category whose morphisms
correspond to cobordisms of fixed finite dimension and whose structure is
based on the notion of cell complexes introduced first. The cobordisms of this
category are ”causal” in a sense used also in the context of Causal Dynamical
Triangulations (CDT), see e.g. [Lol20] for a recent physical review or [DJ15]
for a mathematical treatment. In simple terms, a causal structure here means
that it is decomposed into ”slices”, where each slice can be compared to space
at fixed time according to some choice of (discrete) time coordinate. The
structure of causal cobordisms thus generalizes that of CDT by incorporating
discrete geometries having more general cells than simplices, but it is also
endowed with an additional duality operation we discuss in the next paragraph.

An important feature of causal cobordisms is that they are characterized using
sequences of maps between cell complexes that we call geometric sequences.
This characterization is based on the main result in this work, the
Correspondence Theorem 2.22, which establishes a bijection between certain
geometric sequences and cell complexes we call slices, generalizing the notion
of slice from CDT. The latter result (in its dual version) is key in defining
a composition operation on causal cobordisms and finding an appropriate notion
of state of geometry.

##### A duality is built-in at the base of the formalism.

For the sake of clarity, let us explicit a few elementary facts about the
notion of cell complexes used here. As mentioned earlier, a cell is simply a
subset of a finite set of vertices to which one assigns a non-negative integer
called rank. Cell complexes satisfy certain axioms implying in particular that
vertices have rank 0 and the rank of a cell $x$ in fact corresponds to the
number of cells (including $x$) contained in $x$ and containing any fixed
vertex in $x.$ We will focus on cell complexes satisfying also a number of
basic assumptions well motivated by applications in physics: edges, i.e. cells
of rank 1, contain two vertices, and cells are connected, meaning that any two
vertices in a cell can be connected by a path of edges. Cell complexes
satisfying the latter conditions will be called local if they moreover satisfy
that their underlying graph is connected. We will also assume that all maximal
cells in a cell complex, i.e. cells having no other cell containing it, have
same rank defining the rank or dimension of the cell complex. The image of a
cell $x$ under the duality map is then simply defined as the set
$\overline{x}$ of maximal cells containing $x.$ Figures 2 shows examples of
dual cells in a cell complex of dimension 2.

Figure 2:  Dual cells in a cell complex of rank 2 having $v$ and $w$ as
vertices. The duals of maximal cells are singletons represented by crosses.
The dual cell associated with the vertex $v$ is a set containing three crosses
which intersects the dual cell associated with the vertex $w$ on two crosses
constituting the cell dual to the edge $\\{v,w\\}.$

Using these minimal notions of cell and dual cell, we argue in the first
section that these two collections of cells can be seen as equivalent
encodings of a given discrete geometry. This point of view motivated our
framework in which the duality map is a bijection on a wide class of cell
complexes having empty boundary that we call closed cell complexes and which
play an important role in this work. Consequently, this defines an involution,
i.e. the dual of the dual of closed cell complex is isomorphic to the primal
cell complex. If a closed cell complex is obtained as the face lattice of some
CW-decomposition of a manifold, then its image under the duality map
corresponds to the face lattice of the dual CW-complex leading to Poincaré
Duality, but our framework doesn’t require such geometrical realization.

Not only does this duality emerge at the base of our formalism, but it also
drove its development: as explained in Section 3, we were lead to consider a
notion of cobordism by looking for a structure allowing to extend the duality
map on closed cell complexes to local cell complexes with a non-empty
boundary. It also potentially represents an interesting leverage in order to
make touch with more standard notions of geometry used in physics, as it
allows to single out cell complexes that are self-dual - or locally so - a
property typically satisfied by the square or hypercubic lattices widely used
in statistical mechanics or Lattice QCD. These areas of research include many
applications of the dual lattice or of more general discrete spaces to study
certain transformations on fields defined on it, the simplest example being
arguably the Kramers-Wannier duality of the two-dimensional Ising model. The
latter duality has for example been treated as a generalized Fourier transform
([McK64]) or linked with the electromagnetic duality ([FT19]) and such
connections might shed light on how the duality on cell complexes would couple
with fields defined on causal cobordisms, however this is a topic for further
research.

##### On connections with lattices, quantum logic and other combinatorial
notions of cell complexes.

Our definition of cell complexes is a somewhat simpler and more explicit
formulation of the definition of combinatorial cell complexes introduced by
Basak in [Bas10] (we chose to drop the term ”combinatorial” for brevity). The
main goal of Basak’s work is to show a version of Poincaré duality for
combinatorial cell complexes based on the duality operation discussed above,
hence some of our basic definitions are the same. However he focuses on closed
complexes satisfying certain orientability conditions and no such assumption
will be needed in our context. An other definition of combinatorial cell
complexes was introduced in [Asc96] using a rather general formalism primarily
designed to discuss relations with manifolds and CW-complexes using
assumptions on the topology associated to the cells of the complex.

In order for the duality map to define a bijection on closed cell complexes,
it is imposed as an axiom that the intersection of two cells is a cell (or is
empty). We close part 1.3 by explaining how the latter axiom also leads to a
close relation between cell complexes and lattices, and more specifically the
notion orthocomplemented lattice. Such lattices were considered by Von Newman
and Birkhoff [BvN36] who initiated a line of research, called quantum logic,
aiming to formalize the logical structure underlying possible measurements on
a quantum system. Interestingly, following this connection with quantum logic
(seeing cell complexes as analogous to orthocomplemented lattices) suggest the
interpretation that simplicial complexes are analogous to Boolean lattices (or
”locally Boolean”), and thus describe the subclass of quantum systems that
behave classically. We see this as a motivation to study cell complexes to
look for a natural notion of discrete ”quantum geometry” encompassing more
general structures than triangulations. The field of quantum logic was further
developed in the sixties (see e.g. Mackey [Mac63] or Jauch [Jau68]) but has
also faced a number of criticisms (mentioned for example in [CG01]). We also
remarked that the distinction between cell complexes and orthocomplemented
lattices could potentially solve certain issues raised about the physical
interpretation of the latter. These connections were however only discovered
in a later stage of this research and would deserve further explorations.

##### Why renouncing to a manifold structure while looking for applications to
the physics of Quantum Gravity?

In a nutshell, the search of a theory of Quantum Gravity in the path integral
approach aims to give a precise meaning to the formal notion of integration
over possible space-time geometries with given boundary data. The ordinarily
large size of the space of possible geometries is a major source of
difficulties, it is hence natural to regularize this problem by using discrete
geometries. One could also take the view that a fundamental description of
space-time would require a discrete notion of geometry and that manifolds only
model space-time at larger scales.

Regardless of the choice of point of view, in order to define an integration
over discrete geometries two possible strategies stand out as relatively
tractable: either fixing a discretization and integrate over possible metrics
on it (in which case it is relatively hard to recover the symmetries of
General Relativity) or sum over a class of discrete geometries each of which
is associated a fixed metric. Following the second strategy, one faces the
task of choosing an appropriate class of possible discrete geometries to sum
over. It is often convenient or argued as sufficient to restrict oneself to
triangulations, as in the case of Dynamical Triangulation or Tensor models
([ADJ97], [GR12]). The point is that the study of these models lead to
consider sums over triangulations without constraint on their topology, but
simply constructed from a collection of simplices of a given dimension $d$ by
identifying faces of dimension $d-1.$ Such construction is encoded in the
(colored) ”dual graph”, for which each vertex corresponds to a $d$-dimensional
simplex and two vertices are linked by an edge whenever the corresponding
simplices share a $(d-1)$-dimensional face. Seeing triangulations as duals to
the graphs obtained as the Feynman diagrams associated to a certain field
theory is the idea behing the approach of Tensor models (also called Group
Field Theories, [Fre05]), which aim is to generalize the rigorous results
obtained in dimension two with matrix models ([Fra04]). The geometries
considered in these models hence emerge from purely combinatorial structures
encoded in the terms of a Feynman expansion, therefore restricting to those
having the structure of a manifold does not come as such a natural condition
to impose (especially in dimensions higher than 2).

The cell complexes discussed in the present paper include triangulations (or
more precisely the associated face lattice) and structures dual to that, which
underlying graph corresponds to the generalized Feynman diagrams mentioned
above. Hence our framework allows to consider a rather general notion of field
prior to fixing or recovering a manifold structure, but it also puts
triangulations and their duals on an equal footing, as well as many other
types of discrete geometries such as lattices used in statistical mechanics or
other polyhedral complexes.

##### This paper is self-contained, and organised as follows.

The first section is a step-by-step introduction to the basic notions
underlying this framework and states the simplest version of the duality for
closed cell complexes. We also present a novel combinatorial definition of a
subdivision of a cell complex widely generalizing the common notion of
barycentric subdivision. A subdivision is defined via a map called reduction
and we also introduce maps called collapses that are seen as dual to
reductions.

Section 2 develops the machinery needed for the last section. The notion of
uniformity is introduced, characterizing a large class of cell complexes whose
boundary can be described using a sequence of a reduction and a collapse. Such
sequence is at the base of the definition of geometric sequence that comes
next and the main result of this paper, the Correspondence Theorem 2.22, show
that there is a bijection between certain geometric sequences and a class of
uniform cell complexes called slices.

Section 3 defines cobordisms and their duality, generalizing the duality
obtained for closed cell complexes. It also uses geometric sequences and the
Correspondence Theorem to define a composition operation on cobordisms. This
leads to the definition of a category whose morphisms are cobordisms
constructed using slices called causal cobordisms.

Certain proofs have been alleviated of some details which can be found in the
authors thesis [Sav21].

##  1 Laying down the formalism

This work is based on a combinatorial notion of cell complexes, abbreviated
cc, which is introduced in part 1.2 of this section together with some simple
examples. An other central protagonist of our formalism is the duality map
introduced in part 1.3 for the case of cc with empty boundary, and which would
be extended to cc with boundary in Section 3. The part 1.4 then introduces an
other important cc called midsection used throughout this article. We finish
this first section by defining a notion of subdivision for cc, together with
the operation dual to it, which is the cement for the construction of causal
cobordisms, the end-goal of this work. Let us first start by clarifying some
notation and terminology used in what follows.

###  1.1 Notation

In these notes $\mathbb{N}=\\{0,1,2,\dots\\}$ are the natural numbers and
$\mathbb{N}^{*}=\\{1,2,\dots\\}.$ We usually use the letters $i,j,k,n,r$ to
denote indices in $\mathbb{N}.$ We use $:=$ to denote an equality where the
left hand side is defined by the right hand side and we use italic words
whenever we define a notion for the first time.

We use the notation $\left|S\right|$ for the number of elements in a set $S.$
The set of subsets of $S$ is denoted by $\mathcal{P}\left(S\right)$ and the
disjoint union of the sets $S$ and $S^{\prime}$ is denoted by $S\sqcup
S^{\prime}$ (used purely to emphasize that $S$ and $S^{\prime}$ are disjoint).
The set $S\setminus S^{\prime}$ denotes the difference between $S$ and
$S^{\prime},$ i.e. the elements in $S$ that are not in $S^{\prime}.$ A map
$f:A\longrightarrow B$ is a function with domain equal to $A$ and the
composition of two maps $f:A\longrightarrow B,$ $g:B\longrightarrow C$ is
denoted by $g\circ f.$

Let $(P,\leq)$ be a partially ordered set or poset, which means that $\leq$ is
a reflexive antisymmetric transitive binary relation on the set $P.$ The set
define the sets of cells respectively above and below a cell $x$ in $P:$

$\mathbf{A}(x):=\\{y\in P\leavevmode\nobreak\ |\leavevmode\nobreak\ x\subset
y\\},\quad\mathbf{B}(x):=\\{y\in P\leavevmode\nobreak\ |\leavevmode\nobreak\
y\subset x\\}.$

One sometimes write the latter sets respectively $\mathbf{A}_{P}(x)$ and
$\mathbf{B}_{P}(x)$ for in case of ambiguity on the poset $P.$ An element in
$P$ is maximal if it is not below any other element and minimal if it is not
above any other element. If $T\subset P$ an upper bound of $T$ is an element
$a\in P$ such that $b\leq a$ for all $b\in T$ and a lower bound of $T$ is an
element $a\in P$ such that $a\leq b$ for all $b\in T.$ If there is a unique
upper bound of $T$ which is below all upper bounds of $T$ then this element is
called the least upper bound of $T$ and is denoted by $\bigvee T$. Similarly
if there is a unique lower bound of $T$ which is above all lower bounds of $T$
then this element is called the greatest lower bound of $T$ and is denoted by
$\bigwedge T.$ If $T=\\{a,b\\},$ then we write $\bigvee T=a\vee b$ and
$\bigwedge T=a\wedge b.$

###  1.2 Cell Complexes (cc)

The motivation for considering the following definition of cc can be thought
of as looking for a minimalistic notion of geometry that conserves enough
structure to retain a meaningful notion of field, formalized axiomatically
using cobordisms (or bordisms) as explained in the introduction. That is, one
should in particular be able to identify what the boundary of a given geometry
is, and this is done via a notion of dimension as a non-negative integer.

We thus start with the simplest possible structure, a finite set $S,$ which
elements we call vertices. Vertices can be though of as the fundamental units
of the description of the geometry we are formalizing, whether considered
complete or approximate. In order to build a complex out of these vertices,
one then considers a family of cells, i.e. non-empty subsets of vertices, with
the assumption that every vertex, conventionally identified with the
corresponding singleton, is a cell. The natural way to assign a rank (or
dimension) to these cells is to endow them with the structure of a ranked (or
graded) poset defined as follows.

######  Definition 1.1 (Ranked poset, face, co-face).

Let $(P,\subset)$ be a poset together with a rank function
$\operatorname{rk}=\operatorname{rk}_{P}:P\rightarrow\mathbb{N}$. We define
the set of faces of $x\in P$ by

$\triangle\left(x\right):=\\{y\in K\leavevmode\nobreak\ |\leavevmode\nobreak\
y\subset x,\leavevmode\nobreak\
\operatorname{rk}(y)=\operatorname{rk}(x)-1\\}$

and the set of co-faces of $x$ by

$\bigtriangledown\left(x\right):=\\{y\in K\leavevmode\nobreak\
|\leavevmode\nobreak\ x\subset y,\leavevmode\nobreak\
\operatorname{rk}(y)=\operatorname{rk}(x)+1\\}.$

Then we say that $(P,\subset,\operatorname{rk})$ is a ranked poset if
$\operatorname{rk}(x)=0$ whenever $x$ is a minimal element and the following
axioms are satisfied:

  1. i)

$\operatorname{rk}$ is strictly compatible with the partial order $\subset:$
for all $x,y\in P,$ if $x\subsetneq y$ then
$\operatorname{rk}(x)<\operatorname{rk}(y),$

  2. ii)

there is no gap in the value of the rank function: for all $x,y\in P,$ if
$x\subsetneq y$ then $x$ is a face of a cell contained in $y,$

We denote by $A^{[r]}$ the set of $r$-cells in a subset $A$ of a ranked poset
$(P,\operatorname{rk}).$

In addition to the structure of ranked poset, a fundamental assumption we make
on our collection of cells is that the intersection of two cells is also a
cell or is empty. As explained later in part 1.3, this assumption is key in
order to obtain a duality on posets whose elements are cells, i.e. defined as
subsets of a finite set of vertices.

Finally the ”diamond property”, named after shape of the Hasse diagram of the
four cells it involves, constitutes the last axiom of the definition of cc
stated thereafter. This property is also used in the context of abstract
polytopes ([MS02]) and allows to have a well behaved notion of boundary
$\partial K$ of a cc $K$ (defined in point 4) of Example 1.3) in the sense
that $\partial(\partial K)=\emptyset$ holds for the class of cc satisfying the
assumptions needed for our purposes.

######  Definition 1.2 (Cell complex (cc)).

Let $S$ be a finite set and let
$K\subset\mathcal{P}\left(S\right)\setminus\\{\emptyset\\}$ together with a
rank function
$\operatorname{rk}=\operatorname{rk}_{K}:K\rightarrow\mathbb{N}$. We call an
element of $K$ a cell and an element of $K^{[r]}$ an r-cell. The data
$(K,\operatorname{rk})$ defines a cell complex, or cc for short, if it is a
ranked poset such that the two additional axioms are satisfied:

  1. iii)

$K\cup\\{\emptyset\\}$ is closed under intersection: for all $x,y\in K,$
$x\cap y\in K\cup\\{\emptyset\\}.$

  2. iv)

$K$ satisfies the diamond property: for all $x,y\in K,$ if $x\subset y$ and
$\operatorname{rk}(x)=\operatorname{rk}(y)-2$ then there are exactly two cells
$z_{1},z_{2}$ such that

$\bigtriangledown\left(x\right)\cap\triangle\left(y\right)=\\{z_{1},z_{2}\\}.$

Let us now introduce some terminology related to cc (and more generally to
finite ranked posets). The rank $\operatorname{Rk}(K)$ of a cc $K$ is the
maximum value among the ranks of its cells, and we sometimes call a $R$-cc a
cc of rank $R.$ The $k$-skeleton of a cc $K$ is defined as

$K^{(k)}:=\bigcup_{r=0}^{k}K^{[r]}.$

A cc for which all maximal cells have same rank is called pure. A cc $J$ is a
sub-cc of $K,$ noted $J\leq K,$ if $J\subset K$ and
$\operatorname{rk}_{J}=\operatorname{rk}_{K}|_{J}.$

We called edges the set the cells of rank $1$ in a cc. An edge in a cc
necessarily contains more than two vertices as the definition of cc asserts
that all singletons are vertices (i.e. have rank $0$), and we call graph-based
a cc for which all edges contain no more than two vertices.

In what follows, every cc is pure and graph-based unless otherwise stated.

######  Example 1.3.

We shall discuss a number of basic examples, among which simplicial complexes,
graphs and the boundary of a cc.

  1. 1)

Simplicial complexes: Let $S$ be a finite set. A collection
$\Delta\subset\mathcal{P}\left(S\right)$ is called a simplicial complex if it
is closed under taking the subset of a cell, i.e. if $x\subset y\in\Delta$
then $x\in\Delta.$ An element $x\in\Delta$ is called a simplex and we define
the rank function of $\Delta$ by

$\operatorname{rk}_{\Delta}(x)=\left|x\right|-1.$

It is a good warm up exercise to check that
$(\Delta,\operatorname{rk}_{\Delta})$ satisfies the four axioms of a cell
complex. We could have also defined a simplicial complex to be a cc $K$ with
rank function $\operatorname{rk}_{K}$ satisfying closeness under taking the
subset of a cell. In this case, as a consequence of Axiom ii), the rank
function satisfies $\operatorname{rk}_{K}(x)=\left|x\right|-1.$

The face lattice of a triangulation of a manifold, i.e. the poset defined as
the set of simplices in the triangulation ordered by inclusion, is an example
of an ”abstract” simplicial complex as defined here. But the geometric
realization of such simplicial complexes (defined e.g. in [RS72]) does not in
general have the topology of a manifold.

  2. 2)

Graphs: A graph is defined as a simplicial complex of rank 1. We denote by
$\mathcal{V}$ the set of vertices of a graph and $\mathcal{E}$ its set of
edges. In particular a graph $\mathcal{G}$ is a 1-cc where we use the notation
$\mathcal{V}=\mathcal{G}^{[0]}$ and $\mathcal{E}=\mathcal{G}^{[1]}$ and there
cannot be multiple edges between two vertices nor self-edge (linking a vertex
with itself) in a graph. As expected, if a cc $K$ is graph-based then
$K^{(1)}$ is a graph and we sometimes use the notation $\mathcal{E}$ for
$K^{[1]},$ the set of edges of $K,$ when there is no ambiguity on the cc $K.$

A graph is connected if any two vertices can be linked by a path, i.e. a
sequence of edges such that two consecutive edges intersect on a vertex,
called a crossed vertex. A path is simple if it doesn’t cross a vertex twice
and a cycle is a path starting from and ending at the same vertex. A simple
cycle is called a loop.

  3. 3)

Let $K$ be a cc and let $A\subset K^{[0]}$ (or sometimes $A\subset L^{[0]}$
for some cc $L$ containing $K.$) Then the restriction of $K$ to $A$

$\mathbf{B}(A):=\\{x\in K\leavevmode\nobreak\ |\leavevmode\nobreak\ x\subset
A\\}$

is clearly a cc. In particular, for every cell $x$ in a cc $K,$
$\mathbf{B}(x)$ is a cc of rank $\operatorname{rk}(x)$ having $x$ as its only
maximal cell.

  4. 4)

The boundary of a cc $K$ is defined as the set of cells contained in a sub-
maximal cell having only one maximal cell above itself. In other word the
boundary of a $R$-cc $K$ is defined as

$\partial K:=\bigcup_{\begin{subarray}{c}y\in K^{[R-1]}\\\
\left|\bigtriangledown\left(y\right)\right|=1\end{subarray}}\mathbf{B}(y).$

One can easily check that $(\partial K,\operatorname{rk}_{K}|{\partial K})$ is
a cc and it is explained in Remark 1.18 that under certain hypothesis we have
$\partial(\partial K)=\emptyset$ as a consequence of the diamond property iv).

As we are primarily interested in considering cc in the context of physics, it
is natural to restrict our focus on cc satisfying a certain notion of locality
which is simply defined as follows.

######  Definition 1.4 (Local cell complex (lcc)).

A cc $K$ is local if it is connected, that is to say $K^{(1)}$ is connected as
a graph, and cell-connected, i.e. $\mathbf{B}(x)$ is connected for all $x\in
K.$ Local cc is sometimes be abbreviated lcc.

As we assumed cc to be graph-based, edges are constrained to be sets of two
vertices, and as a consequence of the diamond property iv), one also has that
a 2-cell in a local cc necessarily satisfies that their faces form a loop.
Indeed if $x$ is a 2-cell in a lcc $K,$ the latter axiom implies that for
every vertex $v\in x$ we have

$\bigtriangledown\left(v\right)\cap\triangle\left(x\right)=\\{e,e^{\prime}\\}.$

Iterating this argument on the vertices in $e$ and $e^{\prime}$ let us grow a
path made of faces of $x$ that has no other possibility than being simple and
closing itself into a cycle.

Let us also introduce homomorphisms of cc. In part 1.5 we consider particular
types of homomorphisms of cc that can be interpreted either as defining a
subdivision or a collapse of a cc and which plays a central role in the later
sections.

######  Definition 1.5 (cc-homomorphism, cc-isomorphism).

A map $\phi:K\longrightarrow L$ is called a homomorphism of cc, or cc-
homomorphism, if it is a poset homomorphism, i.e. $\phi(x)\subset\phi(y)$
whenever $x\subset y,$ such that for each cell in the image of $\phi$ there is
a cell in its pre-image having same rank, i.e.
$\phi^{-1}(y)^{[\operatorname{rk}(y)]}\neq\emptyset$ for all $y\in\phi(J).$ A
cc-isomorphism is a bijective cc-homomorphism $\phi$ such that $\phi^{-1}$ is
a cc-homomorphism. We use the notation $K\cong L$ to indicate that $K$ and $J$
are cc-isomorphic.

Figure 3:  Example of a bijective poset homomorphism from a cc of rank 1 to a
cc of rank 2 which is not a cc-homomorphism and such that the inverse is not a
poset homomorphism. The outer edges from the 1-cc are ”mapped to themselves”
and the middle edge is mapped to the 2-cell.

Figure 3 shows an non-example of a cc-homomorphism and illustrates the fact
that a bijective poset homomorphism is not necessarily a cc-isomorphism. Lemma
1.6 is a simple consequence of Axioms i) and ii) and gives two other ways to
characterize a cc-isomorphism that will be of use later.

######  Lemma 1.6.

Let $K,J$ be cc and $\phi:J\longrightarrow K$ be a poset homomorphism. The
following three statements are equivalent.

  1. 1)

$\phi$ is a cc-isomorphism.

  2. 2)

$\phi$ is bijective and $\phi(x)\in K^{[\operatorname{rk}_{J}(x)]}$ for all
$x\in J.$

  3. 3)

$\phi$ is bijective and $\phi^{-1}$ is a poset homomorphism.

###  1.3 Duality map for closed cc

The dual of a poset $P$ is a standard notion defined by $P^{*}=\\{p^{*}|p\in
P\\}$ and $p^{*}\leq_{P^{*}}q^{*}$ if and only if $q\leq_{P}p.$ It provides an
anti-isomorphism of posets, i.e. a bijection reversing the partial order,
which is therefore involutive: $(P^{*})^{*}$ is isomorphic to $P.$ Our goal is
to restrict this duality to the set of cc, which in particular includes the
additional assumption that cells are defined as subsets of a set of vertices,
i.e. are characterized by the minimal elements it contains. Therefore it
should also be the case for dual cells, meaning that dual cells can be defined
as subsets of maximal cells of the ”primal” cc 111Hence it is more than
assuming the poset to be an inclusion order, i.e. defined as the subset-
inclusion relation on a set of objects, as every poset is isomorphic to an
inclusion order. The condition used here to restrict the duality from posets
to cc can be formulated using the notion of 2-dimension of a poset i.e. the
size of the smallest set of objects such that the poset is an inclusion order
on these objects. One then considers duals of posets $P$ such that both the
2-dimension of $P$ is equal to the number of minimal elements in $P$ and the
2-dimension of the dual $P^{*}$ is equal to the number of maximal elements in
$P.$. As we assume cc to be pure, all maximal cells have the same rank equal
to the rank of the cc, hence we can use the following explicit definition of
the dual of a cell, or more generally of any set of vertices.

######  Definition 1.7 (Dual set).

The dual set of a subset $A$ of vertices in a cc $K$ of rank $R$ is defined by

$\overline{A}:=\\{z\in K^{[R]}\leavevmode\nobreak\ |\leavevmode\nobreak\
A\subset z\\}.$

We sometimes use the notation $\overline{A}^{K}$ for $\overline{A}$ to specify
the cc relative to which the dual set is defined.

We explain in details in part 1.5.1 how this definition of dual cell differs
from a more standard definition used in the context of simplicial complexes.
From the latter definition, one is naturally lead to introduce the following
poset.

######  Definition 1.8 (Dual of a cc).

We define the dual of a cc $K$ to be the poset
$\left(\overline{K},\subset\right)$ by

$\overline{K}:=\\{\overline{x}\leavevmode\nobreak\ |\leavevmode\nobreak\ x\in
K\\}.$

In this case $K$ is sometimes called primal with respect to $\overline{K}.$

In other words, $\overline{K}$ is the image of $K$ through the duality map

$x\longmapsto\overline{x}.$

We will more generally use the notation
$\overline{S}:=\\{\overline{x}\leavevmode\nobreak\ |\leavevmode\nobreak\ x\in
S\\}$ for any subset $S$ of cells in $K.$

This duality map can also be seen as a natural operation to define on cc from
the following simple observation. Specifying a cell of a cc consists in
differentiating between vertices contained and those not contained in it,
therefore both sets of vertices in way encode the same data. Looking at the
complement of a cell $x$ in the set of vertices clearly represents a shift
from a local to a non-local description, however restricting our view to
vertices outside $x$ but only ”up to one maximal cell away” from $x,$ i.e.
such that there exists a maximal cell containing both $x$ and a vertex not in
$x,$ can be considered as a local analogue of the notion of complement. In
fact this amounts to specifying the maximal cells containing $x,$ which indeed
is our the definition of the dual cell $\overline{x}.$

As will be shown at the end of this part, the notion of closed cc introduced
next in fact constitutes a first set of assumptions defining a structure on cc
invariant under the duality map.

######  Definition 1.9 (Non-singular and closed cc).

A cc is non-singular if every sub-maximal cell is contained in at most two
maximal cells and a cc is closed if it is non-singular and has empty boundary,
i.e. every sub-maximal cell is contained in exactly two maximal cells.

It’s should be intuitive that, in order for the dual of a cc to be a cc, the
primal cc has to be non-singular, as this assumption is dual to that of graph-
based cc. One can also easily see that if a sub-maximal cell $y$ is included
in only one maximal cell $z$ then $\overline{y}=\overline{z}$ which also
spoils the injectivity of the duality map. In fact introducing the notion of
cobordism in Section 3 will constitute a way around this fact, and will allow
to define a duality map over a class of cc with non-empty boundary.

The fact that cells of a cc are characterized by the vertices it contains in
addition implies that every cell $x$ having two or more faces, i.e. of rank 1
or higher, is the smallest upper bound of its faces, that is to say

$x=\bigvee\triangle\left(x\right)=\bigcup_{w\in\triangle\left(w\right)}y.$ (1)

The identity (1) follows from observation that for all $v\in x$ there is a
face $w$ of $x$ containing $v,$ hence $x\subset y$ for all upper bound $y$ of
$\triangle\left(x\right),$ i.e. $x$ is indeed the smallest upper bound of
$\triangle\left(x\right).$ It isn’t necessarily verified in more general
posets, which explains why the definition of combinatorial cell complexes from
[Bas10] includes (1) as an axiom in the definition of combinatorial cell
complexes.

A closed cc, having empty boundary, satisfies that all non-maximal cells have
at least two cofaces, i.e. the condition dual to that satisfied in a cc that
all cells of rank 1 or higher have two or more faces. This yields the identity
dual to (1) (shown in Lemma A.1), for $x$ a cell in a closed cc:

$x=\bigwedge\bigtriangledown\left(x\right)=\bigcap_{y\in\bigtriangledown\left(x\right)}y.$

It is then seen that the bijectivity of the duality map is achieved under the
assumption of closeness.

######  Lemma 1.10.

If $K$ is a closed cc then the duality map $K\longrightarrow\overline{K}$ is
an anti-isomorphism of posets, i.e. the duality map is bijective and for all
$x,y\in K$ we have

$x\subsetneq y\quad\text{ if and only if
}\quad\overline{y}\subsetneq\overline{x}.$

###### Proof.

It is a simple consequence of the following statement: if $x,y\in K$ are such
that there exists $v\in y\setminus x$ then $\overline{x}$ contains a maximal
cell which does not contain $v,$ meaning
$\overline{x}\not\subset\overline{y}.$ This can be shown by induction on
$n:=R-\operatorname{rk}(x)$ and using the fact that
$x=\bigwedge\bigtriangledown\left(x\right)$ by Lemma A.1. ∎

We can therefore introduce the dual rank function
$\operatorname{rk}_{\overline{K}}:\overline{K}\longrightarrow\mathbb{N}$
defined by

$\operatorname{rk}_{\overline{K}}(\overline{x}):=\operatorname{Rk}(K)-\operatorname{rk}_{K}(x)$

well defined by the previous lemma and such that
$\operatorname{rk}_{\overline{K}}(\overline{z})=0$ for all maximal cells $z\in
K.$

Moreover $(\overline{K},\operatorname{rk}_{\overline{K}})$ is a ranked poset
as the strict compatibility i) and no gap property ii) are easily derived from
the same axioms for $(K,\operatorname{rk}_{K}):$ the first is clear from Lemma
1.10 and the second follows from the observation that $\overline{x}$ is a face
of $\overline{y}$ if and only if $y$ is a face of $x,$ and the (recursive)
application of the statement of ii) for the cells in
$(K,\operatorname{rk}_{K}).$

One can then easily point out why Axiom iii), ensuring that the intersection
of two cells is a cell (or is empty), is required in order for the duality map
to be bijective. This can be seen using an simle example illustrated in Figure
4 where $x,y$ are maximal cells of rank 2 intersecting on two of their faces.
Note that this condition is not necessarily imposed in other notions of cell
complexes (for example the notion of ”ball complex” defined in [RS72]).

Figure 4:  This is an example of two cells $x,y$ of rank 2 with four faces
whose intersection is the union of two edges $e$ and $e^{\prime}.$ The dual
sets associated to $e$ and $e^{\prime}$ as defined for a cc of rank 2 both
equal to the two-set $\\{x,y\\}.$ It therefore spoils the injectivity of the
dual map and explains the inclusion of axiom iii) in the definition of cc.

The next result is the first exhibiting a class of cc on which the duality map
defines an involution. We refine this result in the next part to include
locality of cc and it is generalized in Section 3 to include cc with non-empty
boundary.

######  Proposition 1.11.

If $K$ is a closed $R$-cc then so is $\overline{K}$ and

$\overline{\overline{K}}\cong K.$

###### Proof.

As discussed before $(\overline{K},\operatorname{rk}_{\overline{K}})$ is a
ranked poset, and Axiom iii) for $\overline{K}$ is a consequence of the first
identity in (2). Moreover the diamond property iv) follows from the
observation that by Lemma 1.10 we have
$\overline{\bigtriangledown\left(x\right)}=\triangle\left(\overline{x}\right)$
for all cell $x$ of in a closed cc and this shows that $\overline{K}$ is a cc.

$\overline{K}$ is also pure and satisfies
$\operatorname{Rk}(\overline{K})=\operatorname{Rk}(K)$ for the following
reasons: the maximal cells of $\overline{K}$ of rank $R$ are the images of
vertices of $K$ under the duality map and since every cell of $K$ contains at
least one vertex, we have that every cell of $\overline{K}$ is contained in
$\overline{v}$ for some $v\in K.$ Since $K$ is graph-based, Lemma 1.10 implies
that every sub-maximal cell in $\overline{K}$ is contained in exactly two
maximal cells, hence $\overline{K}$ is non-singular and has empty boundary.

By Lemma 1.10, the map $x\mapsto\overline{\overline{x}}$ is a poset
isomorphism from $K$ to $\overline{\overline{K}},$ which defines a cc-
isomorphism by Lemma 1.6 since
$\operatorname{rk}_{\overline{\overline{K}}}(\overline{\overline{x}})=\operatorname{rk}_{K}(x).$
∎

Let us present a few examples of dual cc.

######  Example 1.12.

We first look at the duals of simplicial complexes and also discuss the case
of the square lattice.

  1. 1)

Simple cc: A closed cc $K$ is said to be simple if $\overline{K}$ is
simplicial. In dimension two, a closed cc is simple if and only if every
vertex is included in 3 edges, which by Axiom iii) implies that every pair of
edges containing a vertex is contained in a 2-cell. One can show that the
boundary of a tetrahedra (i.e. a simplicial cc of rank 3 with one maximal
cell) is the only example of a cc which is both simple and simplicial, and it
is also self-dual.

  2. 2)

Square lattice: Consider a portion of the (face lattice of) the square lattice
$\mathbb{Z}^{2}$ of size $4\times 5,$ seen as a cc $L$ of rank 2 with 2-cells
corresponding to the 12 unit squares it contains. We can turn $L$ into a
closed cc $K$ by identifying opposite parts of the boundary and obtain the cc
pictured in Figure 5. This is also an example of self-dual cc, that is we have
$\overline{K}\cong K.$

Figure 5:  The cc $K$ illustrated here can be seen as the boundary of a 3-cc
having one 3-cell with the shape of a torus.

We close this part by briefly mentioning connections to lattices and quantum
logic. An important consequence of Axiom iii) is that the smallest upper bound
or join $x\vee y$ of two cells $x,y$ in a cc $K$ such that there exists a cell
containing both $x$ and $y$ exists: if both $w$ and $w^{\prime}$ are minimal
elements in $\mathbf{A}(x\cup y)$ then $w\cap w^{\prime}\in\mathbf{A}(x\cup
y)$ and cannot be strictly below $w$ and $w^{\prime}$ hence $w=w^{\prime}.$ We
then have that a maximal cell of $K$ contains $x\vee y$ if and only if it
contains both $x$ and $y.$ Applying a similar reasoning for the case $x\cap
y\neq\emptyset$ we get the identities

$\overline{x}\cap\overline{y}=\overline{x\vee
y},\quad\overline{x}\vee\overline{y}=\overline{x\cap y}.$ (2)

If no cell contains both $x$ and $y,$ then, equivalently, no maximal cell
contains the two cells and therefore $\overline{x}\cap\overline{y}=\emptyset.$
It is thus convenient to introduce a symbol $\overline{\emptyset}$ such that
$x\subset\overline{\emptyset}$ for all $x\in K,$ and

$x\vee
y=\overline{\emptyset}\quad\Leftrightarrow\quad\overline{x}\cap\overline{y}=\emptyset.$

As a result, the poset
$L:=(K\cup\\{\emptyset,\overline{\emptyset}\\},\subset)$ has the structure of
a (ranked) lattice, where $\cap=\wedge$ defines the meet operation and $\vee$
defines the join operation. The dual lattice then corresponds to
$\overline{L}:=(\overline{K}\cup\\{\emptyset,\overline{\emptyset}\\})$ in
which $\emptyset$ and $\overline{\emptyset}$ respectively contains and is
contained in every element and in which the relations (2) hold.

The relation between the lattice structure and formal logic is made by seeing
the inclusion order as the partial order induced by implications between
propositions and where the $\wedge$ corresponds to the conjunction ”AND” and
$\vee$ to the disjunction ”OR”. The relations (2) are then comparable to
DeMorgan’s laws, where the duality operation plays the role of a negation
operation but with an important twist: the meet/AND and join/OR operations are
not defined between a cell in $L$ and a cell in $\overline{L}.$

A similar lattice structure interpreted as a formal logic is at the base of
quantum logic ([BvN36]). Although interestingly, the lattice of quantum logic
does not emerge as a structure associated to the geometry of space-time, but
merely from the relations among propositions associated to a physical system
that does not depend upon the state of the system, in other words, its
kinematic structure (see [Jau68]). In summary, propositions are identified
with subspaces of a Hilbert space and taking the orthogonal complement of a
sub-space is analogous to the negation operation. This yields a lattice
structure with a idempotent automorphism called an orthocomplemented lattice.

The lack of distributivity of $\vee$ over $\wedge$ and vice-versa (and also
its non-modularity) in such general lattices leads to difficulties in truly
interpreting it as describing a logical structure, unlike the case of Boolean
lattices. As mentioned in [CG01], an other criticism comes from the fact that
closeness under intersection (or conjunction) of the lattice of quantum logic
is too strong in that it implies non-physical statements when seen as
propositions associated to a quantum system. The structure of primal and dual
lattices we see in our case could precisely represent a weakening of the
structure of orthocomplemented lattice preventing such incoherences. Seeing
the negation or complement operation as mapping to a different space for
example corroborates a more recent discussion on the relations between
physics, topology, logic and computations [BS10] where, when formulated in the
language of category, the negation is seen as a functor to the opposite
category.

On a more conceptual level, the latter discussion would suggest to see the
relation between primal and dual lattices from our formalism as akin to the
relation between conjugated variables in quantum mechanics, where the Born
duality is defined via a Fourier transform which switches between
complementary representations of a quantum system. This famously yields the
Heisenberg Uncertainty Principle, implying for example that both the position
and momentum representations cannot simultaneously and exactly describe a
quantum system.

###  1.4 Midsections and pinches

In the previous parts we saw several examples of cc constructed from the cells
of an other cc: the restriction of Example 1.3 3), the boundary of Example 1.3
4) and the duality map. We will now see yet an other cc, called midsection,
constructed from the cells intersecting but not included in a given cell or a
given sub-complex of a cc. Midsections are used widely in the next sections
where it will become apparent that it is generalization of an existing notion
of midsection used in the context of CDT (see e.g. [DJ15]).

Let us start by introducing collars, named after an notion used for example in
the context of polyhedra (c.f. [RS72]), but defined here purely in terms of
finite sets.

######  Definition 1.13 (Collar).

Let $A$ be a finite set and $\mathcal{A}$ be a collection of subsets of $A.$
If $B\subset A$ we define the collar of $B$ in $\mathcal{A},$ by

$\mathcal{A}_{B}:=\\{x\in\mathcal{A}\leavevmode\nobreak\ |\leavevmode\nobreak\
\emptyset\neq x\cap B\neq x\\}.$

Moreover, if $C\in\mathcal{A},$ we use the following definition and notation
for the collar of $B$ in $\mathcal{A}$ restricted to $C:$

$\mathcal{A}_{B}^{C}:=\\{x\in\mathcal{A}_{B}\leavevmode\nobreak\
|\leavevmode\nobreak\ x\subset C\\}.$

By abuse of notation, in case $\mathcal{B}$ is also a collection of subsets of
$A,$ the notation $\mathcal{A}_{\mathcal{B}}$ is used to denote
$\mathcal{A}_{B},$ where $B:=\cup_{x\in\mathcal{B}}x$ is the set of elements
of $A$ contained in a set in $\mathcal{B}.$

For instance if $K\in\mathsf{C}$ and $J\leq K$ we have

$K_{J}=\\{x\in K\leavevmode\nobreak\ |\leavevmode\nobreak\ \emptyset\neq x\cap
J^{[0]}\neq x\\}.$

Since we use the notation $\mathcal{E}$ for the set of edges in $K$ when there
is no ambiguity on what the cell complex $K$ is, we often use the following
notation for $x\in K:$

$\mathcal{E}_{J}^{x}=\\{e\in K^{[1]}\leavevmode\nobreak\ |\leavevmode\nobreak\
\emptyset\neq e\cap J^{[0]}\neq e,\quad e\subset x\\}.$

The key fact that allows us to construct cells from the collar of a given
subset of cells is the following (Lemma A.4): if $K$ be a local cell complex
and $J\leq K$ then the map

$x\longmapsto\mathcal{E}_{J}^{x},\quad x\in K_{J}$

is an injective poset homomorphism such that the inverse is a poset
homomorphism. The latter indeed ensures that the midsection introduced in the
following definition is a ranked poset.

######  Definition 1.14 (Midsection).

Let $K$ be a local cell complex and $J\leq K.$ We define the midsection of $J$
in $K$ to be $(M_{J}^{K},\operatorname{rk}_{M_{J}^{K}})$ where

$M_{J}^{K}:=\\{\mathcal{E}_{J}^{x}\leavevmode\nobreak\ |\leavevmode\nobreak\
x\in
K_{J}\\}\quad\text{and}\quad\operatorname{rk}_{M_{J}^{K}}(\mathcal{E}_{J}^{x})=\operatorname{rk}_{K}(x)-1.$

An example of midsection is illustrated in Figure 6, and the next proposition
shows that the midsection associated with a sub-complex is a cc under rather
general assumptions.

Figure 6:  Midsection $M_{J}^{K}$ associated to the boundary part $J$ of a
3-cc $K$ composed of three maximal cells (the center one having dots on its
vertices).

######  Proposition 1.15.

Let $K$ be a local cell complex and $J\lneq K$ such that $x\not\subset
J^{[0]}$ for all $x\in K\setminus J$ and such that $K_{J}\neq\emptyset.$ Then
$(M_{J}^{K},\operatorname{rk}_{M_{J}^{K}})$ is a cc and if $K$ is pure then so
is $M_{J}^{K}$ and $\operatorname{Rk}(M_{J}^{K})=\operatorname{Rk}(K)-1.$
Moreover if $J\cap x$ is connected for all $x\in K^{[2]}_{J}$ then $M_{J}^{K}$
is graph-based.

###### Proof.

Since $K$ is cell-connected, $J\neq K,$ and by the condition $x\not\subset
J^{[0]}$ for all $x\in K\setminus J,$ there is at least one edge of $K$ having
one vertex in $J^{[0]}$ and one vertex in $K^{[0]}\setminus J^{[0]},$
therefore $K_{J}\neq\emptyset.$

By definition of $\mathcal{E}_{J}^{x}$ and $\operatorname{rk}_{M_{J}^{K}},$
the elements in $M_{J}^{K}$ of rank $0,$ i.e. vertices, are indeed 1-element
sets and every set $\mathcal{E}_{J}^{x}$ is indeed a set of vertices of
$M_{J}^{K}.$ By the fact that $x,y\in K,$ if $x\in K_{J}$ and $x\subset y$
then $y\in K_{J},$ the axioms of cc for $M_{J}^{K}$ follow as a direct
consequence of Lemma A.4 and the axioms for $K.$

Let $R=\operatorname{Rk}(K).$ By purity of $K,$ every cell in $x\in K_{J}$ is
contained in a $R$-cell of $K,$ which must then belong to $K_{J}$ and
constitute an $R-1$ cell in $M_{J}^{K}$ containing $\mathcal{E}_{J}^{x},$
showing that $M_{J}^{K}$ is also pure.

As mentioned when introducing local cc, we have that $\mathbf{B}(x)$ is a
cycle for every 2-cell $x$ of $K.$ Therefore, for every edge
$\mathcal{E}^{x}_{J}$ of $M_{J}^{K},$ i.e. for every $x\in K^{[2]}_{J},$ if
$\mathbf{B}(x)$ is connected then there are exactly two edges in $K^{[1]}_{J}$
included in $\mathbf{B}(x),$ and therefore there are exactly two vertices of
$M_{J}^{K}$ in $\mathcal{E}_{J}^{x}.$ ∎

From the previous result, one has that the midsection $M_{\mathbf{B}(x)}^{K}$
associated with the restriction of a local cc $K$ to a non-maximal cell $x$ is
a graph based cc, that we call the local figure of $x$ in $K$ (in analogy with
the notion of vertex figure defined in the context of polytopes). Indeed the
condition that every cell in $K\setminus\mathbf{B}(x)$ cannot have all its
vertices in $x$ is trivially verified and the condition that
$\mathbf{B}(x)\cap y$ is connected for all $y\in K^{[2]}$ is a direct
consequence of Axiom iii).

Let us observe that in a non-singular cc $K$ the cells in $K\setminus\partial
K$ satisfy that every non-maximal cell has 2 or more co-faces. We can
therefore use the same arguments as in the proof of Lemma 1.10 to show that
the duality map is bijective on $K\setminus\partial K$ and see that
$(\overline{K\setminus\partial
K},\operatorname{rk}_{\overline{K}}|_{\overline{K\setminus\partial K}})$
defines a ranked poset. The non-singularity of $K$ also guarantees that the
1-skeleton of $\overline{K\setminus\partial K}$ is a graph. We can thus
introduce the following notion, dual to that of connectedness of a cc.

######  Definition 1.16.

A cc $K$ is strongly connected if $\overline{K\setminus\partial K}$ is
connected.

And the following notion of pinch allows to define the notion dual to that of
cell-connectedness.

######  Definition 1.17 (Pinch and non-pinching cc).

A pinch, or $K$-pinch, in a local cc $K$ is a cell $x\in K$ whose local figure
is not strongly-connected. $K$ is said to be non-pinching if it contains no
$K$-pinch and no $\partial K$-pinch.

Figure 7 illustrates a simple example of a pinch. As explained in the caption,
this illustration can also help understand the reason for adding in our
definition of a non-pinching cc $K$ the condition that $K$ also do not contain
any $\partial K$-pinch.

Figure 7:  There are three different ways to look at this illustration. First,
one can see it as a portion of a closed 2-cell complex $K$ and the central
vertex $v$ gives an example of a $K$-pinch. The dual local figure
$\overline{M_{\mathbf{B}(v)}^{K}}$ (which is a closed cc in this example) is
then constituted of two disconnected simplices of rank 2 (i.e. triangles)
whose vertices correspond to the 2-cells of $K$ containing $v.$ Second, one
could look at this illustration as a portion of a non-singular 3-cell complex
$L,$ in which $v$ is included in two simplices of rank 3 (i.e. tetrahedra)
considered as two different 3-cells of $L.$ In this case $v$ is both a
$L$-pinch and a $\partial L$-pinch. The third way to look at this illustration
is to consider that the two tetrahedra of $L$ containing $v$ form a single
cell of a non-singular 3-cell complex $L^{\prime}.$ In this case $v$ is a
$\partial L^{\prime}$-pinch, but not a $L^{\prime}$-pinch.

The next remark explains why the notion of non-pinching cc constitutes a class
of cc well suited for our purposes.

######  Remark 1.18.

Let us note the two following facts about non-pinching cc.

  1. 1)

It follows from the observation that a maximal cell belongs to the local
figure of all its vertices that if a closed cc is connected and non-pinching
then it is strongly connected.

  2. 2)

One can show (Lemma A.2) that if $K$ is a non-pinching cc then $\partial K$ is
closed. This means in particular that in this case $\partial(\partial
K)=\emptyset.$

Let us finally introduce sets of cc we often use in the remaining of this
work, one reason being, as mentioned afterwards, that the duality we showed in
the previous part naturally restricts to this class of cc.

######  Definition 1.19 ($\mathsf{C}$ and $\mathsf{B}$).

We define $\mathsf{C}$ to be the set of non-singular non-pinching local (hence
strongly connected by Remark 1.18) cc and $\mathsf{C}^{R}$ for the set of
elements in $\mathsf{C}$ of rank $R.$

We define $\mathsf{B}$ to be the set of closed elements in $\mathsf{C}$ and
similarly $\mathsf{B}^{R}$ is the set of elements in $\mathsf{B}$ of rank $R.$

We use the notation $\mathsf{B}_{\sqcup}$ (respectively
$\mathsf{B}^{R}_{\sqcup}$) to denote the set of (possibly empty) disjoint
unions of elements in $\mathsf{B}$ (respectively $\mathsf{B}^{R}$).

As a consequence, the boundary of an element in $\mathsf{C}^{R}$ is in
$\mathsf{B}_{\sqcup}^{R-1},$ and conversely one can show (Lemma A.3) that a
closed $(R-1)$-sub-cc of the boundary of an element in $\mathsf{C}^{R}$ is a
dijoint union of connected component of its boundary. We finish with the
following corollary:

######  Corollary 1.20.

If $K\in\mathsf{B}^{R}$ then so is $\overline{K}.$

###### Proof.

If $x\in K$ then the 1-skeleton of the cc
$\mathbf{B}_{\overline{K}}(\overline{x})$ is connected if and only if the
local figure of $x$ is strongly connected, hence a closed cc is non-pinching
if and only if $\overline{K}$ is cell connected. By remark 1.18 $K$ is
strongly connected, hence $\overline{K}$ is connected. ∎

###  1.5 Subdividing and collapsing closed cc

The goal of this part is to give meaning to the operation of subdividing a cc,
i.e. a purely combinatorial structure not defined from a discretization of a
continuous space like a manifold where the topology of the latter space put
constraints on what a valid subdivision can be. We start by giving an abstract
definition of barycentric subdivision to give a foretaste of how such an
operation on purely combinatorial structure can be defined. The more general
definition of a subdivision of a closed cc is given in part 1.5.2 via a map
called reduction. The map dual to that of a reduction will be introduced as a
collapse and these two maps are in fact cc-homomorphisms that have just the
right properties to describe the structure near the boundary of a large class
of cc, called uniform cc, as shown in Section 2.

####  1.5.1 Barycentric subdivision and dual cells

Let us start by recalling that a totally ordered subset of a poset $P$ is a
sub-poset $T$ of $P$ such that for all $x,y\in T,$ either $x\leq y$ or $y\leq
x.$ The barycentric subdivision of a poset (and by extension of a any cc),
also called order poset in some contexts, is then simply defined as follows.

######  Definition 1.21 (Barycentric subdivision).

Let $(P,\subset)$ be a poset. The barycentric subdivision of $P$ is the poset,
noted $\widecheck{P},$ of totally ordered subsets of $P.$

We give simple examples of cells in the barycentric subdivision of a 2-cc in
Figure 8. For the case of a simplicial complex obtained from a triangulations
of a manifold, this definition corresponds to the face lattice of the (first
derived) barycentric subdivision of the triangulation as defined for example
in [Bry02]. The barycentric subdivision of a cc is a simplicial complex (and
therefore also a cc) a subset of a totally ordered set is also totally
ordered.

As a direct consequence of Lemma 1.10, the barycentric subdivision is
isomorphic to the dual barycentric subdivision via the cc-isomorphism

$\widecheck{K}\ni\\{x_{1}\subsetneq\dots\subsetneq
x_{k}\\}\longmapsto\\{\overline{x_{k}}\subsetneq\dots\subsetneq\overline{x_{1}}\\}\in\widecheck{\overline{K}}.$
(3)

Figure 8: This illustration represents certain cells of the barycentric
subdivision $\widecheck{K}$ of a 2-cc $K$ and gives three examples of cells in
$\overline{K}^{B}.$ Being a simplicial complex, each 2-cell of the barycentric
subdivision is a triangle and the vertices of the barycentric subdivision are
represented using three different symbols depending of the rank of the cells
it represent: a vertex of $\widecheck{K}$ of the form $\\{C\\}$ where $C\in
K^{[2]}$ is represented by a cross, a vertex of the form $\\{e^{\prime}\\}$
where $e^{\prime}\in K^{[1]}$ is represented by a dash and a vertex of the
form $\\{v\\}$ simply corresponds to a vertex of $K.$ There are two 2-cells of
$\overline{K}^{B},$ the first $\overline{v}^{B}$ is filled the black wavy
pattern, the other $\overline{w}^{B}$ is filled with the white wavy pattern.
We also have the 1-cell $\overline{e}^{B},$ the dual of $e=\\{v,w\\},$
represented by the two thick dashed lines. As a comparison, the dual cell
$\overline{v}$ and $\overline{w}$ used in our framework are illustrated in
Figure 2.

The barycentric subdivision is sometimes used to define a notion of dual cell,
as for example in [Bry02]. This definition can also be defined in a purely
combinatorial manner, but it corresponds to different cells than those from
Definition 1.7 we chose to use here, as it leads to the involution property
from Proposition 1.11. Let us therefore explain how this other notion of dual
cell is defined, and why it is less suited for our framework. Let $K$ be a
closed cc. Consider the sets $\mathbf{A}_{L}(w),\mathbf{B}_{L}(w)$ in the
cases where $L=\widecheck{K}$ and $w=\\{x\\}\in\widecheck{K}^{[0]}$ and the
case $L=\widecheck{\overline{K}}$ and
$w=\\{\overline{x}\\}\in\widecheck{\overline{K}}^{[0]}$ respectively. These
sets can be more explicitly expressed as

$\mathbf{A}_{\widecheck{K}}(\\{x\\})=\\{\\{y_{1}\subsetneq\dots\subsetneq
y_{k}\\}\in\widecheck{K}\leavevmode\nobreak\ |\leavevmode\nobreak\ x\subset
y_{1}\\},$

and

$\mathbf{B}_{\widecheck{\overline{K}}}(\\{\overline{x}\\})=\\{\\{\overline{y_{1}}\subsetneq\dots\subsetneq\overline{y_{k}}\\}\in\widecheck{\overline{K}}\leavevmode\nobreak\
|\leavevmode\nobreak\ \overline{y_{k}}\subset\overline{x}\\}.$

Using the map (3) we get that $\mathbf{A}_{\widecheck{K}}(\\{x\\})$ and
$\mathbf{B}_{\widecheck{\overline{K}}}(\\{\overline{x}\\})$ are isomorphic
ranked posets, hence

$\mathbf{A}_{\widecheck{K}}(\\{x\\})\cong\mathbf{B}_{\widecheck{\overline{K}}}(\\{\overline{x}\\})=\widecheck{\mathbf{B}_{\overline{K}}(\overline{x})}.$

For the case of a simplicial complex $K,$ the map
$K\longrightarrow\mathcal{P}\left(\widecheck{K}\right)$ defined by

$x\longmapsto\overline{x}^{B}:=\widecheck{\mathbf{B}_{\overline{K}}(\overline{x})}\cong\\{\\{y_{1}\subsetneq\dots\subsetneq
y_{k}\\}\in\widecheck{K}\leavevmode\nobreak\ |\leavevmode\nobreak\ x\subset
y_{1}\\}$

is equivalent to the definition of dual cell used in [Bry02] (hence our choice
for the letter $B$ in the notation $\overline{x}^{B}$). The dual cell
$\overline{x}^{B}$ is therefore a sub-simplicial complex of $\widecheck{K}.$

We can then define the poset
$\overline{K}^{B}:=\\{\overline{x}^{B}\leavevmode\nobreak\
|\leavevmode\nobreak\ x\in K\\}$ with rank function

$\operatorname{rk}_{\overline{K}^{B}}(\overline{x}^{B}):=\operatorname{Rk}(\widecheck{\mathbf{B}_{\overline{K}}(\overline{x})})=\operatorname{rk}_{\overline{K}}(\overline{x}).$

It is possible to show that the map $\phi(\overline{x}^{B}):=\overline{x}$
defines an isomorphism of posets from $\overline{K}^{B}$ to $\overline{K}$
such that
$\operatorname{rk}_{\overline{K}}(\phi(\overline{x}^{B}))=\operatorname{rk}_{\overline{K}^{B}}(\overline{x}^{B}).$
Nevertheless $\overline{K}^{B}$ is not a cc since the cells in
$\overline{K}^{B}$ are not subsets of $(\overline{K}^{B})^{[0]}$ as these
cells also contain elements in
$\widecheck{K}^{[0]}\setminus(\overline{K}^{B})^{[0]}.$

####  1.5.2 Reductions and collapses

In order to define a generalization of the notion of barycentric subdivision
including cc which are not simplicial complexes, we use a map
$\rho:J\longrightarrow K$ satisfying certain conditions allowing to interpret
$J$ as a subdivision of $K.$ For simplification, we only define $\rho$ in the
case where $J$ and $K$ are closed cc but it can be generalized to arbitrary
cc, see [Sav21] for a more general treatment. The meaning of each condition in
the following definition of reduction is illustrated in Figure 9.

######  Definition 1.22 (Reduction and subdivision).

Let $J,K,$ be closed cc. We say that a map $\rho:J\longrightarrow K$ is a
reduction if $\rho$ is a surjective poset homomorphism satisfying the
following conditions.

  1. (r1)

$\left|\rho^{-1}(v)\right|=1,$ for all $v\in K^{[0]}$ (i.e. $\rho|^{K^{[0]}}$
is injective).

  2. (r2)

$\mathbf{A}(\rho(x))^{[r]}\subset\rho(\mathbf{A}(x)^{[r]}),$ for all $x\in
J,\leavevmode\nobreak\ r\geq 0;$

  3. (r3)

if $x\in J$ satisfies
$\operatorname{rk}_{J}(x)=\operatorname{rk}_{K}(\rho(x))-1$ then
$\left|\bigtriangledown\left(x\right)\cap\rho^{-1}(\rho(x))\right|=2;$

  4. (r4)

if $x\in J$ satisfies
$\operatorname{rk}_{J}(x)=\operatorname{rk}_{K}(\rho(x))$ then

$\left|\bigtriangledown\left(x\right)\cap\rho^{-1}(y)\right|=1,\quad\forall
y\in\bigtriangledown\left(\rho(x)\right).$

We shall use the notation $J\mathrel{\raisebox{1.0pt}{$\prec$}}_{\rho}K$ if
$\rho:J\longrightarrow K$ is a reduction and call $J$ a subdivision of $K$ or
equivalently say that $K$ is a reduction of $J,$ written
$J\mathrel{\raisebox{1.0pt}{$\prec$}}K,$ if there exists a reduction
$\rho:J\longrightarrow K.$ A cell $y\in K$ is called subdivided by $\rho$ if
$\left|\rho^{-1}(y)\right|>1.$ For example, a subdivision does not subdivide
vertices.

Figure 9:  In this picture, we illustrate what undesirable features the
conditions (r1), (r2), (r3) and (r4) of a reduction are preventing, in the
case of a reduction $\rho:J\longrightarrow K$ between two 1-dimensional cell
complexes. The vertical arrows indicate which cells in $J$ are mapped to each
cell in $K,$ for example all the cells contained in the grey areas are mapped
to an edge in $K.$ More explicitly, condition (r1) prevents more than two
vertices to be mapped to the same vertex. Condition (r2) ensures that the
vertex $v\in J$ is contained in an edge in the pre-image of the edge $e\in K$
since $\rho(v)\in e.$ Condition (r3) does not allow branchings or boundaries
in the pre-image of a cell and condition (r4) with $x=w\in J^{[0]}$ ensures in
particular that the type of branching on $w$ shown on the figure does not
occur or that a cell is not subdivided multiple times.

If $\phi:J\longrightarrow K$ is a map between closed cc, then one can define
the dual of $\phi$ to be the map

$\displaystyle\overline{\phi}:\overline{J}$
$\displaystyle\longrightarrow\overline{K}$ $\displaystyle\overline{x}$
$\displaystyle\longmapsto\overline{\phi}(\overline{x}):=\overline{\phi(x)},$
(4)

well defined thanks to Lemma 1.10.

By simply checking that each condition in the following definition of collapse
is indeed the statement dual to the corresponding condition for reductions,
one can see that collapses are indeed duals of reductions.

######  Definition 1.23 (Collapse).

Let $J,K$ be closed cc. We say that a map $\pi:J\longrightarrow K$ is a
collapse it is a surjective poset homomorphism and

  1. (c1)

$\left|\pi^{-1}(z)\right|=1,$ for all maximal cell $z\in K;$

  2. (c2)

$\mathbf{B}(\pi(x))^{[r]}\subset\pi(\mathbf{B}(x)^{[r]})$ for all $x\in
J,\leavevmode\nobreak\ r\geq 0;$

  3. (c3)

if $x\in J$ satisfies
$\operatorname{rk}_{J}(x)=\operatorname{rk}_{K}(\pi(x))+1$ then
$\left|\triangle\left(x\right)\cap\pi^{-1}(\pi(x))\right|=2;$

  4. (c4)

if $x\in J$ satisfies $\operatorname{rk}_{J}(x)=\operatorname{rk}_{K}(\pi(x))$
then

$\left|\triangle\left(x\right)\cap\pi^{-1}(y)\right|=1,\quad\forall
y\in\triangle\left(\pi(x)\right).$

We use the notation $J\vdash_{\pi}K$ if $\pi:J\longrightarrow K$ is a collapse
and say that  $J$ is an expansion of $K$ or equivalently that $K$ is a
collapse of $J,$ written $J\vdash K,$ if there exists a collapse
$\pi:J\longrightarrow K.$

Figure 10 includes a simple example of reduction and collapse, and show that a
cc can be both a subdivision and an expansion of a given cc. Also,
$\widecheck{K}\mathrel{\raisebox{1.0pt}{$\prec$}}K.,$ i.e. the barycentric
subdivision is indeed an example of subdivision (Lemma A.5).

Figure 10:  Example of cell complexes $J$ and $K$ such that
$J\mathrel{\raisebox{1.0pt}{$\prec$}}K$ and $J\vdash K.$

An important property of reductions of collapses which supports their relation
with the notion of subdivision is the following result.

######  Proposition 1.24.

$\mathrel{\raisebox{1.0pt}{$\prec$}}$ and $\vdash$ are partial orders on the
set of closed cc, up to cc-isomorphism.

###### Proof.

In Appendix B. ∎

We close this part with two remarks that will be referred to later on,
pointing particular at the fact that reductions and collapses are cc-
homomorphisms between cc of same rank.

######  Remark 1.25.

A consequence of condition (r2) is that
$\operatorname{rk}_{K}(\rho(x))\geq\operatorname{rk}_{J}(x),$ for a reduction
$\rho:J\longrightarrow K$ and (c2) implies that
$\operatorname{rk}_{K}(\pi(x))\leq\operatorname{rk}_{J}(x)$ for a collapse
$\pi:J\longrightarrow K.$ This can be seen for example for the case of a
reduction $\rho$ by considering the instance
$r=\operatorname{rk}_{K}(\rho(x))$ in condition (r2) according to which

$\\{\rho(x)\\}=\mathbf{A}(\rho(x))^{[r]}\subset\rho(\mathbf{A}(x)^{[r]}).$

This implies $\mathbf{A}(x)^{[r]}\neq\emptyset$ and therefore
$\operatorname{rk}_{J}(x)\leq r=\operatorname{rk}_{K}(\rho(x))$ for all $x\in
J.$ The case of a collapse is similar. This argument also shows that
conditions (r2) and (c2) in particular imply that reductions and collapses are
cc-homomorphisms.

######  Remark 1.26.

If $J\mathrel{\raisebox{1.0pt}{$\prec$}}K$ or $J\vdash K$ then
$\operatorname{Rk}(J)=\operatorname{Rk}(K).$ Indeed, for the case
$J\mathrel{\raisebox{1.0pt}{$\prec$}}K,$ $\rho=\rho_{J}^{K}$ is in particular
a cc-homomorphism hence there exists a $\operatorname{Rk}(K)$-cell in
$\rho^{-1}(z),$ for all $z\in K^{[\operatorname{Rk}(K)]}$ and therefore
$\operatorname{Rk}(J)\geq\operatorname{Rk}(K).$ Remark 1.25 also implies
$\operatorname{rk}_{J}(x)\leq\operatorname{rk}_{K}(\rho(x))$ for all $x\in J$
and we get $\operatorname{Rk}(J)=\operatorname{Rk}(K).$

##  2 Structure near the boundary and Correspondence Theorem

This section focuses on the study of the relation between the cells in the
boundary of a cc and the cells in their collar, i.e. linking the boundary to
the associated midsection. This relation will be essential for Section 3 where
we discuss the duality of cells on the boundary, leading to the definition of
cobordism, and give sufficient conditions for the union of two cobordisms
sharing a boundary component to form a cell complex.

It is therefore sometimes convenient to use relative cc, i.e. couples $(K,J)$
where $J\leq K,$ to formulate the notions needed for our purposes in this and
the next sections, which are discussed to a large extend in part 2.1.

Part 2.2 defines a poset called the transition made of certains cells arising
”between” the boundary and the midsection. This poset is then used to
formulate the assumption of uniformity, under which the transition is a cc. We
finish this part with an important result of this section involving uniformity
(Proposition 2.10) and making explicit the relation between a boundary
component and the associated midsection and transition via a reduction and a
collapse.

It is shown in part 2.3 that the reduction and collapse from Proposition 2.10
satisfy additional conditions that lead us to the definition of geometric
sequence. We finish in part 2.4 by introducing slices and showing the main
theorem of this work (Theorem 2.22) establishing a bijection between slices
and some geometric sequences. This result is what allows to define the
composition of cobordisms introduced in the last section and slices are a
basic building block in the definition of causal cobordisms.

###  2.1 Relative cc

A relative cc is simply one for which we single out a sub-cc and it will be
mostly used in the case where the sub-cc is a connected component of the
boundary. We will also need a relative notion of homomorphism defined as
follows.

######  Definition 2.1 (Relative cc and homomorphism).

A couple $(K,J)$ of cc is called a relative cc if $J\leq K.$ A relative cc-
homomorphism $\phi:(K,J)\longrightarrow(K^{\prime},L)$ (or a homomorphism of
$J$ relative to $K$) is a cc-homomorphism $\phi:K\longrightarrow K^{\prime}$
such that $\phi(J)=L$ and $\phi|_{K\setminus J}:K\setminus J\longrightarrow
K^{\prime}\setminus L$ is a poset isomorphism.

Typically a relative cc-homomorphism is used to define a map on a boundary
component of a cc $K$ such that its image can still be seen as a sub-cc of
$K,$ and we will primarily use the two following types of relative cc-
homomorphisms.

######  Definition 2.2 (Relative reduction and collapse).

Let $(K,J),(H,L)$ be relative cc. A  relative reduction (respectively a
relative collapse) or a reduction (respectively collapse) of $J$ relative to
$K$ is a cc-homomorphism $\phi:(K,J)\longrightarrow(H,L)$ of $J$ relative to
$K$ such that $\phi:K\longrightarrow H$ is a reduction (respectively a
collapse). We will also write $(K,J)\mathrel{\raisebox{1.0pt}{$\prec$}}(H,L)$
and say that $(K,J)$ is a subdivision of $L$ relative to $H$ if there exists a
reduction $\rho:(K,J)\longrightarrow(H,L)$ of $J$ relative to $K.$ Similarly
we will write $(K,J)\vdash(H,L)$ and say that $(K,J)$ is an expansion of $L$
relative to $H$ if there exists a collapse $\pi:(K,J)\longrightarrow(H,L)$ of
$J$ relative to $K.$

Clearly Definition 2.2 implies in particular that a relative cc-homomorphism
$\phi:(K,J)\longrightarrow(H,L)$ is a relative reduction (respectively a
relative collapse) if and only if $\phi|_{J}:J\longrightarrow L$ is a
reduction (respectively a collapse). By abuse of notation, we will sometimes
also use the notation $\phi$ to designate $\phi|_{J}$ when
$\phi:(K,J)\longrightarrow(H,L)$ is a relative homomorphism.

The property that a relative cc-homomorphism
$\phi:(K,J)\longrightarrow(K^{\prime},L)$ is a poset isomorphism when
restricted to $K\setminus J$ reveals the fact that most of the emphasis on a
relative cc $(K,J)$ is put on the sub-cc $J.$ Hence $K$ can be merely
considered as a ”background” in which $J$ is included, and this enables the
following definition that will be convenient towards the end of this section.

######  Definition 2.3 ($K_{\phi}$ and $K^{\phi}$).

Let $\phi:J\longrightarrow L$ be a surjective cc-homomorphism and let $H$ be a
cc such that $L\leq H.$ If there exists a cc $K\geq J$ and a relative cc-
homomorphism $\phi^{H}:(K,J)\longrightarrow(H,L)$ such that
$\phi^{H}|_{J}=\phi$ then $K$ is uniquely determined by $H$ and
$\phi:J\longrightarrow L$ up to cc-isomorphism. We will therefore use the
notation $K=:H^{\phi}.$

Similarly, let $\phi:J\longrightarrow L$ be a cc-homomorphism and let $K$ be a
cc such that $J\leq K.$ If there exists a cc $H\geq L$ and a relative cc-
homomorphism $\phi_{K}:(K,J)\longrightarrow(H,L)$ such that
$\phi_{K}|_{J}=\phi$ then $H$ is uniquely determined by $K$ and
$\phi:J\longrightarrow L$ up to cc-isomorphism. We will therefore use the
notation $H=:K_{\phi}.$

Let us make the following remark about the previous definitions considering
the example of a reduction $\rho:J\longrightarrow L.$ In general it is wrong
that given a relative cc $(H,L)$ there exists a cc $K$ such that
$(K,J)\mathrel{\raisebox{1.0pt}{$\prec$}}_{\rho}(H,L)$ is a relative reduction
or that given a relative cc $(K,J)$ there exists a cc $H$ such that
$(K,J)\mathrel{\raisebox{1.0pt}{$\prec$}}_{\rho}(H,L)$ is a relative
reduction.

One reason that this is wrong for the latter case is that it is possible for
the reduction $\rho$ to reduce two $r$-cells $x_{1},x_{2}$ in $J$ each sharing
a face $y_{1},y_{2}\in J^{[r-1]}$ with a $r$-cell in $y\in K_{J}$ into one
cell $x=\rho(x_{1})=\rho(x_{2}),$ hence $\\{x_{1},x_{2}\\}\subset x\cap y,$
violating the intersection Axiom iii) for $H.$

Similarly, a reason for the first case is that a relative subdivision of
$(H,L)$ can subdivide a cell $x$ of $L$ contained in a cell of $y\in H_{L}$
only if every cell $w$ of $L$ strictly containing $x$ is also subdivided in
such a way that any intersection of an element in $\rho^{-1}(y)$ with an
element in $\rho^{-1}(w)$ contains at most one cell, which is not the case in
general.

As mentioned before, we are particularly interested in relative cc $(K,J)$
where $J$ is a connected component of the boundary of $K$ but, even in this
case, there can be cells in the collar $K_{J}$ having all their vertices in
$J.$ For example all vertices in a cc having a unique maximal cell are on the
boundary. Hence it will be valuable to have the following notion of non-
degenerate relative cc at hand when studying the midsection associated with a
boundary component.

######  Definition 2.4 (Non-degenerate relative cc).

We say that a relative cc $(K,J)$ is degenerate if there exists $x\in
K\setminus J$ such that $x\subset J^{[0]}.$ We say that $(K,J)$ is non-
degenerate if it is not degenerate.

When discussing the duality of cobordisms in the next section, the midsection
associated with a boundary component will play an important role. As we are
interested in preserving a framework where cc are local, one would like to
ensure that the midsection is a local cc. However, even in a local and non-
pinching cc, the midsection associated with a boundary component can be a non-
local cc (such an example can be constructed using a cell as in Figure 7 in
the collar of a boundary component). We thus also need a refined relative
notion of locality as defined next.

######  Definition 2.5 (Local relative cc).

We say that a relative cc $(K,J)$ is local if it is non-degenerate, $K$ is
local, every connected component $J_{0}$ of $J$ satisfies that
$\mathbf{B}_{J_{0}}(x)$ is connected for all $x\in K_{J_{0}}$ and
$M_{J_{0}}^{K}$ is local. In particular $(K,\emptyset)$ is local for any lcc
$K.$

Finally, a particular type of relative cc we will use in the next section is
the following notion of exactly collared relative cc. It will allow us to
obtain a class of cobordisms on which the duality map acts as a involution.

######  Definition 2.6 (Exactly collared relative cc).

Let $(K,J)$ be a non-degenerate cc. We say that $(K,J)$ is exactly collared if
the map from $M_{J}^{K}$ to $J$ defined by $\mathcal{E}_{J}^{x}\mapsto J\cap
x$ is a cc-isomorphism. In particular an exactly collared relative cc $(K,J)$
such that $K$ is local is a local relative cc and $(K,\emptyset)$ is also
exactly collared for all lcc $K.$

###  2.2 Transition and uniformity

Consider for example a cc $K\in\mathsf{C}$ of rank 3 and a boundary component
$J\leq\partial K.$ If we consider a maximal cell $y$ in the collar $K_{J}$ as
illustrated in Figure 11, we notice that the edges in $y$ and in the collar
$K_{J}$ define a subset $w$ of the vertices in $y\cap J^{[0]}$ ($w$ contains
the four vertices linked by thicker lines in the example considered). These
types of subsets of vertices in $J$ is what defines the transition.

######  Definition 2.7 (Transition).

Let $(K,J)$ be a relative cc. For $x\in K_{J},$ we define the reduced cell of
$x$ to be

$x_{J}:=\\{e\cap J^{[0]}\leavevmode\nobreak\ |\leavevmode\nobreak\
e\in\mathcal{E}_{J}^{x}\\}.$

We then define the transition of $(K,J)$ to be the poset

$J(K):=\\{x_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ x\in K_{J}\\}.$

The whole idea of this part will be to identify sufficient assumptions to
prove Proposition 2.10 showing that there is a reduction from $J$ to $J(K)$
and a collapse from the $M_{J}^{K}$ to $J(K).$

Figure 11:  Example of a 3-cell $y$ in the collar of the boundary component
$J$ of a 3-cc.

In order to associate a rank to a cell $w$ of the transition there are two
possibilities: either we consider a maximal cell in $\mathbf{B}_{J}(y)$ (e.g.
$x_{i},\leavevmode\nobreak\ i=1,2,3$) or we take $\operatorname{rk}_{K}(y)-1,$
considering that $y$ is minimal with the property of containing $w.$ In order
for the two choices to agree, we need an additional relative notion of purity
introduced next.

Once the transition admits the structure of a ranked poset, one formulates the
uniformity assumption as enforcing in particular that it is a cc. We will show
that uniformity is sufficient to prove Proposition 2.10.

In this part the two following subsets of the collar $K_{J}$ will be used
repeatedly. First we will use the notation

$K_{J}(A):=\\{y\in K_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ A\subset
y\\}$

where $A$ is simply a subset of $J^{[0]}.$ We also define $K^{-}_{J}(A)$ to be
the set of minimal elements in $K_{J}(A).$ Second, it will also be convenient
to refer to the set

$K_{J}[x]=\\{y\in K_{J}\leavevmode\nobreak\ |\leavevmode\nobreak\ y_{J}=x\\}$

associated to a cell $x\in J(K)$ and denote by $K_{J}^{-}[x]$ the set of
minimal elements in $K_{J}[x].$

######  Definition 2.8 (Pure relative cc and relative rank).

A relative cc $(K,J)$ is said to be pure if $J\cap x$ is pure for all $x\in
K_{J}$ and, dually, for all $x\in J,$ all cells in $K^{-}_{J}(x)$ have equal
rank $r(x).$ It is clear that in this case $r(x)\geq 1$ for all $x\in J$ and
we define $\operatorname{rk}_{J}^{K}(x):=r(x)-1$ to be the relative rank of
$x\in J$ in $K.$

Counterexamples of pure relative cc $(K,J)$ are illustrated in Figures 13 and
13 (using as setup where the cc $K$ and $J$ are similar to those used in
Figure 11).

Figure 12:  An example of a 3-cell $x\in K_{J}$ of a relative cc that does not
satisfy the first condition for pure relative cc since $\mathbf{B}_{J}(x),$
whose edges are represented with thicker lines, is not pure.

Figure 13:  An example of a cell $x\in J$ that does not satisfy the second
condition of purity for the relative cc $(K,J),$ i.e. $x$ is the central
vertex (black dot) and is both included in an edge (thicker line) and a 2-cell
(wavy triangle) in $K_{J}.$ There are five 3-cells in this picture: four
tetrahedra and one 3-cell below the wavy triangle having seven vertices.

Reduced cells are yet an other example of cells derived from the cell
structure of a cc and in order for the transition to satisfy the axioms of cc,
we need to ensure that reduced cells contain ”enough vertices” in some sense.
Indeed if $x\in K_{J}$ then $x_{J}\subset x\cap J^{[0]},$ but in general there
can be vertices in $x\cap J^{[0]}$ not contained in an edge in
$\mathcal{E}_{J}^{x}.$

We therefore add an additional condition to our framework, namely that the
cells of a relative cc $(K,J)$ satisfy what we will refer to as the uniformity
condition:

$J^{[0]}\cap x\subset J^{[0]}\cap y\quad\text{if and only if}\quad
x_{J}\subset y_{J},\quad\forall x,y\in K_{J}.$ (U)

This rather weak additional condition implies in particular (Lemma A.6) that
the two ways to associated a rank to a cell of the transition discussed above
are equivalent. More explicitly if $(K,J)$ be a pure relative cc satisfying
(U) and $x\in J(K)$ then for all $x^{\prime}\in K_{J}^{-}[x]$ and all maximal
cells $w\in\mathbf{B}_{J}(x^{\prime})$ we have

$\operatorname{rk}_{J}^{K}(x):=\operatorname{rk}_{K}(x^{\prime})-1=\operatorname{rk}_{K}(w)$

As a consequence the map

$J(K)\ni x_{J}\longmapsto\operatorname{Rk}(\mathbf{B}_{J}(x))$

is well defined and we have
$\operatorname{Rk}(\mathbf{B}_{J}(x))=\operatorname{rk}_{K}(x)-1$ whenever
$x\in K_{J}^{-}[x_{J}].$ We can therefore associate the rank function

$\operatorname{rk}_{J(K)}(x_{J}):=\operatorname{Rk}(\mathbf{B}_{J}(x))$

to $J(K).$

######  Definition 2.9 (Uniform relative cc).

A relative cc $(K,J)$ is called uniform if the following four conditions are
fulfilled:

  * •

$(K,J)$ non-degenerate,

  * •

$(K,J)$ is pure,

  * •

the uniformity condition (U) holds for $(K,J),$

  * •

$(J(K),\operatorname{rk}_{J(K)})\in\mathsf{B}.$

We will say that a cc $K\in\mathsf{C}$ is uniform if so is $(K,J)$ for all
connected component $J$ of $\partial K.$

One can easily check that these conditions are satisfied if $K\in\mathsf{C}$
has rank two or lower. It is however not clear to us whether the uniformity
condition (U) and the assumption that $J(K)\in\mathsf{B}$ are independent from
the assumptions of purity and non-degeneracy in dimension 3 or higher.

A few more technical manipulations (Lemma A.7) show that if $(K,J)$ is uniform
and $x\in J$ then there exists a unique cell $x_{J}\in J(K)$ such that

$K_{J}^{-}[x_{J}]=K_{J}^{-}(x)\quad\text{and}\quad\operatorname{rk}_{J(K)}(x_{J})=\operatorname{rk}_{J}^{K}(x).$

This provides an generalization of the notion of reduced cell $x_{J}$ for
cells $x\in J.$

We close this part with the following important result, confirming that
uniformity indeed is the assumption that unlocks the ability to characterize a
boundary component of a cc and its neighbourhood using a reduction and a
collapse.

######  Proposition 2.10.

Let $K\in\mathsf{C}^{R+1}$ and $J\leq\partial K,$ $J\in\mathsf{B}^{R}$ such
that $(K,J)$ is uniform and local. Then one has the two following results.

  1. 1)

We have $J\mathrel{\raisebox{1.0pt}{$\prec$}}J(K)$ as the poset homomorphism
$\rho_{J}^{K}:J\longrightarrow J(K)$ defined by

$\rho_{J}^{K}(x)=x_{J}$

is a reduction.

  2. 2)

We have $M_{J}^{K}\vdash J(K)$ as the poset homomorphism
$\pi_{J}^{K}:M_{J}^{K}\longrightarrow J(K)$ defined by

$\pi_{J}^{K}(\mathcal{E}^{x}_{J})=x_{J}$

is a collapse.

###### Proof.

In Appendix C. ∎

###  2.3 Geometric sequences

In the previous part we saw that assuming uniformity allows to relate the
cells in a boundary component $J$ of a cc $K$ with the cells in the
corresponding midsection $M_{J}^{K}$ using the reduction
$\rho_{J}^{K}:J\longrightarrow J(K)$ and the collapse
$\pi_{J}^{K}:M_{J}^{K}\longrightarrow J(K).$ In this part we show that these
to maps satisfy additional conditions inherited from the axioms of the cc $K$
and we present technical results in preparation to the next part. An example
of such condition can be obtained by considering two disjoint cells in the
midsection $M_{J}^{K}$ having identical images through $\pi_{J}^{K}$ in the
transition. These correspond to two cells in the collar intersecting only on
the boundary, therefore the image of these two cells cannot be subdivided by
$\rho_{J}^{K}$ as it would contradict the intersection axiom iii). Similar
scenarios of intersections between cells in the collar and in the boundary
leads to the following notion of compatible cc-homomorphisms, a property that
is shown to hold for the case of $\rho_{J}^{K}$ and $\pi_{J}^{K}$ in the
corollary that comes next.

######  Definition 2.11 (Compatible homomorphisms, $\leftthreetimes$).

Let $j:J\longrightarrow I$ and $l:L\longrightarrow I$ be cc-homomorphisms.
Then we say that $j$ is compatible with $l,$ noted $j\leftthreetimes l,$ if
the three following conditions are satisfied:

  1. 1)

For all $w\in I$ either $\left|j^{-1}(w)\right|=1$ or
$\left|l^{-1}(w)\right|=1.$

  2. 2)

If $x,x^{\prime}\in J$ are such that $x\vee
x^{\prime}\neq\overline{\emptyset},$ then

$j(x\vee x^{\prime})=j(x)\vee j(x^{\prime}).$

  3. 3)

If $y,y^{\prime}\in L$ are such that $y\cap y^{\prime}\neq\emptyset$ then

$l(y\cap y^{\prime})=l(y)\cap l(y^{\prime}).$

Note that if $j$ and $l$ are poset isomorphisms then $j$ is compatible with
$l.$

######  Corollary 2.12.

Let $K\in\mathsf{C}^{R}$ and $J\leq\partial K,$ $J\in\mathsf{B}^{R-1}$ such
that $(K,J)$ is uniform and local then

$J\mathrel{\raisebox{1.0pt}{$\prec$}}_{\rho_{J}^{K}}J(K)\dashv_{\pi_{J}^{K}}M_{J}^{K}\quad\text{and}\quad\rho_{J}^{K}\leftthreetimes\pi_{J}^{K}.$

###### Proof.

The first statement is a direct consequence of Proposition 2.10. As for the
first condition of compatibility, suppose by contradiction that there exists
$x\in J(K)$ such that there are two distinct elements $x_{1},x_{2}\in J$
satisfying $(x_{1})_{J}=(x_{2})_{J}=x,$ as well as two distinct elements
$\mathcal{E}_{J}^{y_{1}},\mathcal{E}_{J}^{y_{2}}\in M_{J}^{K}$ such that
$(y_{1})_{J}=(y_{2})_{J}=x.$ By the conditions (r2), (c2), (r3) and (c3) we
can assume that

$r:=\operatorname{rk}_{J(K)}(x)=\operatorname{rk}_{J}(x_{i})=\operatorname{rk}_{M_{J}^{K}}(\mathcal{E}_{J}^{y_{i}})=\operatorname{rk}_{K}(y_{i})-1,$

for $i=1,2$ and where the two last equalities come from the definition of the
midsection $M_{J}^{K}.$ This implies that $y_{1},y_{2}\in K_{J}^{-}[x]$ and
therefore $y_{1}\cap y_{2}\in J^{[r]}.$ But we have
$\\{x_{1},x_{2}\\}\subset\mathbf{B}_{J}((y_{1}\cap y_{2}))$ so this
contradicts the intersection axiom iii) of $K.$

In order to prove the condition 2) of compatibility, take $x,x^{\prime}\in J$
such that $x\vee x^{\prime}\neq\overline{\emptyset}.$ We need to show that

$\rho_{J}^{K}(x\vee x^{\prime})=(x\vee x^{\prime})_{J}=x_{J}\vee
x^{\prime}_{J}=\rho_{J}^{K}(x)\vee\rho_{J}^{K}(x^{\prime}),$

where the map $J\ni x\mapsto x_{J}\in J(K)$ is defined in Lemma A.7, which
also states that $K_{J}^{-}[x_{J}]=K_{J}^{-}(x)$ and
$\operatorname{rk}(y)-1=\operatorname{rk}(x)$ for all $y\in K_{J}^{-}(x).$

The inclusion $x_{J}\vee x^{\prime}_{J}\subset(x\vee x^{\prime})_{J}$ is clear
(and in particular $x_{J}\vee x^{\prime}_{J}\neq\overline{\emptyset}$) since
if $y\in K_{J}^{-}(x\vee x^{\prime})$ then in particular $x\subset y$ and
$x^{\prime}\subset y$ hence the definition of reduced cells easily leads
$x_{J}\subset y_{J}$ and $x^{\prime}_{J}\subset y_{J}$ and we have

$x_{J}\vee x^{\prime}_{J}\subset y_{J}=(x\vee x^{\prime})_{J}.$

Now consider $w\in K_{J}^{-}[x_{J}\vee x^{\prime}_{J}],$ hence we have that
$x_{J}\subset w_{J}$ and $x^{\prime}_{J}\subset w_{J}.$ The uniformity
condition (U) implies that $J^{[0]}\cap y^{\prime}\subset J^{[0]}\cap w$ for
all $y^{\prime}\in K_{J}^{-}(x)\cup K_{J}^{-}(x^{\prime})$ which in particular
implies that $x\subset w$ and $x^{\prime}\subset w.$ Hence we have $x\vee
x^{\prime}\subset w$ and this implies that $y_{J}=(x\vee
x^{\prime})_{J}\subset w_{J}$ which leads the desired equality.

The condition 3) of compatibility is a direct consequence of Lemma A.4 since
if $\mathcal{E}_{J}^{y},\mathcal{E}_{J}^{y^{\prime}}\in M_{J}^{K}$ are such
that $\mathcal{E}_{J}^{y}\cap\mathcal{E}_{J}^{y^{\prime}}\neq\emptyset,$ then

$\displaystyle\pi_{J}^{K}(\mathcal{E}_{J}^{y}\cap\mathcal{E}_{J}^{y^{\prime}})$
$\displaystyle=\pi_{J}^{K}(\mathcal{E}_{J}^{y\cap y^{\prime}})=(y\cap
y^{\prime})_{J}$ $\displaystyle=\\{e\cap J^{[0]}\leavevmode\nobreak\
|\leavevmode\nobreak\ e\in\mathcal{E}_{J}^{y\cap y^{\prime}}\\}=\\{e\cap
J^{[0]}\leavevmode\nobreak\ |\leavevmode\nobreak\
e\in\mathcal{E}_{J}^{y}\cap\mathcal{E}_{J}^{y^{\prime}}\\}$
$\displaystyle=\\{e\cap J^{[0]}\leavevmode\nobreak\ |\leavevmode\nobreak\
e\in\mathcal{E}_{J}^{y}\\}\cap\\{e\cap J^{[0]}\leavevmode\nobreak\
|\leavevmode\nobreak\ e\in\mathcal{E}_{J}^{y^{\prime}}\\}$
$\displaystyle=x_{J}\cap
y_{J}=\pi_{J}^{K}(\mathcal{E}_{J}^{y})\cap\pi_{J}^{K}(\mathcal{E}_{J}^{y^{\prime}}).$

∎

Conversely, the compatibility condition allows to determine when a subdivision
of a boundary component of a cc $K$ allows to define a relative subdivision of
$K,$ as shown in the next proposition. This result is more involved than the
previous corollary and constitutes an important simplification step in the
proof of the Correspondence Theorem 2.22 shown in the next part.

######  Proposition 2.13.

Let $K\in\mathsf{C}$ and let $J^{\prime}$ be a connected component of
$\partial K$ such that $(K,J^{\prime})$ is uniform. If $\rho:J\longrightarrow
J^{\prime}$ is a reduction such that
$\rho\circ\rho_{J^{\prime}}^{K}:J\longrightarrow J^{\prime}(K)$ is compatible
with $\pi_{J^{\prime}}^{K}:M_{J^{\prime}}^{K}\longrightarrow J^{\prime}(K)$
and $J$ is disjoint from $K$ then the cc $K^{\rho}$ (introduced in Definition
2.3) is an element in $\mathsf{C}$ and $J$ is a connected component of
$\partial K^{\rho},$ in other words

$(K^{\rho},J)\mathrel{\raisebox{1.0pt}{$\prec$}}_{\rho}(K,J^{\prime})$

defines a relative reduction. Moreover, we have that
$\rho_{J}^{K^{\rho}}=\rho\circ\rho_{J^{\prime}}^{K}$ and $(K^{\rho},J)$ is
uniform.

###### Proof.

In Appendix D. ∎

As we are also interested in seeing a cc $J\in\mathsf{B}$ as the boundary
component of two different cc $K$ and $K^{\prime}$ in order to study in which
case their union is a cc, this leads to having two reductions of the form

$J(K)\mathrel{\raisebox{1.0pt}{$\succ$}}_{\rho_{J}^{K}}J\mathrel{\raisebox{1.0pt}{$\prec$}}_{\rho_{J}^{K^{\prime}}}J(K^{\prime}).$
(5)

The main idea from here is to show that the following notion of reflectivity
is indeed a sufficient assumption on $\rho_{J}^{K}$ and
$\rho_{J}^{K^{\prime}}$ in order for the union $K\cup K^{\prime}$ to become a
cc (as done in part 3.2). For this we will pass by the dual notion to that
called orthogonality also defined below.

######  Definition 2.14 (Reflective ($\mathbin{\top}$) and Orthogonal
($\perp$) homomorphisms).

Let $j:I\longrightarrow J$ and $l:I\longrightarrow L$ be two cc-homomorphisms.

  1. 1)

We say that $j$ and $l$ are reflective, noted $j\mathbin{\top}l,$ if whenever
two elements $w,w^{\prime}\in I$ satisfy that both $j(w)\vee
j(w^{\prime})\neq\overline{\emptyset}$ and $l(w)\vee
l(w^{\prime})\neq\overline{\emptyset}$ then we have that $w\vee
w^{\prime}\neq\overline{\emptyset},$ as well as the two equalities

$j(w\vee w^{\prime})=j(w)\vee j(w^{\prime})\quad\text{ and }\quad l(w\vee
w^{\prime})=l(w)\vee l(w^{\prime}).$

  2. 2)

Dually, we say that $j$ and $l$ are orthogonal, noted $j\perp l,$ if whenever
two elements $w,w^{\prime}\in I$ satisfy that both $j(w)\cap
j(w^{\prime})\neq\emptyset$ and $l(w)\cap l(w^{\prime})\neq\emptyset$ then we
have that $w\cap w^{\prime}\neq\emptyset,$ as well as the two equalities

$j(w\cap w^{\prime})=j(w)\cap j(w^{\prime})\quad\text{ and }\quad l(w\cap
w^{\prime})=l(w)\cap l(w^{\prime}).$

We also have that if $j$ and $l$ are poset isomorphisms then $j$ and $l$ are
reflective and orthogonal.

The key point about the latter definitions, that will allow to go back and
forth between the primal and dual perspectives, is that they satisfy the
following properties.

######  Lemma 2.15.

Let $I,J,L\in\mathsf{B}^{R}.$ Then the two following statements hold.

  1. A)

If $j:J\longrightarrow I$ and $\l:L\longrightarrow I$ are cc-homomorphisms
then

$j\leftthreetimes l\quad\text{if and only
if}\quad\overline{j}\rightthreetimes\overline{l}.$

  2. B)

If $j:I\longrightarrow J$ and $l:I\longrightarrow L$ are cc-homomorphisms then

$j\perp l\quad\text{if and only
if}\quad\overline{j}\mathbin{\top}\overline{l}.$

###### Proof.

For the claim A), the first condition of the compatibility of $\overline{l}$
with $\overline{j}$ is obtained from the fact that if $A$ is subset of cells
in a closed cc then $\left|\overline{A}\right|=\left|A\right|$ by Lemma 1.10.
As for the other conditions for compatibility and the claim B), they follow
from the ”DeMorgan” identities (2). ∎

From the notions of compatibility, reflectivity and orthogonality, it is
natural to combine them together into the following notion of geometric
sequence. These will turn out to be essentially the main constituents in the
definition of the category of causal cobordisms in part 3.3.

######  Definition 2.16 (Geometric sequence).

Let $K_{1},\dots,K_{N}$ be distinct cc in $\mathsf{B}.$ We call a sequence of
reductions and collapses a geometric sequence if is is of the form

$K_{1}\dots\longleftarrow K_{i}\longrightarrow K_{i+1}\longleftarrow
K_{i+2}\longrightarrow\dots K_{N}$ (6)

without constraint on whether the first and last map have respectively $K_{1}$
and $K_{N}$ as domain or codomain, and if it satisfies the conditions

  1. a)

if two maps have same codomain, then it is a reduction and a collapse and they
are compatible,

  2. b)

if two cc-homomorphisms have same domain, then either they are two orthogonal
collapses or two reflective reductions.

As noted in remark 1.26, the fact that the $K_{i}$’s belong to the same
geometric sequence implies that they all have the same rank. Moreover, we have
the following corollary.

######  Corollary 2.17.

The dual of a geometric sequence (6), i.e. where each cc $K_{i}$ is replaced
by $\overline{K_{i}}$ and each map is replaced by the dual map defined by
(1.5.2), is also a geometric sequence.

###### Proof.

This is a simple consequence of Lemma 2.15 and the fact that reductions and
collapses are dual to one another. ∎

As mentioned before, we are going to switch from two reflective reductions as
in (5) to the dual sequence then composed of two orthogonal collapses. The
next lemma is an important technical result that leads to the definition of
augmented (ranked) poset, named after the cells of rank one higher than their
pre-image it allows to define in the proof of the Correspondence Theorem 2.22.

######  Lemma 2.18.

Let $M,J,L$ be mutually disjoint cc and let $\phi_{J}:M\longrightarrow J$ and
$\phi_{L}:M\longrightarrow L$ be two orthogonal cc-homomorphisms. Then the map
from $M$ to $\mathcal{P}\left(J^{[0]}\sqcup L^{[0]}\right)$ defined by

$x\mapsto\phi_{J}(x)\sqcup\phi_{L}(x)$

is an injective poset homorphism such that the inverse is a poset
homomorphism. The poset

$(\phi_{J}\sqcup\phi_{L})(M):=\\{\phi_{J}(m)\sqcup\phi_{L}(m)\leavevmode\nobreak\
|\leavevmode\nobreak\ m\in M\\}$

will be called the augmented poset associated to $\phi_{J}$ and $\phi_{L}$ and
will be assigned the rank function

$\operatorname{rk}_{(\phi_{J}\sqcup\phi_{L})(M)}(x\sqcup
y)=\operatorname{rk}_{M}(m(x,y))+1,$

where $m(x,y)$ is the unique element in $M$ such that

$\phi_{J}(m(x,y))\sqcup\phi_{L}(m(x,y))=x\sqcup y.$

###### Proof.

Follows directly from the orthogonality condition 2). ∎

###  2.4 Slices and Correspondence Theorem

We are now going to introduce slices as a certain cc with all its vertices on
the boundary and see in the following proposition that slices have exactly two
connected components. We then introduce the notion of slice sequence and show
our main result of this section, the Correspondence Theorem, proving that
slice sequences are in bijective correspondence with slices up to cc-
isomorphism.

######  Definition 2.19 (Slice).

A cc $S\in\mathsf{C}$ if $(S,J)$ is uniform and local for all connected
component $J\leq\partial S$ and such that

$S^{[0]}=(\partial S)^{[0]}.$

We denote the set of slices by $\mathsf{C}_{s}.$

Figure 6 is in fact an illustration of a portion of a slice of rank 3. The
next result is also used to introduce the midsection associated to a slice and
is a good demonstration of how the assumptions of (relative) locality and non-
degeneracy can be combined. Note also that the proof does not make use of the
uniformity of the slice except for the non-degeneracy condition.

######  Proposition 2.20.

Let $S$ be a slice. Then $\partial S=J\sqcup L$ where $J$ and $L$ are
connected components, and $M_{L}^{S}=M_{J}^{S}.$ We thus define the midsection
associated to a slice $S$ by

$M^{S}:=M_{J}^{S}=M_{L}^{S}.$

###### Proof.

Let $J$ be a connected component of $S.$ Since $(S,J)$ is in particular non-
degenerate, we have that $L:=\partial S\setminus J$ is non-empty.

We next show that $L$ is connected. Suppose by contradiction that
$L=L_{1}\sqcup L_{2}$ where $L_{i}\in\mathsf{B}$ for $i=1,2$ and let
$R=\operatorname{Rk}(L)=\operatorname{Rk}(J).$ The case $R=0$ is clear since
the connectedness of $S$ implies that there is a path from $L_{1}$ to $L_{2}$
going through $J,$ which means that there is a vertex in $J$ included in two
edges and this is a contradiction with the definition of a boundary cell, i.e.
that a sub-maximal cell of the boundary - here a 0-cell - is included in only
one maximal cell. The strategy of the proof for the case $R\geq 3$ is to show
that our hypothesis implies that there exists an edge of $S$ having one vertex
in $L_{1}$ and the other vertex in $L_{2},$ which contradicts the non-
degeneracy of $(S,L).$

By connectedness of $S,$ there exists a path $p$ in $S$ from $v_{1}\in
L_{1}^{[0]}$ to $v_{2}\in L_{2}^{[0]}$ and we may suppose that $p$ is simple
and intersects $L$ only on two vertices. Let $\\{v_{1},w\\}$ be the first edge
of $p.$ If $w\in L_{2}$ we are done, so we suppose that $w\in J^{[0]}.$ By our
assumptions, $p$ enters $J$ from the vertex $v$ and leaves $J$ through an edge
of the form $\\{w^{\prime},v_{2}\\},$ where $w\in J^{[0]}.$ By locality of
$(S,J)$ the midsection $M:=M_{J_{0}}^{S}$ is connected, hence there exists a
path $p_{M}$ in $M$ from $\\{w,v_{1}\\}$ to $\\{w^{\prime},v_{2}\\}$ (seen as
vertices of $M$). Each edge of the path $p_{M}$ corresponds to a 2-cell in
$S_{J}$ which by locality intersects $J$ in a simple path $p_{0},$ or a single
vertex $v,$ in which case we set $p_{0}$ to be the empty path (based at the
vertex $v$). So using each such 2-cell $C$ in $p_{M}$ we can successively
deform $p$ using a path move of the form $m_{C}^{p_{1}}$ where $p_{1}\subset
p$ is the simple path with first edge $\\{v_{1},w\\}$ followed by the edges in
$p_{0}.$ At each iteration the resulting path might cross $L$ in more than two
vertices. If it does so by also crossing through $L_{2}$ this would imply that
there is an edge of $S$ crossing from $L_{1}$ to $L_{2}$ which would produce
the desired contradiction. We can therefore assume that deforming $p$ can only
increase the number of vertices it crosses in $L_{1},$ in which case we can
simply re-define $v_{1}$ to be the last vertex the path crosses in $L_{1}.$ In
this way we can successively deform and shorten the path $p$ until the last
step, corresponding to the last edge in $p_{M}$ which again produces an edge
of $S$ crossing from $L_{1}$ to $L_{2},$ which concludes our argument.

Proving that $M_{L}^{S}=M_{J}^{S}$ amounts to prove that $S_{L}=S_{J}.$ The
latter is clear from the property $S^{[0]}=(\partial S)^{[0]}$ which implies
that $\emptyset\neq x\cap J^{[0]}\neq x$ if and ony if $\emptyset\neq x\cap
L^{[0]}\neq x,$ for all $x\in S$ and this concludes the proof. ∎

In [DJ15], a notion of slice is defined and corresponds to a restriction of
the present definition to simplicial complexes satisfying the condition of
being homeomorphic to a cylinder (where the topology is defined via a notion
of geometric realisation). A notion of midsection is also defined and studied
essentially in dimension 2, for which it is a cell complexes made only of
triangles and quadrangles (2-cell containing 4 vertices). Interestingly, in
our framework both the boundary components of a slice and the associated
midsection belong to the same class of cc. Moreover, in the context of causal
cobordisms discussed in part 3.3, boundary components and midsections will be
put on the same footing, being both a cc in a geometric sequence.

In part 3.3 we also see that general geometric sequences can be seen as
constituted of two types of sub-sequences: the slice sequence introduced next,
and the dual notion of connecting sequence introduced in part 3.2 when
discussing the composition of cobordisms.

######  Definition 2.21 (Slice sequence).

A geometric sequence of the form

$J\mathrel{\raisebox{1.0pt}{$\prec$}}_{\rho_{J}}J^{\prime}\dashv_{\pi_{J}}M\vdash_{\pi_{L}}L^{\prime}\mathrel{\raisebox{1.0pt}{$\succ$}}_{\rho_{L}}L$

will be called a slice sequence. More explicitly this implies that
$J,J^{\prime},M,L^{\prime},L$ are elements in $\mathsf{B}^{R}$ for some $R\leq
0$ and that $\rho_{J},\pi_{J},\rho_{L},\pi_{L}$ satisfy

$\rho_{J}\leftthreetimes\pi_{J},\quad\rho_{L}\leftthreetimes\pi_{L},\quad\pi_{J}\perp\pi_{L}.$

We can finally turn to our main result.

######  Theorem 2.22 (Correspondence).

There is a one-to-one correspondence between slice sequences and slices, up to
cc-isomorphism. More precisely, a sequence

$J\mathrel{\raisebox{1.0pt}{$\prec$}}_{\rho_{J}}J^{\prime}\dashv_{\pi_{J}}M\vdash_{\pi_{L}}L^{\prime}\mathrel{\raisebox{1.0pt}{$\succ$}}_{\rho_{L}}L$

is a slice sequence if and only if the poset $(S,\operatorname{rk}_{S})$
defined by

$S:=(\pi_{J}\sqcup\pi_{L})(M)\sqcup J\sqcup L,$
$\operatorname{rk}_{S}:=\operatorname{rk}_{(\pi_{J}\sqcup\pi_{L})(M)}+\operatorname{rk}_{J}+\operatorname{rk}_{L},$

is an element in $\mathsf{C}_{s}$ such that

$\partial S=J\sqcup L,\quad M=M^{S},\quad J(S)=J^{\prime},\quad
L(S)=L^{\prime},$ $\phi_{I}=\phi_{I}^{S}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ for all }\leavevmode\nobreak\ \leavevmode\nobreak\
\phi\in\\{\rho,\pi\\},\leavevmode\nobreak\ \leavevmode\nobreak\
I\in\\{J,L\\}.$

###### Proof.

We first show that a slice sequence is uniquely defined from a slice. This can
be obtained directly as a consequence of Corollary 2.12 if we also show that
if $S$ is a slice with boundary components $J$ and $L$ then

$\pi_{J}^{S}\perp\pi_{L}^{S}.$

In order to prove that $\pi_{J}^{S}$ and $\pi_{L}^{S}$ are orthogonal, i.e.
satisfy the point 2) of Definition 2.14, let us consider $m_{1},m_{2}\in
M^{S}$ such that $\pi_{J}^{S}(m_{1})\cap\pi_{J}^{S}(m_{2})\neq\emptyset$ and
$\pi_{L}^{S}(m_{1})\cap\pi_{L}^{S}(m_{2})\neq\emptyset.$ By Proposition 2.20
we have that $m_{i}=\mathcal{E}_{J}^{y_{i}}=\mathcal{E}_{L}^{y_{i}}$ for some
$y_{i}\in S_{J}=S_{L},$ for $i=1,2,$ and the definition of $\pi_{J}^{S}$ in
Proposition 2.10 implies that $\pi_{J}^{S}(m_{i})=(y_{i})_{J}$ for $i=1,2.$ By
assumption we therefore have that $(y_{1})_{J}\cap(y_{2})_{J}\neq\emptyset$
and $(y_{1})_{L}\cap(y_{2})_{L}\neq\emptyset,$ hence $y_{1}\cap
y_{2}\neq\emptyset.$ Lemma A.4 implies $\mathcal{E}_{J}^{y_{1}\cap
y_{2}}=\mathcal{E}_{J}^{y_{1}}\cap\mathcal{E}_{J}^{y_{2}}$ hence we get the
following inclusion for $I=J,L:$

$\displaystyle(y_{1}\cap y_{2})_{I}$ $\displaystyle=\\{e\cap
J^{[0]}\leavevmode\nobreak\ |\leavevmode\nobreak\
e\in\mathcal{E}_{J}^{y_{1}\cap y_{2}}\\}=\\{e\cap J^{[0]}\leavevmode\nobreak\
|\leavevmode\nobreak\
e\in\mathcal{E}_{J}^{y_{1}}\cap\mathcal{E}_{J}^{y_{2}}\\}$
$\displaystyle\subset\\{e\cap J^{[0]}\leavevmode\nobreak\
|\leavevmode\nobreak\ e\in\mathcal{E}_{J}^{y_{1}}\\}\cap\\{e\cap
J^{[0]}\leavevmode\nobreak\ |\leavevmode\nobreak\
e\in\mathcal{E}_{J}^{y_{2}}\\}=(y_{1})_{I}\cap(y_{2})_{I}.$

To prove the reverse inclusion, and conclude that $\pi_{J}^{S}$ and
$\pi_{L}^{S}$ are orthogonal, suppose by contradiction that there exists a
vertex $v$ in $(y_{1})_{I}\cap(y_{2})_{I}\setminus(y_{1}\cap y_{2})_{I}$ for
$I=J$ or $I=L.$ If $\operatorname{rk}(y_{1}\cap y_{2})=1$ we directly obtain a
contradiction as in this case $v\in y_{1}\cap y_{2}=(y_{1}\cap y_{2})_{I}.$ If
$\operatorname{rk}(y_{1}\cap y_{2})\geq 2,$ we have in particular that $v$ is
by assumption contained in an edge $e\subset\mathcal{E}_{I}^{y_{1}}$ and since
$v\notin(y_{1}\cap y_{2})_{I},$ there is no edge in
$\mathcal{E}_{I}^{y_{1}\cap y_{2}}$ containing $v.$ As a consequence, if $x$
is a minimal cell in $\mathbf{A}(v)\cap S_{I}^{y_{1}\cap y_{2}}$ then
$\operatorname{rk}(x)\geq 2.$ But $x$ is then also a minimal cell in $S(v)$
and this contradicts the assumption that $(S,\partial S)$ is pure, since both
$e$ and $x$ are minimal cells and
$1=\operatorname{rk}(e)\neq\operatorname{rk}(x).$

We then show how to obtain a slice from a slice sequence. By Proposition 2.13,
it is sufficient to consider the case $J=J^{\prime}$ and $L=L^{\prime}$ since
if, in this case, we obtain a slice $S$ (satisfying the desired conditions)
the latter result using $\rho=\rho_{I}$ for $I=J,L$ indeed leads
$\rho_{I}^{S}=\rho_{I}$ and $I(S)=I^{\prime}.$

Our goal is then to prove that $(S,\operatorname{rk}_{S})\in\mathsf{C}_{s}$
where we recall that $(S,\operatorname{rk}_{S})$ is defined by

$S:=(\pi_{J}\sqcup\pi_{L})(M)\sqcup J\sqcup L,$
$\operatorname{rk}_{S}:=\operatorname{rk}_{(\pi_{J}\sqcup\pi_{L})(M)}+\operatorname{rk}_{J}+\operatorname{rk}_{L},$

and $(\pi_{J}\sqcup\pi_{L})(M)$ is the augmented poset associated to $\pi_{J}$
and $\pi_{L}.$ As usual we start by proving that $S$ is a cc. By Lemma 2.18
the Axioms i) and ii) are directly derived from the same axioms for the cc
$J,L$ and $M.$

As for Axiom iii), it is sufficient to show that if $m,m^{\prime}\in M,$ then

$x:=(\pi_{J}\sqcup\pi_{L})(m)\cap(\pi_{J}\sqcup\pi_{L})(m^{\prime})\in
S\cup\\{\emptyset\\}.$

If either $(\pi_{J}(m)\cap\pi_{J}(m^{\prime})=\emptyset$ or
$(\pi_{L}(m)\cap\pi_{L}(m^{\prime})=\emptyset$ then Axiom iii) for
respectively $L$ and $J$ implies that $x\in L$ or $x\in J.$ If both
$(\pi_{J}(m)\cap\pi_{J}(m^{\prime})$ and $(\pi_{L}(m)\cap\pi_{L}(m^{\prime})$
are non-empty, then as a consequence of the orthogonality of $\pi_{J}$ and
$\pi_{L}$ we have that $m\cap m^{\prime}\neq\emptyset$ and

$x=(\pi_{J}\sqcup\pi_{L})(m\cap m^{\prime})\in S.$

In order to show Axiom iv) for $S,$ the only cases that are not directly
obtained from Axiom iv) for $J,L$ or $M$ are inclusions of the form
$w\subsetneq(\pi_{J}\sqcup\pi_{L})(m),$ for some $m\in M$ and $w\in J\sqcup L$
such that

$\operatorname{rk}_{S}(w)=\operatorname{rk}_{S}((\pi_{J}\sqcup\pi_{L})(m))-2=\operatorname{rk}_{M}(m)-1.$

To deal with this cases we set $(\pi_{J}\sqcup\pi_{L})(m):=x\sqcup y$ and
assume without loss of generality that $w\in J,$ hence $w\subset
x=\pi_{J}(m).$ As noted in Remark 1.25, condition (c2) implies that
$\operatorname{rk}_{J}(\pi_{J}(m))\leq\operatorname{rk}_{M}(m).$ Therefore we
have

$\operatorname{rk}_{S}(w)=\operatorname{rk}_{J}(w)\leq\operatorname{rk}_{J}(x)\leq\operatorname{rk}_{M}(m)$

and this leaves us with two possibilities: either
$\operatorname{rk}_{J}(x)=\operatorname{rk}_{M}(m)-1$ or
$\operatorname{rk}_{J}(x)=\operatorname{rk}_{M}(m).$ In the first case we have
$w=x,$ so that (c3) implies

$\left|\triangle\left(m\right)\cap(\pi_{J})^{-1}(x)\right|=2.$

By Lemma 2.18 an element $m^{\prime}\in M$ belongs to
$\triangle\left(m\right)\cap(\pi_{J})^{-1}(x)$ if and only if it satisfies
that
$\operatorname{rk}_{S}(\pi_{J}(m^{\prime})\sqcup\pi_{L}(m^{\prime}))=\operatorname{rk}_{S}(x)+1$
and $\pi_{J}(m^{\prime})=x.$ Moreover any such $m^{\prime}$ satisfies
$\pi_{L}(m^{\prime})\subset y$ hence
$m^{\prime}\in\triangle\left(m\right)\cap(\pi_{J})^{-1}(x)$ if and only if

$\pi_{J}(m^{\prime})\sqcup\pi_{L}(m^{\prime})\in\bigtriangledown\left(x\right)\cap\triangle\left(x\sqcup
y\right).$

Therefore we indeed have

$\left|\triangle\left(m\right)\cap(\pi_{J})^{-1}(x)\right|=\left|\bigtriangledown\left(x\right)\cap\triangle\left(x\sqcup
y\right)\right|$

which concludes the case when $x=w.$

The second case is when $w\subsetneq x=\pi_{J}(m)$ and therefore
$\operatorname{rk}_{J}(\pi_{J}(m))=\operatorname{rk}_{M}(m).$ In this case
(c4) implies

$\left|\triangle\left(m\right)\cap(\pi_{J})^{-1}(x^{\prime})\right|=1\quad\forall
x^{\prime}\in\triangle\left(x\right).$

Since $w\in\triangle\left(x\right)$ we have
$\triangle\left(m\right)\cap(\pi_{J})^{-1}(w)=\\{w^{\prime}\\}$ and therefore

$\bigtriangledown\left(w\right)\cap\triangle\left(x\sqcup
y\right)=\\{x,\pi_{J}(w^{\prime})\sqcup\pi_{L}(w^{\prime})\\}.$

This concludes the proof that $S$ is a cc.

$S$ is graph-based since every edge in $(\pi_{J}\sqcup\pi_{L})(M)$ is of the
form $\pi_{J}(v)\sqcup\pi_{L}(v)$ for some vertex $v\in M^{[0]}$ and
$\operatorname{rk}_{J}(\pi_{J}(v))=\operatorname{rk}_{L}(\pi_{L}(v))=0$ by
Remark 1.25.

$S$ is also clearly connected and cell-connected, as every cell
$\pi_{J}(m)\sqcup\pi_{L}(m)\in(\pi_{J}\sqcup\pi_{L})(M)$ satisfies that
$\mathbf{B}_{J}(\pi_{J}(m))$ and $\mathbf{B}_{L}(\pi_{L}(m))$ are connected
and can be linked by an edge of the form $\pi_{J}(v)\sqcup\pi_{L}(v)$ where
$v\in m.$

By Remark 1.26 we have
$\operatorname{Rk}(J)=\operatorname{Rk}(M)=\operatorname{Rk}(L)=R.$ It follows
that $S$ is pure of rank $R+1$ since the surjectivity of $\pi_{J}$ and
$\pi_{L}$ implies that every cell in $J\sqcup L$ is contained in a cell of the
form $\pi_{J}(m)\sqcup\pi_{L}(m)$ for some $m\in M^{R}$ and hence

$\operatorname{rk}_{S}(\pi_{J}(m)\sqcup\pi_{L}(m))=\operatorname{rk}_{M}(m)+1=R+1.$

$S$ is non-branching as a consequence of Lemma 2.18 for the cells in
$(\pi_{J}\sqcup\pi_{L})(M)$ and since $J$ and $L$ are also non-branching.

By Lemma 2.18 and the closeness of $M$ every $R$-cell in $S\setminus(J\sqcup
L)$ is contained in two $(R+1)$-cells. Therefore every $R$ cell contained in
one $(R+1)$-cell must be contained in either $J$ or $L$ and by the
surjectivity of $\pi_{J}$ and $\pi_{L}$ it is the case for all maximal cells
of $J$ and $L,$ hence $\partial S=J\sqcup L.$

$S$ is non-pinching since an $S$-pinch would imply an $M$-pinch by Lemma 2.18
and a $\partial S$-pinch would imply a $J$-pinch or an $L$-pinch by the
previous derivation, all of which are excluded since $M,J,L$ are non-pinching.

As a consequence we indeed have that $S\in\mathsf{C}.$ It remains to show that
$S$ is a slice. First, we have that $K^{[0]}=J^{[0]}\sqcup L^{[0]}$ since
every cell in $(\pi_{J}\sqcup\pi_{L})(M)$ has rank higher than or equal to
one. Clearly $S_{J}=S_{L}=(\pi_{J}\sqcup\pi_{L})(M)$ and by Lemma 2.18 this
implies $M=M^{S}.$ $(S,\partial S)$ is also clearly non-degenerate and it is
pure by the following argument. If, say, $y\in S_{J}$ then
$y=\pi_{J}(m)\cap\pi_{L}(m)$ for some $m\in M$ and
$\mathbf{B}_{J}(y)=\mathbf{B}_{J}(\pi_{J}(m))$ is a pure cc. Hence if $y$ has
minimal rank among elements in $S_{J}(x)$ then $\pi_{J}(m)=x.$ By condition
(c2) we can assume that $\operatorname{rk}_{M}(m)=\operatorname{rk}_{J}(x)$
therefore

$\operatorname{rk}_{S}(y)=\operatorname{rk}_{M}(m)+1=\operatorname{rk}_{J}(x)+1=\operatorname{rk}_{J}^{S}(x)+1,$

and this shows that all elements in $S^{-}(x)$ have same rank.

Concerning uniformity, for all
$\pi_{J}(m)\sqcup\pi_{L}(m)\in(\pi_{J}\sqcup\pi_{L})(M)=S_{\partial S}$ we
have that

$\displaystyle(\pi_{J}(m)\sqcup\pi_{L}(m))_{J}$
$\displaystyle=\\{v\in\pi_{J}(m)\leavevmode\nobreak\ |\leavevmode\nobreak\
\exists w\in m,\leavevmode\nobreak\ \pi(w)=v\\}$
$\displaystyle=\pi_{J}(m)=\left(\pi_{J}(m)\sqcup\pi_{L}(m)\right)\cap J^{[0]}$

and similarly for $L.$ Since we also assumed that $J(S)=J\in\mathsf{B}$ and
$L(S)=L\in\mathsf{B}$ it follows that $(S,\partial S)$ is indeed uniform.

Finally, we have that $\pi_{I}=\pi_{I}^{S}$ for $I\in\\{J,L\\}$ by the
following argument. It is sufficient to show this for the case $I=J$ in which
for any $m\in M$ we have
$\mathcal{E}_{J}^{x}=\\{\pi_{J}(v)\sqcup\pi_{L}(v)\leavevmode\nobreak\
|\leavevmode\nobreak\ v\in m\\},$ where $x=\pi_{J}(m)\sqcup\pi_{L}(m).$ By
identifying each vertex $v\in m$ with the edge
$\pi_{J}(v)\sqcup\pi_{L}(v)\in(M^{S})^{[0]}$ we obtain $\mathcal{E}_{J}^{x}=m$
and therefore

$\pi_{J}^{S}(\mathcal{E}_{J}^{x})=x_{J}=x\cap J^{[0]}=\pi_{J}(m),$

where in the second equality we used the assumption $J=J(S).$ ∎

##  3 Cobordisms

This final section is based on the results of the previous sections and is
organised as follows. We first define cobordisms as a type of relative cc with
boundary on which one can generalize the duality map of closed cc, as shown in
Theorem 3.5, an other major result of this work. This is appears to be the
most general way to define such a duality that acts as an involution on a sub-
class of cobordisms, and in particular it doesn’t use the assumption of
uniformity.

In part 3.2 we define connecting sequences and show how this notion can be
used to define the union of two cobordisms sharing a boundary component. This
operation however relies on the assumption of uniformity, hence this
composition operation is not compatible with the duality map since the dual of
a uniform cobordism is not necessarily uniform.

The part 3.3 finally introduces a natural class of cobordisms on which both
the duality map and the composition operation are defined. These are called
causal cobordisms as they are decomposable into a sequence of slices as for
CDT. We notice that the resulting category exhibits interesting symmetries.

###  3.1 Definition and duality

In order to obtain an involutive duality map on a certain sub-class of cc in
$\mathsf{C}$, we have to focus on how this duality acts on the cells of the
boundary (this property being already satisfied for cells not on the boundary,
as a consequence of Lemma 1.10). The idea here is to look for a canonical way
to define a dual set for cells of the boundary, i.e. a notion of dual set
which solely depends on the structure of the neighbouring cells. For this
purpose, we will use a different type of dual set, noted $\widetilde{A}$ for
$A$ a set of vertices. If $K\in\mathsf{C}^{R}$ and $y\in(\partial K)^{[R-1]}$
is a maximal cell in the boundary, one must have in particular that the dual
of $y$ is an edge of the dual of $K,$ so that it becomes again a sub-maximal
cell of the double dual of $K.$ Since $\widetilde{y}$ therefore has to contain
at least two elements, it is natural to define

$\widetilde{y}=\\{y,z\\},$

where $z$ is the unique element of $K^{[R]}$ containing $y.$ In fact, the
”generalized dual” of a cell $x\in\partial K$ will be defined as

$\widetilde{x}:=\\{y\in(\partial K)^{[R-1]}\leavevmode\nobreak\
|\leavevmode\nobreak\ x\subset y\\}\sqcup\\{z\in K^{[R]}\leavevmode\nobreak\
|\leavevmode\nobreak\ x\subset z\\}=\overline{x}^{\partial
K}\sqcup\overline{x}^{K},$

where this definition also applies to any set $A\subset K^{[0]},$ with the
convention that $\overline{A}^{\partial K}=\emptyset$ whenever
$A\not\subset\partial K^{[0]}.$

The first issue one then encounters is that $\widetilde{K}$ is not a cc, as
for example in the case of $y\in(\partial K)^{[R-1]},$ the poset
$\widetilde{K}$ does not contain the vertex $\\{y\\}=\overline{y}^{\partial
K}.$ One can easily fix this by considering the dual of $K$ to be
$\widetilde{K}\sqcup\overline{\partial K}$ but then this notion of dual is not
involutive as it ”adds” more cells to the primal cc: for example the dual of a
graph made of one edge, therefore having two boundary components each made of
one vertex, is a graph made of two edges forming a simple path of length 2.
The double dual would then be a graph forming a simple path of length 3, etc.
The radically opposite alternative would be to ”ignore” the cells in $\partial
K$ and define the dual of $K$ to be $\overline{K\setminus\partial K},$ with
obvious drawbacks when it comes to the involution property, as this definition
”removes” cells. It turns out that alternating between these two definitions
is essentially the idea behind the construction of the duality for cobordisms:
the dual of a path of length 1 is a path of length 2 and the double dual is
again a path of length 1. But then for each boundary component, one has the
choice of either applying one or the other definition of dual cell. Relative
cc therefore become a handy solution, as one can define the dual of a relative
cc $(K,J)$ where $J\leq\partial K$ using the definition that adds cells on the
cells in $\partial K\setminus J$ and use the definition that removes cell on
the cells in $J.$

This is where the picture with cobordisms arises: an element $(K,J)$ where
$K\in\mathsf{C}^{R}$ and $J\in\mathsf{B}^{[R-1]},\leavevmode\nobreak\
J\leq\partial K,$ can be interpreted as a cobordism from some ”ingoing
components” $J$ to some ”outgoing components” $L:=\partial K\setminus J.$ In
this picture, the duality map switches the locations of the ingoing components
and the outgoing components.

Let us now make this more precise by introducing our combinatorial notion of
cobordism. For this we will use the notation $(K-J)$ to distinguish such an
element from a usual relative cc. The sign ”$-$” is used to signal that cells
in $J$ can be thought of as ”removed” in the Definition 3.4 of the duality
map, since it acts on $(K-J)$ as if the cells in $J$ are not in $K.$

######  Definition 3.1 (Cobordism).

The set of cobordisms is defined as

$\mathsf{Cob}:=\\{(K-J)\leavevmode\nobreak\ |\leavevmode\nobreak\
K\in\mathsf{C},\leavevmode\nobreak\ J\leq\partial K,\leavevmode\nobreak\
J\in\mathsf{B}^{\operatorname{Rk}(K)-1}_{\sqcup},\leavevmode\nobreak\
(K,J)\text{ non-deg. and local}\\}.$

We use $\mathsf{Cob}^{R}$ to denote elements in $\mathsf{Cob}$ such that
$\operatorname{Rk}(K)=R,$ in which case $R$ is also called the rank or
dimension of the cobordisms in $\mathsf{Cob}^{R}.$ The boundary of a cobordism
is simply defined as

$\partial(K-J):=\partial K.$

The boundary components $J$ of a cobordism $(K-J)$ are called ingoing
(boundary) components or removed components of the cobordism and the cells in
$J$ are sometimes called the removed cells. The boundary components in
$\partial K\setminus J$ are called outgoing (boundary) components of $(K-J).$
We say that $(K-J)\in\mathsf{Cob}$ is exactly collared if $(K,J)$ is exactly
collared.

We will discuss some examples of cobordisms below, as those shown in Figure 14
and 15. But let us point out that this is a rather large class of objects
including in particular all elements $(K-J)$ where $K$ is a simplicial
complex, as the assumption of non-degeneracy and locality are directly
satisfied in this cases.

Our next goal is to introduce the duality map on cobordisms. The following
definition constitutes a generalization of the Definition 1.7 of dual cell to
the cells on the boundary of a non-singular cc.

######  Definition 3.2.

Let $K$ be a cc and let $A\subset K^{[0]}$ be non-empty. We define the
$\sim$-dual of $A$ to be

$\widetilde{A}^{K}:=\overline{A}^{K}\sqcup\overline{A}^{\partial K},$

where by convention $\overline{A}^{\partial K}=\emptyset$ if
$A\not\subset(\partial K)^{[0]}.$ We also write $\widetilde{x}$ for
$\widetilde{x}^{K}$ when there is no ambiguity on $K.$

For a collection of sets $\mathcal{A}\subset\mathcal{P}\left(K^{[0]}\right),$
we use the notation

$\widetilde{\mathcal{A}}^{K}:=\\{\widetilde{A}^{K}\leavevmode\nobreak\
|\leavevmode\nobreak\ A\in\mathcal{A}\\}.$

For example if $(K-J)\in\mathsf{Cob},$ we obtain $\widetilde{K\setminus
J}^{K}=\\{\widetilde{x}^{K}\leavevmode\nobreak\ |\leavevmode\nobreak\ x\in
K\setminus J\\}.$ We sometimes simply write $\widetilde{K\setminus J}$ instead
of $\widetilde{K\setminus J}^{K}.$ Note that if $K$ is closed then
$\widetilde{K}=\widetilde{K\setminus\emptyset}=\overline{K}.$

We would also like to define a rank function for elements in $\widetilde{K}$
when $K\in\mathsf{C}.$ This can be done by mean of the following lemma.

######  Lemma 3.3.

Let $K$ be a non-singular cc and let $x,y\in K.$ Then

$\widetilde{x}^{K}\subsetneq\widetilde{y}^{K}\quad\text{ if and only if }\quad
y\subsetneq x.$

###### Proof.

This is a simple consequence of Lemma 1.10 applied to cells in
$K\setminus\partial K$ and $\partial K.$ ∎

As a consequence of Lemma 3.3, the map $x\mapsto\widetilde{x}$ is a bijection
from a non-singular cc $K$ to $\widetilde{K}$ and the rank function

$\operatorname{rk}_{\widetilde{K}}(\widetilde{x}):=\operatorname{Rk}(K)-\operatorname{rk}_{K}(x)$

where $x\in\widetilde{K}$ is well defined for all
$\widetilde{x}\in\widetilde{K}.$

Let $K,K^{\prime}$ be disjoint posets with rank functions
$\operatorname{rk}_{K}$ and $\operatorname{rk}_{K^{\prime}}.$ We define the
disjoint union $K\sqcup K^{\prime}$ to be a ranked poset with rank function

$\operatorname{rk}_{K\sqcup
K^{\prime}}:=\operatorname{rk}_{K}+\operatorname{rk}_{K^{\prime}}.$

For an element $(K-J)\in\mathsf{Cob},$ this definition allows us to have a
rank function on the poset defined as the disjoint union

$\widetilde{K\setminus J}\sqcup\overline{\partial K\setminus
J}=\widetilde{K\setminus J}^{K}\sqcup\overline{\partial K\setminus
J}^{\partial K}$

used below in our definition of dual cobordism. Note that the latter union is
indeed disjoint by the non-degeneracy of $(K,J):$ every cell $x\in\partial
K\setminus J$ satisfies $\overline{x}^{\partial
K}\subsetneq\widetilde{x}^{K},$ which implies in particular that
$\overline{x}^{\partial K}\neq\widetilde{x}^{K}$ for all such $x$ and
therefore $\widetilde{K\setminus J}^{K}\cap\overline{\partial K\setminus
J}^{\partial K}=\emptyset.$

More explicitly, if $(K,J)$ is non-degenerate our notation leads to the
following expressions for the rank function on $\widetilde{K\setminus
J}\sqcup\overline{\partial K\setminus J}:$

$\operatorname{rk}_{\widetilde{K\setminus J}\sqcup\overline{\partial
K\setminus
J}}(x^{\prime})=\begin{cases}\operatorname{rk}_{\overline{K}}(x^{\prime})=R-\operatorname{rk}_{K}(x)&\textit{if
}x^{\prime}=\overline{x}^{K}\textit{ for }x\in K\setminus\partial K,\\\
\operatorname{rk}_{\widetilde{K}}(x^{\prime})=R-\operatorname{rk}_{K}(x)&\textit{if
}x^{\prime}=\widetilde{x}^{K}\textit{ for }x\in\partial K,\\\
\operatorname{rk}_{\overline{\partial
K}}(x^{\prime})=(R-1)-\operatorname{rk}_{\partial K}(x)&\textit{if
}x^{\prime}=\overline{x}^{\partial K}\textit{ for }x\in\partial K.\end{cases}$

where $R=\operatorname{Rk}(K).$ The particular case $J=\emptyset$ yields

$(\widetilde{K}\sqcup\overline{\partial
K})^{[0]}=\overline{K^{[R]}}^{K}\sqcup\overline{(\partial
K)^{[R-1]}}^{\partial K},$

for the dual sets of rank $0.$ For dual sets of rank $1\leq r\leq R-1$ we have

$(\widetilde{K}\sqcup\overline{\partial
K})^{[r]}=\overline{(K\setminus\partial
K)^{[R-r]}}^{K}\sqcup\overline{\partial K^{[R-r-1]}}^{\partial
K}\sqcup\widetilde{(\partial K)^{[R-r]}}^{K}.$

and the dual sets of rank $R$ are

$(\widetilde{K}\sqcup\overline{\partial
K})^{[R]}=\overline{(K\setminus\partial
K)^{[0]}}^{K}\sqcup\widetilde{(\partial K)^{[0]}}^{K}.$

For example, if $e^{\prime}\in(\widetilde{K}\sqcup\overline{\partial
K})^{[1]}$ then $e^{\prime}$ is a dual set having one of the three following
forms:

  * •

the ”usual” dual of a sub-maximal cell of $K$ i.e.
$e^{\prime}=\overline{y}^{K}$ for $y\in K^{[R-1]},$

  * •

the ”usual” dual of a sub-maximal cell of $\partial K,$ i.e.
$e^{\prime}=\overline{y}^{\partial K}$ for $y\in(\partial K)^{[R-2]},$

  * •

the $\sim$-dual of a maximal cell in $\partial K$, i.e.
$e^{\prime}=\widetilde{y}^{K}=\\{y,z\\}$ where $y\in(\partial K)^{[R-1]}$ and
$z\in K^{[R]}$ is the unique maximal cell containing $y.$

With this in mind, the next definition of duality map for cobordism is more
transparent.

######  Definition 3.4 (Dual cobordism).

Let $(K-J)\in\mathsf{Cob},$ we define the dual cobordism to be

$\overline{(K-J)}:=\left((\widetilde{K\setminus J}\sqcup\overline{\partial
K\setminus J})-\overline{\partial K\setminus J}\right).$

This definition, above all, raises the following question: is
$\overline{(K-J)}$ also a cobordism? This question is answered positively by
the following theorem.

######  Theorem 3.5.

If $(K-J)\in\mathsf{Cob}^{R}$ then:

  * •

$\overline{(K-J)}\in\mathsf{Cob}^{R},$

  * •

$\overline{(K-J)}$ is exactly collared,

  * •

$\overline{K_{J}}\in\mathsf{B}^{R-1}_{\sqcup}$ and

$\partial\overline{(K-J)}=\overline{\partial K\setminus
J}\sqcup\overline{K_{J}}.$

###### Proof.

In Appendix E. ∎

In the proof of Theorem 3.5 it is seen that, as a consequence of Lemma A.4, we
have $\overline{K_{J}}\cong\overline{M_{J}^{K}}.$ In other words, the dual
outgoing component $\overline{K_{J}}$ is better understood as the dual of the
midsection associated to the ingoing components $J.$ We point out that
although $\overline{K_{J}}$ is a closed cc, it is not the case of $K_{J}$ as
cells in the collar are not defined as subsets of edges in the collar (i.e.
minimal elements in the collar).

Definition 3.4 leads to the interpretation that the action of the duality map
on cobordisms in effect exchanges the role of the ingoing and outgoing
boundary components. This can be seen particularly well in the case of the
cylinder-like cobordism depicted in Figure 15.

As an example, we can also look at what Definition 3.4 implies for the special
cases where $J=\emptyset$ and $J=\partial K.$ In the first case we get:

$\overline{(K-\emptyset)}=\left((\widetilde{K}\sqcup\overline{\partial
K})-\overline{\partial K}\right).$

We can think of this case as duality acting on a non-singular cc $K,$ as no
other data is specified. Such an example is depicted in Figure 14. As a
consequence of Theorem 3.5, we have that
$\widetilde{K}\sqcup\overline{\partial K}\in\mathsf{C}$ and

$\partial(\widetilde{K}\sqcup\overline{\partial K})=\overline{\partial K}.$

The dual of $(K-\emptyset)$ is a cobordism with ingoing components equal to
the entire boundary. For such cobordisms, corresponding to our second case
$J=\partial K,$ the duality gives

$\overline{(K-\partial K)}=\left(\widetilde{K\setminus\partial
K}-\emptyset\right).$

Another consequence of Theorem 3.5 is therefore that
$\widetilde{K\setminus\partial K}\in\mathsf{C}$ for all $(K-\partial
K)\in\mathsf{Cob}.$

Finally, the following corollary concerns the special case of exactly collared
cobordisms, for which the involution property holds. To state this result, we
defined the isomorphism $(K-J)\cong(K^{\prime}-J^{\prime})$ between cobordisms
as $K\cong K^{\prime}$ with an isomorphism mapping $J$ to $J^{\prime}.$

######  Corollary 3.6.

If $(K-J)\in\mathsf{Cob}$ is exactly collared 222A consequence of this
corollary is that $(K-J)$ is exactly collared if and only if there exists
$(K^{\prime}-J^{\prime})\in\mathsf{Cob}$ such that
$\overline{(K^{\prime}-J^{\prime})}=(K-J),$ a property that motivated us to
include the notion of ”exactness” in the term used to denote such cobordisms.
then we have

$\partial\overline{(K-J)}\cong\overline{J}\sqcup\overline{\partial K\setminus
J}\quad\text{and}\quad\overline{\overline{(K-J)}}\cong(K-J).$

###### Proof.

By Theorem 3.5 and using the assumption that $(K-J)$ is exactly collared, i.e.
that $J\cong M_{J}^{K},$ and the identity
$\overline{M_{J}^{K}}\cong\overline{K_{J}},$ we directly obtain that
$\partial\overline{(K-J)}\cong\overline{J}\sqcup\overline{\partial K\setminus
J}.$

The identity $\overline{\overline{(K-J)}}\cong(K-J)$ is a consequence of the
following computation, using the previous point and the assumption that
$(K-J)$ is exactly collared:

$\displaystyle\overline{\overline{(K-J)}}$
$\displaystyle=\overline{(\widetilde{K\setminus J}\sqcup\overline{\partial