# Duality for positive opetopes and tree complexes

Marek Zawadowski

###### Abstract

We show that the (positive) zoom complexes, here called tree complexes, with
fairly natural morphisms, form a dual category to the category of positive
opetopes with contraction epimorphisms. We also show how this duality can be
slightly generalized to thicket complexes and opetopic cardinals.

###### Contents

  1. 1 Introduction
  2. 2 The category of tree complexes
    1. 2.1 Trees
    2. 2.2 Constellations
    3. 2.3 Tree complexes
  3. 3 Duality
    1. 3.1 From positive opetopes to tree complexes
    2. 3.2 From tree complexes to positive opetopes
    3. 3.3 The main theorem
  4. 4 Thicket complexes
    1. 4.1 Thicket constellations
    2. 4.2 Thicket complexes and duality
  5. 5 Appendix: Positive opetopes and positive opetopic cardinals
    1. 5.1 Positive hypergraphs
    2. 5.2 Positive opetopic cardinals
    3. 5.3 Positive opetopes
    4. 5.4 The embedding of ${\bf pOpeCard}$ into ${\bf{\omega}Cat}$
    5. 5.5 $\iota$-maps of positive opetopes
    6. 5.6 The embedding of ${\bf pOpe}_{\iota}$ into ${\bf{\omega}Cat}$

##  1 Introduction

Opetopes are higher dimensional shapes that were originally invented in [BD]
as shapes that can be used to define a notion of higher dimensional category.
By now there are more than a dozen of definitions of opetopes. Some
definitions use very abstract categorical machinery [Bu], [Z3], some are more
concrete using one way or another some kinds of operads and/or polynomial or
analytic monads [BD], [HMP], [Lei], [Ch], [Z3], [SZ], [FS], some definitions
describe the ways opetopes can be generated [HMP], [CTM], [T], and finally
there are also some purely combinatorial definitions [P], [KJBM], [Z1], [Z2],
[St].

So it is not surprising that it is easier to show a picture of an opetope then
to give a simple definition that will leave the reader with no doubts as to
what an opetope is. In this paper we will deal with positive opetopes
(p-opetopes, for short) only. This means that each face in opetopes we
consider has at least one face of codimension $1$ in its domain. As a
consequence such opetopes have no loops.

The definitions of opetopes mentioned above seem to agree in a ‘reasonable
sense’. However, the morphisms between opetopes are not always treated the
same way. In some approaches even face maps between opetopes do not seem to be
natural. In [Z1],[Z3] it is shown how opetopes can be treated as some special
kinds of ${\omega}$-categories and therefore all the ${\omega}$-functors (i.e.
all face maps and all degeneracies) between opetopes can be considered. Of all
the definitions of opetopes, the one given in [KJBM], through so called zoom
complexes, seem to be very different from any other. In fact, to describe this
definition even pictures are of a very different kind, as the reader can
notice below. In this paper we shall show that there is an explanation of this
phenomenon. Namely, the (positive) zoom complexes, here called tree complexes,
with fairly natural morphisms, form a dual category to the category of
positive opetopes with contraction epimorphisms (often called $\iota$-epis).
The contraction epimorphisms are some kind of degeneracies of opetopes that
can send a face only to a face but possibly of a lower dimension (still
preserving usual constraints concerning both domains and codomains).

Below we draw some pictures of positive opetopes of few low dimensions and
corresponding dual111The fact that these structures are dual one to another is
actually the main claim of the paper. tree complexes, a simplified version of
zoom complexes in [KJBM].

An opetope $O_{1}$ of dimension 1

$t_{2}\vphantom{t_{1}}$$\vphantom{t_{2}}t_{1}$$y$

and its dual tree complex $T_{1}$

dim 0dim 1$\bullet_{t_{2}}$${}_{t_{1}}$$\bullet_{y}$

An opetope $O_{2}$ of dimension 2

$t_{3}\vphantom{t_{1}}$$\vphantom{t_{3}}t_{1}$$y_{1}$$t_{2}$$y_{3}$$y_{2}$$\Downarrow\\!b$

and its dual tree complex $T_{2}$

dim 0dim 1dim
2$\bullet_{t_{3}}$${}_{t_{2}}$${}_{t_{1}}$$\bullet_{y_{2}}$$\bullet_{y_{3}}$${}_{y_{1}}$$\bullet_{b}$

An opetope $O_{3}$ of dimension 3

$t_{3}\vphantom{t_{1}}$$\vphantom{t_{3}}t_{1}$$y_{3}$$t_{2}$$y_{5}$$y_{4}$$\Downarrow\\!b_{3}$$t_{4}\vphantom{}$$y_{6}$$\vphantom{}t_{0}$$y_{1}$$y_{0}$$y_{2}$$\Downarrow\\!b_{2}$$\Downarrow\\!b_{1}$$\Longrightarrow$$\beta$$t_{3}\vphantom{t_{1}}$$\vphantom{t_{3}}t_{1}$$t_{2}$$y_{5}$$y_{4}$$\Downarrow\\!b_{0}$$t_{4}\vphantom{}$$y_{6}$$\vphantom{}t_{0}$$y_{1}$$y_{0}$

and its dual tree complex $T_{3}$

dim 0dim 1dim 2dim
3$\bullet$${}_{t_{5}}$${}_{t_{4}}$${}_{t_{3}}$${}_{t_{2}}$${}_{t_{1}}$$\bullet_{y_{1}}$$\bullet^{y_{4}}$$\bullet_{y_{5}}$$\bullet^{y_{6}}$${}_{y_{3}}$${}_{y_{2}}$${}_{y_{0}}$$\bullet_{b_{1}}$$\bullet_{b_{2}}$$\bullet_{b_{3}}$${}_{b_{0}}$$\bullet_{\beta}$

The drawings of the above tree complexes are, in fact, drawings of consecutive
(non-empty) constellations of tree complexes. In particular, it is not an
accident that the partial order of nesting of circles in one constellation is
isomorphic to the partial order vertices of the next constellation.

Opetopic cardinals still consist of cells that can be meaningfully composed in
a unique way, but have a bit more general shape. Here is an example

$s_{6}\vphantom{s_{4}}$$\vphantom{s_{6}}s_{4}$$x_{6}$$s_{5}$$x_{8}$$x_{7}$$\Downarrow
a_{2}$$\Downarrow
a_{1}$$s_{7}\vphantom{\phantom{s_{7}}}$$x_{4}$$x_{9}$$x_{5}$$s_{2}\vphantom{s_{0}}$$\vphantom{s_{2}}s_{0}$$x_{0}$$s_{1}$$x_{2}$$x_{1}$$\Downarrow
a_{0}$$S:$$s_{3}\vphantom{\phantom{s_{2}}}$$x_{3}$

and its dual thicket222The name ‘thicket’ was suggested to me by Karol
Szumiło. complex

dim 0dim 1dim
2$\bullet$${}_{s_{7}}$${}_{s_{6}}$$\vdots$${}_{s_{1}}$${}_{s_{0}}$$\bullet^{x_{1}}$$\bullet_{x_{2}}$$\bullet_{x_{3}}$$\bullet^{x_{5}}$$\bullet^{x_{7}}$$\bullet_{x_{8}}$$\bullet_{x_{9}}$${}_{x_{0}}$${}_{x_{4}}$${}_{x_{6}}$$\bullet_{a_{0}}$$\bullet_{a_{1}}$$\bullet_{a_{2}}$

To illustrate how the duality works on morphisms is a bit more involved. We
shall present a $\iota$-epimorphism from $O_{3}$ to $O_{2}$ by naming faces of
$O_{3}$

$t_{2}\vphantom{t_{1}}$$\vphantom{t_{2}}t_{1}$$y_{2}$$t_{2}$$t_{2}$$y_{2}$$\Downarrow\\!y_{2}$$t_{3}\vphantom{}$$y_{3}$$\vphantom{}t_{1}$$t_{1}$$y_{1}$$y_{1}$$\Downarrow\\!b$$\Downarrow\\!y_{1}$$\Longrightarrow$$b$$t_{2}\vphantom{t_{1}}$$\vphantom{t_{2}}t_{1}$$t_{2}$$t_{2}$$y_{2}$$\Downarrow\\!b$$t_{3}\vphantom{}$$y_{3}$$\vphantom{}t_{1}$$t_{1}$$y_{1}$

by the cells on $O_{2}$

$t_{3}\vphantom{t_{1}}$$\vphantom{t_{3}}t_{1}$$y_{1}$$t_{2}$$y_{3}$$y_{2}$$\Downarrow\\!b$

they are sent to. For example, there are three faces in $O_{3}$ sent to the
$1$-face $y_{2}$. Two of them are $1$-faces and one of them is a $y_{2}$ face.
The dual of this morphism is a collection of three embeddings of trees sending
leaves (vertices) to leaves, and inner nodes (circles) to inner nodes and
respecting the relation between circles at one dimension and the vertices on
the next dimension. We present the dual morphism of tree complexes from
$T_{2}$ to $T_{3}$ drawing $T_{3}$ and naming its nodes

dim 0dim 1dim 2dim
3$\bullet$${}_{t_{3}}$${}_{t_{2}}$${}_{t_{1}}$$\bullet$$\bullet^{y_{2}}$$\bullet$$\bullet^{y_{3}}$${}_{y_{1}}$$\bullet$$\bullet_{b}$$\bullet$$\bullet$

by the names of the nodes of the tree complex $T_{2}$

dim 0dim 1dim
2$\bullet_{t_{3}}$${}_{t_{2}}$${}_{t_{1}}$$\bullet_{y_{2}}$$\bullet_{y_{3}}$${}_{y_{1}}$$\bullet_{b}$

that are sent to those faces.

This duality could be compared to a restricted version of the duality between
simple categories and discs [Be], [Ou], [MZ]. In that duality we have on one
side some pasting diagrams described in terms of (simple)
${\omega}$-categories and all ${\omega}$-functors, and on the other some
combinatorial structures called (finite) discs and some natural morphisms of
disks. Roughly speaking, a finite disc is a finite planar tree extended by
dummy/sink nodes at the ends of any linearly ordered set of sons of each node.
These sink nodes do not bring any new information about the object but they
are essential to get the right notion of a morphism between ‘such structures’,
i.e. all those that correspond to ${\omega}$-functors on the other side. If we
were to throw away the sink nodes, i.e. we would consider trees instead of
discs, we could still have a duality but we would need to revise the notion of
a morphism on both sides. In fact, we would have duality for degeneracy maps
only. On the planar tree side we would not be able to dump a ‘true’ node onto
a dummy node. This corresponds on the side of simple categories to the fact
that we consider only some ${\omega}$-functors. If we think about
${\omega}$-functors between simple ${\omega}$-categories as a kind of ‘partial
composition of a part of the pasting diagram’, we would need to restrict to
those ${\omega}$-functors that represent partial composition of the whole
pasting diagram. In other words, if we have trees and do not have sink nodes
around, our operations cannot drop any part of the pasting diagram before they
start to compose them. The duality presented in this paper can be understood
through this analogy. Namely, at the level of objects positive opetopes
correspond to tree complexes but when we look at the morphism, the natural
morphisms of tree complexes correspond only to degeneracies (contraction
epimorphisms) on the side of positive opetopes. This leaves of course an open
question of whether we can extend tree complexes one way or the other,
introducing some kind of sink nodes, so that we could have duality for more
maps, e.g. all $\iota$-maps, and not only $\iota$-epimorphisms, or even all
${\omega}$-functors?

The paper is organized as follows. In Section 2 we define a simplification of
both constellations and zoom complex that were originally introduced in
[KJBM], here called tree complexes, and the maps of both constellations and
tree complexes. In Section 3, we describe duality for the category ${\bf
pOpe}_{\iota}$ of opetopes with contraction epimorphisms on one side and the
category ${\bf cTree}$ of tree complexes and tree complex maps. In Section 4,
we present the extension of this duality to larger categories of ${\bf
pOpeCard}_{\iota}$ of opetopic cardinals with contraction epimorphisms and
${\bf cThicket}$ of thicket complexes and thicket complex maps. The paper ends
with an appendix where the relevant notions and facts concerning positive
opetopes are recalled from [Z1] and [Z4].

##  2 The category of tree complexes

###  2.1 Trees

A tree is a non-empty finite poset with binary sups and no infs of non-
linearly ordered subsets. In particular, tree can be empty but if it is not,
it has the largest element, called root. A tree embedding is a one-to-one
function that preserves and reflects order. $\bf Tree$ is the category of
trees and tree embeddings. We will also consider other kinds of (monotone)
morphisms of trees: sup-morphisms (= preserving suprema), monotone maps
(automatically preserving infs), onto maps.

Construction. Let $S$, $T$ be trees. Let $S_{\perp}$ denotes the poset
obtained by adding bottom element to $S$. Then $S_{\perp}$ has both sups and
infs, i.e. it is a lattice. If $D\subseteq S$ is a proper downward closed
subset of $S$, then $S-D$ is again a tree. Any monotone map $f:S-D\rightarrow
T$ can be extended to an infs preserving map $f_{*}:S_{\perp}\rightarrow
T_{\perp}$ (sending $D$ and $\perp$ to $\perp$). $f_{*}$ has a left adjoint
$f^{*}:T_{\perp}\rightarrow S_{\perp}$. If $f$ is onto, then $f^{*}$ reflects
$\perp$, and hence it restricts to a sup-preserving morphism again named
$f^{*}:T\rightarrow S$.

Some notions and notation concerning trees. Let $T$ be a tree,
$t,t^{\prime}\in T$.

  1. 1.

$t\prec t^{\prime}$ means that $t$ is a son of $t^{\prime}$ or $t^{\prime}$ is
the successor of $t$.

  2. 2.

The suprema (infima) of a subset $X$ of a poset $T$ will be denoted by
$\rm{sup}^{T}(X)$ or $\rm{sup}(X)$, ($\inf^{T}(X)$ or $\inf(X)$) and if
$X=\\{t,t^{\prime}\\}$, we can also write $t\vee^{T}t^{\prime}$ of $t\vee
t^{\prime}$ ($t\wedge^{T}t^{\prime}$ of $t\wedge t^{\prime}$).

  3. 3.

A subposet $X$ of a tree $T$ is a convex subtree of $T$ iff $X$ has the
largest element, and whenever $x,x^{\prime}\in X$, $s\in T$ and
$x<s<x^{\prime}$, then $s\in X$. Clearly a convex subset of a tree is in
particular a non-empty tree. Let ${\bf St}(T)$ denote the poset of the convex
sub-trees of the tree $T$.

  4. 4.

$t\perp t^{\prime}$ means that either $t\leq t^{\prime}$ or $t^{\prime}\leq
t$.

  5. 5.

$t$ is a leaf in $T$ iff the set $\\{s\in T\,:\,s\prec t\\}$ is empty.
${\rm{lvs}}(T)$ denotes the set of leaves of the tree $T$.

  6. 6.

${\rm{lvs}}^{T}(t)$ is the set of leaves of the tree $T$ over the element $t$,
i.e.

${\rm{lvs}}^{T}(t)=\\{s\in{\rm{lvs}}(T)\,:\,s\leq t\\}.$

  7. 7.

$\rm{cvr}^{T}(t)$ is the cover of the element $t$ in the tree $T$

$\rm{cvr}^{T}(t)=\\{s\in T\,:\,s\prec t\\},$

i.e. the set of elements of the tree $T$ whose successor is $t$.

  8. 8.

If $X$ is a convex subtree of $T$, then the cover of $X$ in the tree $T$ is
the set

$\rm{cvr}^{T}(X)=\bigcup_{x\in X}\rm{cvr}^{T}(x)-X.$

Note that $\rm{cvr}^{T}(t)=\rm{cvr}^{T}(\\{t\\})$, so the notation for value
of $\rm{cvr}^{T}$ on elements and convex subsets is compatible.

We have an easy Lemma establishing some relation between the above notions. It
will be needed for the proof of the duality.

######  Lemma 2.1.

Let $S$ be a tree.

  1. 1.

Let $T$ be a convex subtree of $S$ not containing leaves. Then the family set
$\\{{\rm{lvs}}^{S}(s)\\}_{s\in\rm{cvr}^{S}(T)}$ is a partition of the set
${\rm{lvs}}^{S}(\rm{sup}(T))$.

  2. 2.

Let $T$ be a convex subtree of $S$ not containing leaves and let
$\\{T_{i}\\}_{i\in I}$ be a partition of $T$. Then, for $t\in S$

    1. (a)

$t\in\rm{cvr}^{S}(T_{i})$, for some $i\in I$, iff either $t\in cvr^{S}(T)$ or
there is ${j\in I}$ such that $t=\rm{sup}(T_{j})$;

    2. (b)

$t=\rm{sup}^{S}(T_{i})$ iff either $t=sup(T)$ or there is $j\in I$ such that
$t\in\rm{cvr}^{S}(T_{j})$. $~{}~{}\Box$

###  2.2 Constellations

A constellation is a triple $(S_{1},\sigma,S_{0})$, where $S_{0}$ and $S_{1}$
are trees and $\sigma$ is a monotone function

$\sigma:S_{1}\rightarrow{\bf St}(S_{0})$

such that

  1. 1.

it preserves top element;

  2. 2.

if $s,s^{\prime}\in S_{1}$ and $\sigma(s)\cap\sigma(s^{\prime})\neq\emptyset$,
then $s\perp s^{\prime}$.

Let $\sigma:S_{1}\rightarrow{\bf St}(S_{0})$ be a constellation. Then the
constellation tree $(S_{1}\lhd_{\sigma}S_{0},\leq^{co})$ is the tree arising
by extension of the tree $S_{1}$ by nodes of the tree $S_{0}$ added as leaves,
so that if $s_{0}\in S_{0}$ and $s_{1}\in S_{1}$, then $s_{0}$ is a leaf over
$s_{1}$ iff $s_{0}\in\sigma(s_{1})$.

Formally, the set $S_{1}\lhd_{\sigma}S_{0}$ is a disjoint sum of $S_{1}$ and
$S_{0}$. If $s_{0}\in S_{0}$, and $s_{1}\in S_{1}$, then the corresponding
elements in $S_{1}\lhd_{\sigma}S_{0}$ are denoted by $s_{0}^{\bullet}$ and
$s_{1}^{\circ}$, respectively. The constellation order $\leq^{co}$ in
$S_{1}\lhd_{\sigma}S_{0}$ is defined as follows. If $s_{0},t_{0}\in S_{0}$ and
$s_{1},t_{1}\in S_{1}$, then

  1. 1.

$s_{1}^{\circ}\leq^{co}t_{1}^{\circ}$ iff $s_{1}\leq^{T}t_{1}$;

  2. 2.

$s_{0}^{\bullet}\leq^{co}s_{1}^{\circ}$ iff $s_{0}\in\sigma(s_{1})$;

  3. 3.

$s_{0}^{\bullet}\leq^{co}t_{0}^{\bullet}$ iff $s_{0}=t_{0}$;

  4. 4.

$s_{1}^{\circ}\leq^{co}s_{0}^{\bullet}$ never holds.

Clearly $(S_{1}\lhd_{\sigma}S_{0},\leq^{co})$ is again a tree. We often drop
the index $\sigma$ in $S_{1}\lhd_{\sigma}S_{0}$ when it does not lead to a
confusion.

Let $(S_{1},\sigma,S_{0})$ and $(T_{1},\tau,T_{0})$ be two constellations. Any
pair of tree embeddings $f_{0}:S_{0}\rightarrow T_{0}$,
$f_{1}:S_{1}\rightarrow T_{1}$ induces a tree embedding of constellation
orders

$f_{1}\lhd f_{0}:S_{1}\lhd_{\sigma}S_{0}\longrightarrow
T_{1}\lhd_{\tau}T_{0}.$

Such a pair is a morphism of constellations

$(f_{1},f_{0}):(S_{1},\sigma,S_{0})\longrightarrow(T_{1},\tau,T_{0})$

iff the induced map $f_{1}\lhd f_{0}$ preserves binary sups. The category of
constellations will be denoted by $\bf Constell$.

Remarks and notation.

  1. 1.

The fibers of any constellation $\sigma$ are linearly ordered.

  2. 2.

Let $\sigma_{\perp}:(S_{1})_{\perp}\rightarrow{\bf St}(S_{0})_{\perp}$ be the
extension of the function $\sigma$ sending $\perp$ to $\perp$. Then $\sigma$
is a constellation iff $\sigma_{\perp}$ preserve all infs iff $\sigma$
preserves top element and preserves and reflects binary infs, i.e.

    1. (a)

$\sigma(sup(S_{1}))=S_{0}$;

    2. (b)

and for $s,s^{\prime}\in S_{1}$,
$\sigma(s)\cap\sigma(s^{\prime})\neq\emptyset$ iff $s\wedge s^{\prime}$ is
defined and then $\sigma(s)\cap\sigma(s^{\prime})=\sigma(s\wedge s^{\prime})$.

  3. 3.

One can look at a single constellation $(S_{1},\sigma,S_{0})$ as a way of
gluing elements of a poset $S_{0}$ as new leaves in the posted $S_{1}$ along
the function $\sigma$. Thus the order of $S_{0}$ is not essential for building
one constellation order. The constellation order can be graphicly drawn with
leaves from $S_{0}$, also called vertices, marked as dots, and inner nodes
from $S_{1}$, also called circles, marked as circles, enclosing all the leaves
under them and all the smaller circles. General elements of constellation
orders are often called nodes.

  4. 4.

Let $(T_{1},\tau_{0},T_{0})$ and $(T_{2},\tau_{1},T_{1})$ be two (consecutive)
constellations. We can form a diagram

$T_{2}\vphantom{T_{2}\lhd T_{1}}$$\vphantom{T_{2}}T_{2}\lhd
T_{1}$$\stackrel{{{}_{\circ}}}{{(-)}}$$\vphantom{\phantom{T_{2}\lhd
T_{1}}}T_{1}$$\stackrel{{{}_{\bullet}}}{{(-)}}$$\vphantom{\phantom{T_{1}}}T_{1}\lhd
T_{0}$$\stackrel{{{}_{\circ}}}{{(-)}}$$\vphantom{\phantom{T_{1}\lhd
T_{0}}}T_{0}$$\stackrel{{{}_{\bullet}}}{{(-)}}$

consisting of two coproduct diagrams.

    1. (a)

If $t\in T_{1}$, then $t^{\bullet}\in T_{1}\lhd T_{0}$ and $t^{\circ}\in
T_{2}\lhd T_{1}$. So the node $t$, depending on the order in which we consider
it, can be either a vertex (leaf) or a circle (inner node). As we will deal
with this situation very often, we will usually be careful to distinguish
these two roles, when it may cause confusions, by putting either circle of dot
over the node considered.

    2. (b)

Moreover, for $X\subset T_{1}$ we use very often the notation
$X^{\bullet}\subseteq T_{1}\lhd T_{1}$ and
$X^{\circ}=X^{\bullet\circ}\subseteq T_{1}\lhd T_{0}$.

###  2.3 Tree complexes

A tree complex $(T,\tau)$ is a sequence of constellations:

$\tau_{0}:T_{1}\rightarrow{\bf St}(T_{0}),$ $\tau_{1}:T_{2}\rightarrow{\bf
St}(T_{1}),$ $\ldots$ $\tau_{i}:T_{i+1}\rightarrow{\bf St}(T_{i}),$ $\ldots$

for $i\in{\omega}$, with almost all sets $T_{i}$ empty. The dimension
$(T,\tau)$ is $n$ iff $T_{n}$ be the last non-empty set. We write $dim(T)$ for
dimension of the tree complex $(T,\tau)$. $T_{0}$ as well as $T_{dim(T)}$ are
required to be singletons. We sometimes tacitly assume that
$T_{0}=\\{\bullet_{T}\\}$ and $T_{n}=\\{m_{T}\\}$.

A morphism of tree complexes $f:(S,\sigma)\rightarrow(T,\tau)$ is a family of
tree embeddings $f_{i}:S_{i}\rightarrow T_{i}$, for $i\in{\omega}$, such that,
for $i\in{\omega}$,

$(f_{i+1},f_{i}):(S_{i+1},\sigma_{i},S_{i})\longrightarrow(T_{i+1},\tau_{i},T_{i})$

is a morphism of constellations, i.e. the tree embeddings

$\vec{f}_{i}=f_{i+1}\lhd f_{i}:S_{i+1}\lhd_{\sigma_{i}}S_{i}\longrightarrow
T_{i+1}\lhd_{\tau_{i}}T_{i}$

that preserve binary sups.

The category of tree complexes and their morphisms will be denoted by ${\bf
cTree}$.

##  3 Duality

###  3.1 From positive opetopes to tree complexes

In this section we define a functor

${\bf pOpe}_{\iota,epi}\vphantom{{\bf cTree}^{op}}$$\vphantom{{\bf
pOpe}_{\iota,epi}}{\bf cTree}^{op}$$(-)^{*}$

Let $P$ be a p-opetope. We shall define a tree complex $(P^{*},\pi)$. For
$i\in{\omega}$, the set

$P^{*}_{i}=(\delta\gamma^{(i+1)}(m_{P}),\leq^{-})=(P_{i}-\gamma(P_{i+1}),\leq^{-}),$

is the set of faces of the $i$-th tree of the tree complex $(P^{*},\pi)$. The
$i$-th constellation map

$\pi_{i}:P^{*}_{i+1}\longrightarrow{\bf St}(P^{*}_{i})$

is given, for $p\in P^{*}_{i+1}$, by

$\pi_{i}(p)=\\{s\in P^{*}_{i}:s<^{+}\gamma(p)\\}.$

NB. If $p,p^{\prime}\in P^{*}_{i+1}$, then $p<^{-}p^{\prime}$ iff
$\gamma(p)<^{+}\gamma(p^{\prime})$.

With the notation as above, we have

######  Proposition 3.1.

For $i\in{\omega}$, the triple $(P^{*}_{i+1},\pi_{i+1},P^{*}_{i})$ defined
above is a constellation. Thus $(P^{*},\pi)$ is a tree complex.

Proof. First we show that, for $p\in P_{i+1}^{*}$, $\pi_{i}(p)$ is a convex
subtree of $(P^{*}_{i},<^{-})$. The $<^{+}$-least element in the
$\gamma$-pencil of $\gamma\gamma(p)$ is the largest element of $\pi_{i}(p)$.
Let $p_{1},p_{3}\in\pi_{i}(p)$ and $p_{2}\in P^{*}_{i+1}$ such that
$p_{1}<^{-}p_{2}<^{-}p_{3}$. Thus there is a maximal lower
$P^{*}_{i}=P_{i}-\gamma(P^{*}_{i+1})$-path containing $p_{1},p_{2},p_{3}$.
Thus, by Path Lemma (cf. [Z1] or Appendix), $p_{2}<^{+}\gamma(p)$.

Next we show that $\pi_{i}:P_{i+1}^{*}\rightarrow P_{i}^{*}$ is monotone. Let
$p,p^{\prime}\in P_{i+1}^{*}$ so that $p<^{-}p^{\prime}$. Then
$\gamma(p)<^{+}\gamma^{+}(p^{\prime})$ and hence

$\pi(p)=\\{s\in P^{*}_{i}:s<^{+}\gamma(p)\\}\subseteq\\{s\in
P^{*}_{i}:s<^{+}\gamma(p^{\prime})\\}=\pi(p^{\prime}),$

as required.

Finally, we will show that if $p,p^{\prime}\in P_{i+1}^{*}$ $s\in P^{*}_{i}$
and $s\in\pi_{i}(p)\cap\pi_{i}(p^{\prime})$, then $p\perp^{-}p^{\prime}$. Thus
$s<^{+}\gamma(p)$ and $s<^{+}\gamma(p^{\prime})$. Let $r_{1},\ldots,r_{k}$ be
a maximal lower $P^{*}_{i+1}$-path such that, for some $j$,
$s\in\delta(r_{j})$. Then both $p$ and $p^{\prime}$ must occur in this path.
So $p\perp^{-}p^{\prime}$, as required. $~{}~{}\Box$

We define a poset morphism

$\varepsilon_{P,i}:(P^{*}_{i+1}\lhd
P^{*}_{i},<^{co})\longrightarrow(P_{i},<^{+})$

as follows. For $p\in P^{*}_{i+1}\lhd P^{*}_{i}$, we put

$\varepsilon_{P,i}(p)\;=\;\left\\{\begin{array}[]{ll}s&\mbox{ if
}p=s^{\bullet}\mbox{ for some }s\in P^{*}_{i},\\\ \gamma(s)&\mbox{ if
}p=s^{\circ}\mbox{ for some }s\in P^{*}_{i+1}.\\\ \end{array}\right.$

######  Lemma 3.2.

The morphism $\varepsilon_{P,i}$ defined above is an order isomorphism.

Proof. If $p_{1},p_{2}\in P^{*}_{i+1}=P_{i+1}-\gamma(P_{i+2})$ and $p_{1}\neq
p_{2}$, then $\gamma(p_{1})\neq\gamma(p_{2})$. Moreover, if $p\in
P^{*}_{i}=P_{i}-\gamma(P_{i+1})$, then
$\varepsilon_{P,i}(p^{\bullet})=p\neq\gamma(p_{1})=\varepsilon_{P,i}(p_{1}^{\circ})$.
Thus $\varepsilon_{P,i}$ is one-to-one. It is onto as well, since
$P_{i}=(P_{i}-\gamma(P_{i+1})\cup(\gamma(P_{i+1}-\gamma(P_{i+2}))$.

It remains to show that $\varepsilon_{P,i}$ preserves and reflects order. We
have that

$p^{\bullet}\;<^{co}\;p_{1}^{\circ}\;\;{\rm
iff}\;\;p\;<^{+}\;\gamma(p_{1})\;\;{\rm
iff}\;\;\varepsilon_{P,i}(p^{\bullet})\;<^{+}\;\varepsilon_{P,i}(p_{1}^{\circ}).$

Moreover

$p_{1}^{\circ}\;<^{co}\;p_{2}^{\circ}\;\;{\rm
iff}\;\;p_{1}\;<^{-}\;p_{2}\;\;{\rm
iff}\;\;\gamma(p_{1})\;<^{+}\;\gamma(p_{2})\;\;{\rm
iff}\;\;\varepsilon_{P,i}(p_{1}^{\circ})\;<^{+}\;\varepsilon_{P,i}(p_{2}^{\circ}).$

The other cases are obvious. $~{}~{}\Box$

Let $f:P\rightarrow Q$ be a $\iota$-epimorphism of p-opetopes. We define a
tree complex map

$f^{*}=\\{f^{*}_{i}\\}_{i\in{\omega}}:(Q^{*},\pi)\longrightarrow(P^{*},\pi).$

For $i\in{\omega}$, the map $f^{*}_{i}:Q^{*}_{i}\longrightarrow P^{*}_{i}$ is
defined as follows. Let $q\in Q^{*}_{i}=Q_{i}-\gamma(Q_{i+1})$, $p\in
P^{*}_{i}=P_{i}-\gamma(P_{i+1})$, $0\leq i$. Then

$f^{*}_{i}(q)=p$

iff $p$ is the unique element of $P_{i}-\gamma(P_{i+1})$ so that $f(p)=q$.
Such an element exists since $f_{i}$ is epi.

We can also describe the above map using the construction from Section 2.1, as
follows. We have a monotone onto map
$f_{i}:(P_{i}-\ker(f),\leq^{+})\rightarrow(Q_{i},\leq^{+})$. As
$P_{i}\cap\ker(f)$ is a proper downward closed subset of $P_{i}$, $f_{i}$
extends to an all infs preserving map $f_{i,*}:P_{i,\perp}\rightarrow
Q_{i,\perp}$ sending $P_{i}\cap\ker(f)$ to $\perp$. Thus it has a left adjoint
$\bar{f}_{i}:Q_{i,\perp}\rightarrow P_{i,\perp}$. Clearly, $\bar{f}_{i}$,
defined this way, preserves sups. For $q\in Q_{i}$, $\bar{f}_{i}(q)$ picks the
least element in the fiber of the function $f_{i}$ over element $q$. We have

######  Lemma 3.3.

With the notation as above, the following diagram

$(P_{i})_{\perp}\vphantom{(Q_{i})_{\perp}}$$\vphantom{(P_{i})_{\perp}}(Q_{i})_{\perp}$$\bar{f}_{i}$$(P^{*}_{i+1}\lhd
P^{*}_{i})_{\perp}\vphantom{(Q^{*}_{i+1}\lhd
Q^{*}_{i})_{\perp}}$$\vphantom{(P^{*}_{i+1}\lhd
P^{*}_{i})_{\perp}}(Q^{*}_{i+1}\lhd Q^{*}_{i})_{\perp}$$(f^{*}_{i+1}\lhd
f^{*}_{i})_{\perp}$$(\varepsilon_{P,i})_{\perp}$$(\varepsilon_{Q,i})_{\perp}$

commutes. In particular, $f^{*}_{i+1}\lhd f^{*}_{i}$ preserves binary sups.

Proof. Let $q^{\bullet}\in Q^{*}_{i+1}\lhd Q^{*}_{i}$ and let $f^{*}_{i+1}\lhd
f^{*}_{i}(q^{\bullet})=p^{\bullet}$. Thus $f_{i,*}(p)=q\in Q^{*}_{i+1}$ and,
as $p$ is a leaf, $\bar{f}_{i}(q)=p$. Hence the square commutes in this case.

Now, let $q^{\circ}\in Q^{*}_{i+1}\lhd Q^{*}_{i}$ and let $f^{*}_{i+1}\lhd
f^{*}_{i}(q^{\circ})=p^{\circ}$. Thus $f_{i+1,*}(p)=q$. So $p$ is indeed in
the fiber of $f_{i+1,*}$ over $q$. We need to show that $p$ is minimal in this
fiber. Suppose to the contrary that there is $p^{\prime}\in P_{i}$ such that
$p^{\prime}<^{+}\gamma(p)$ and yet $f_{i+1,*}(p^{\prime})=q$. Then, as $p\in
P_{i+1}-\gamma(P_{i+2})$ and $p^{\prime}<^{+}\gamma(p)$, there is a
$p^{\prime\prime}\in\delta(p)$ such that $p^{\prime}\leq^{+}p^{\prime\prime}$.
We have
$q=f_{i+1,*}(p^{\prime})\leq^{+}f_{i+1,*}(p^{\prime\prime})\leq^{+}f_{i+1,*}(\gamma(p))=q$.
But then $f_{i+1,*}(p^{\prime\prime})=q=f_{i+1,*}(\gamma(p))$ and hence
$p\in\ker(f)$, contrary to the supposition. Thus the diagram commutes in this
case, as well. $~{}~{}\Box$

As a corollary of Lemma 3.3, we get

######  Proposition 3.4.

Let $f:(P,\gamma,\delta)\rightarrow(Q,\gamma,\delta)$ be an epi $\iota$-map of
positive opetopes. Then the family of maps
$f^{*}=\\{f^{*}_{i}\\}_{i\in{\omega}}:(Q^{*},\pi)\longrightarrow(P^{*},\pi)$
defined above is a morphism of tree complexes. $~{}~{}\Box$

Examples. We explain below in more detail the correspondence from the
introduction between the $\iota$-epimorphism $f:O_{3}\rightarrow O_{2}$ and
its dual tree complex embedding.

  1. 1.

The dual of the opetope $Q=O_{2}$ of dimension 2

$t_{3}\vphantom{t_{1}}$$\vphantom{t_{3}}t_{1}$$y_{1}$$t_{2}$$y_{3}$$y_{2}$$\Downarrow\\!b$

is the tree complex $Q^{*}=T_{2}$ with nodes

$Q^{*}_{0}=\\{t_{3}\\},$ $Q^{*}_{1},=\\{y_{2}>y_{3}\\},$ $Q^{*}_{2},=\\{b\\},$

and the constellation maps

$\pi_{0}(y_{2})=\\{t_{3}\\},$ $\pi_{0}(y_{3})=\\{t_{3}\\},$

$\pi_{1}(b)=\\{y_{2},y_{3}\\}.$

Such a tree complex $Q^{*}$ can be drawn as follows:

dim 0dim 1dim
2$\bullet_{t_{3}^{\bullet}}$${}_{y_{3}^{\circ}}$${}_{y_{2}^{\circ}}$$\bullet_{y_{2}^{\bullet}}$$\bullet_{y_{3}^{\bullet}}$${}_{b^{\circ}}$$\bullet^{b^{\bullet}}$

  2. 2.

The dual of the opetope $P=O_{3}$ of dimension 3

$t_{3}\vphantom{t_{1}}$$\vphantom{t_{3}}t_{1}$$y_{3}$$t_{2}$$y_{5}$$y_{4}$$\Downarrow\\!b_{3}$$t_{4}\vphantom{}$$y_{6}$$\vphantom{}t_{0}$$y_{1}$$y_{0}$$y_{2}$$\Downarrow\\!b_{2}$$\Downarrow\\!b_{1}$$\Longrightarrow$$\beta$$t_{3}\vphantom{t_{1}}$$\vphantom{t_{3}}t_{1}$$t_{2}$$y_{5}$$y_{4}$$\Downarrow\\!b_{0}$$t_{4}\vphantom{}$$y_{6}$$\vphantom{}t_{0}$$y_{1}$$y_{0}$

is a tree complex $P^{*}$ with nodes

$P^{*}_{0}=\\{t_{4}\\},$ $P^{*}_{1},=\\{y_{1}>y_{4}>y_{5}>y_{6}\\},$
$P^{*}_{2},=\\{b_{1}>b_{2}>b_{3}\\},$ $P^{*}_{3},=\\{\beta\\},$

and the constellation maps

$\pi_{0}(y_{1})=\\{t_{4}\\},$ $\pi_{0}(y_{4})=\\{t_{4}\\},$
$\pi_{0}(y_{5})=\\{t_{4}\\},$ $\pi_{0}(y_{6})=\\{t_{4}\\},$

$\pi_{1}(b_{1})=\\{y_{1},y_{4},y_{5},y_{6},\\},$
$\pi_{1}(b_{2})=\\{y_{4},y_{5},y_{6},\\},$
$\pi_{1}(b_{3})=\\{y_{4},y_{5},\\},$

$\pi_{2}(\beta)=\\{b_{1},b_{2},b_{3}.\\}.$

Such a tree complex $P^{*}$ can be drawn as follows:

dim 0dim 1dim 2dim
3$\bullet$${}_{t_{4}^{\bullet}}$${}_{y_{6}^{\circ}}$${}_{y_{5}^{\circ}}$${}_{y_{4}^{\circ}}$${}_{y_{1}^{\circ}}$$\bullet_{y_{1}^{\bullet}}$$\bullet^{y_{4}^{\bullet}}$$\bullet_{y_{5}^{\bullet}}$$\bullet^{y_{6}^{\bullet}}$${}_{b_{3}^{\circ}}$${}_{b_{2}^{\circ}}$${}_{b_{1}^{\circ}}$$\bullet_{b_{1}^{\bullet}}$$\bullet_{b_{2}^{\bullet}}$$\bullet_{b_{3}^{\bullet}}$${}_{\beta^{\circ}}$$\bullet^{\beta^{\bullet}}$

  3. 3.

The dual of the $\iota$-epimorphism $f:P\rightarrow Q$ from opetope $P=O_{3}$
to the opetope $Q=O_{2}$ given by

$f_{0}(t_{0})=f_{1}(y_{1})=f_{0}(t_{1})=t_{1},$
$f_{0}(t_{2})=f_{1}(y_{5})=f_{0}(t_{3})=t_{2},$ $f_{0}(t_{4})=t_{3},$

$f_{1}(y_{2})=f_{2}(b_{1})=f_{1}(y_{0})=y_{1},$
$f_{1}(y_{4})=f_{2}(b_{3})=f_{1}(y_{3})=y_{2},$ $f_{1}(y_{6})=y_{3},$

$f_{2}(b_{2})=f_{3}(\beta)=f_{2}(b_{0})=b,$

is a morphism of tree complexes $f^{*}:Q^{*}\rightarrow P^{*}$ such that

$f^{*}_{0}(t_{3})=t_{4},$

$f^{*}_{1}(y_{2})=y_{4},$ $f^{*}_{1}(y_{3})=y_{6},$

$f^{*}_{2}(b)=b_{2}.$

###  3.2 From tree complexes to positive opetopes

In this section we define a functor

${\bf pOpe}_{\iota,epi}\vphantom{{\bf cTree}^{op}}$$\vphantom{{\bf
pOpe}_{\iota,epi}}{\bf cTree}^{op}$$(-)^{*}$

Let $(S,\sigma)=\\{S_{i},\sigma_{i}\\}_{i\in{\omega}}$ be a tree complex. We
define the p-opetope $\\{S^{*}_{i},\gamma^{i},\delta^{i}\\}_{i\in{\omega}}$,
as follows. We put

$(S^{*}_{i},<^{co})=(S_{i+1}\lhd_{\sigma_{i}}S_{i},<^{co}),$

i.e. the set $S^{*}_{i}$ of $i$-dimensional faces of the p-opetope $S^{*}$ is
the universe of the $i$-th constellation poset of $(S,\sigma)$. Later we shall
prove that the constellation order $<^{co}$ agree with the upper order
$<^{+}$, defined using the operations $\gamma$ and $\delta$ below.

Let $i\in{\omega}$. The $(i-$th) codomain operation

$\gamma:S^{*}_{i+1}\longrightarrow S^{*}_{i}$

is defined, for $p\in S^{*}_{i+1}$, as follows

$\gamma(p)\;=\;\rm{sup}\\!\,^{S^{*}_{i}}({\rm{lvs}}^{S^{*}_{i+1}}(p)^{\circ}).$

In words

  1. 1.

if the face $p$ is a node, i.e. $p=t^{\bullet}$, then its codomain
$\gamma(t^{\bullet})=t^{\circ}$, i.e. it is ‘the same’ $t$ but considered as a
circle one dimension below;

  2. 2.

if the face $p$ is a circle, i.e. $p=t^{\circ}$, then its codomain $\gamma(p)$
is the circle $s^{\circ}$ whose corresponding node $s^{\bullet}$ is the
supremum $\rm{sup}^{S^{*}_{i}}({\rm{lvs}}^{S^{*}_{i+1}}(t^{\circ})^{\circ})$
in $S^{*}_{i}$ of the leaves in $S^{*}_{i+1}$ over $t^{\circ}$ considered as
circles one dimension below, in $S^{*}_{i}$.

The $(i-$th) domain operation

$\delta:S^{*}_{i+1}\longrightarrow{{\cal P}}_{\neq\emptyset}(S^{*}_{i})$

is defined, for $p\in S^{*}_{i+1}$, as follows

$\delta(p)\;=\;\rm{cvr}^{S^{*}_{i}}({\rm{lvs}}^{S^{*}_{i+1}}(p)^{\circ}).$

In words

  1. 1.

if the face $p$ is a node, i.e. $p=t^{\bullet}$, then its domain
$\delta(t^{\bullet})=\rm{cvr}^{S^{*}_{i}}(t^{\circ})$, i.e. it is the cover of
‘the same’ $t$ but considered as a circle one dimension below;

  2. 2.

if the face $p$ is a circle, i.e. $p=t^{\circ}$, then its domain
$\delta(t^{\circ})$ is the sum of $\delta$’s applied to the leaves over
$t^{\circ}$ in $S^{*}_{i+1}$ considered as circles one dimension below in
$S^{*}_{i}$ minus these leaves considered as circles.

######  Lemma 3.5.

Let $(S,\sigma)$ be a tree complex. Then the face structure
$(S,\sigma)^{*}=(S^{*},\gamma,\delta)$ defined above is a positive opetope.

Proof. Let $(S^{*},\gamma,\delta)$ a face structure as defined above. We shall
check that it satisfies the axioms of positive opetopes.

Globularity. We shall use Lemma 2.1.

Fix $s\in S^{*}_{i+2}$, for some $i\geq 0$. Then ${\rm{lvs}}^{i+2}(s)^{\circ}$
is a convex subtree of $S^{*}_{i+1}$ not containing leaves. Thus, by Lemma
2.1.1, the family of sets

$\\{{\rm{lvs}}^{i+1}(r)\\}_{r\in\rm{cvr}^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ})}$

is a partition of the set

${\rm{lvs}}^{i+1}(\rm{sup}\\!^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ})).$

If $r=r^{\prime\bullet}\in\rm{cvr}^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ})$, for
some $r^{\prime}\in S_{i+2}$, then
${\rm{lvs}}^{i+1}(r)^{\circ}=r^{\prime\circ}$. If
$r=r^{\prime\circ}\in\rm{cvr}^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ})$, for some
$r^{\prime}\in S_{i+2}$, then ${\rm{lvs}}^{i+1}(r)^{\circ}=\sigma_{i+1}(r)$.
Thus in any case ${\rm{lvs}}^{i+1}(r)^{\circ}$ is a convex subtree, and hence
the partition, give rise to the partition of a convex subtree

${\rm{lvs}}^{i+1}(\rm{sup}\\!^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ}))^{\circ},$

into a family of convex subtrees

$\\{{\rm{lvs}}^{i+1}(r)\\}_{r\in\rm{cvr}^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ})}^{\circ}.$

.

Then, using Lemma 2.1.2, we get

$\gamma\gamma(s)=$

$=\rm{sup}\\!^{i}({\rm{lvs}}^{i+1}(\rm{sup}\\!^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ}))^{\circ})=$

$=\bigcup_{r\in\rm{cvr}^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ})}\rm{sup}\\!^{i}({\rm{lvs}}^{i+1}(r)^{\circ})-\bigcup_{r\in\rm{cvr}^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ})}\rm{cvr}^{i}({\rm{lvs}}^{i+1}(r)^{\circ})=$

$=\gamma\delta(s)-\delta\delta(s),$

and

$\delta\gamma(s)=$

$=\rm{cvr}^{i}({\rm{lvs}}^{i+1}(\rm{sup}\\!^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ}))^{\circ})=$

$=\bigcup_{r\in\rm{cvr}^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ})}\rm{cvr}^{i}({\rm{lvs}}^{i+1}(r)^{\circ})-\bigcup_{r\in\rm{cvr}^{i+1}({\rm{lvs}}^{i+2}(s)^{\circ})}\rm{sup}\\!^{i}({\rm{lvs}}^{i+1}(r)^{\circ})=$

$=\delta\delta(s)-\gamma\delta(s),$

as required.

Strictness.

We shall show that the transitive relation $<^{+}$, defined using $\gamma$’s
and $\delta$’s, coincides with the constellation order $<^{co}$.

Let $s,s^{\prime\circ}\in S_{i+1}\lhd S_{i}$, for some $0\leq i\leq dim(S)$.
Then $\gamma(s^{\prime\bullet})=s^{\prime\circ}$. Moreover,
$s\prec^{co}s^{\prime\circ}$ iff
$s\in\rm{cvr}^{i}(s^{\prime\circ})=\delta(s^{\prime\bullet})$. The latter
condition means that $s<^{+}s^{\prime\circ}$. Thus $\prec^{co}\subseteq<^{+}$.

It remains to show that $<^{+}\subseteq<^{co}$. Assume $s,s^{\prime}\in
S_{i+1}\lhd S_{i}$ and that there is $r\in S_{i+2}\lhd S_{i+1}$ such that
$s\in\delta(r)$ and $\gamma(r)=s^{\prime}$. We shall show that
$s<^{co}s^{\prime}$.

If $r=r^{\prime\bullet}$, for some $r^{\prime}\in S_{i+1}$, then
$s^{\prime}=r^{\prime\circ}$ and $s\in\rm{cvr}^{i}(r^{\prime\circ})$ so
$s\prec^{co}s^{\prime}$ indeed.

Now assume that $r=r^{\prime\circ}$, for some $r^{\prime}\in S_{i+2}$. Let
$s=s_{0}\in\delta(r^{\prime\circ})=\rm{cvr}^{i}({\rm{lvs}}^{i+1}(r^{\prime\circ})^{\circ})$.
Let $s_{1}$ be the $<^{co}$-successor of $s_{0}$, i.e. $s_{0}\prec^{co}s_{1}$.
Since ${\rm{lvs}}^{i+1}(r^{\prime\circ})$ is a convex tree, there is a path
$s_{1},\ldots,s_{k}$ in ${\rm{lvs}}^{i+1}(r^{\prime\circ})^{\circ}$ such that
$s_{i}\prec^{co}s_{i+1}$ and

$s_{k}=\rm{sup}^{i}({\rm{lvs}}^{i+1}(r^{\prime\circ})^{\circ})=\gamma(r^{\circ})=s^{\prime}.$

Thus $s=s_{0}<^{co}s_{k}=s^{\prime}$, as required.

Disjointness.

Let $s,t\in S^{*}_{i+1}$ such that $s<^{+}t$. Then
${\rm{lvs}}^{i+1}(s)\subseteq{\rm{lvs}}^{i+1}(t)$. Hence

$\gamma(s)=\rm{sup}\\!^{i}({\rm{lvs}}^{i+1}(s)^{\circ})\not\in\rm{cvr}^{i}({\rm{lvs}}^{i+1}(t)^{\circ})=\delta(t),$

i.e. $s\not<^{-}t$.

On the other hand, if $s^{\prime}\in\delta(s)$, then

$s^{\prime}<^{+}\gamma(s)=\rm{sup}\\!^{i}({\rm{lvs}}^{i+1}(s)^{\circ})\leq^{+}\rm{sup}\\!^{i}({\rm{lvs}}^{i+1}(t)^{\circ})=\gamma(t),$

i.e. $\gamma(t)\not\in\delta(s)$ and $t\not<^{-}s$, as well.

Thus if $s<^{+}t$, then $s\not\perp^{-}t$, as required.

Pencil linearity.

Let $s,t\in S^{*}_{i}$, for some $0\leq i\leq dim(S)$.

Assume that $s\neq t$ and $\gamma(s)=\gamma(t)$. Then $s$ and $t$ cannot be
leaves at the same time. If $s$ is a leaf, then $s<^{co}t$ and hence
$s<^{+}t$, by the above. If both $s$ and $t$ are inner nodes, then

$\sigma_{i}(s)^{\circ}\ni\rm{sup}^{i-1}({\rm{lvs}}^{i}(s)^{\circ})=\rm{sup}^{i-1}({\rm{lvs}}^{i}(t)^{\circ})\in\sigma_{\it
i}(t)^{\circ}.$

Thus $\sigma_{i}(s)\cap\sigma_{i}(t)\neq\emptyset$, and, as $\sigma_{i}$ is a
constellation, we have $s\perp^{+}t$.

Now assume that there is $r\in\delta(s)\cap\delta(t)$. Let $s_{1}^{\circ}$ be
the successor of $r$, i.e. $r\prec^{co}s_{1}^{\circ}$. Hence
$s_{1}^{\circ}\in{\rm{lvs}}^{i}(s)^{\circ}\cap{\rm{lvs}}^{i}(t)^{\circ}$ and
hence

$s_{1}^{\bullet}\in{\rm{lvs}}^{i}(s)\cap{\rm{lvs}}^{i}(t).$

If $s$ and $t$ were leaves, then we would have $s=t$.

If $s$ is a leaf and $t$ is an inner node then $s\in{\rm{lvs}}^{i}(t)$ and
hence $s<^{+}t$.

If both $s$ and $t$ are inner nodes then

$s_{1}^{\bullet}\in{\rm{lvs}}^{i}(s)\cap{\rm{lvs}}^{i}(t)=\sigma_{i+1}(s)^{\bullet}\cap\sigma_{i+1}(t)^{\bullet},$

and as $\sigma_{i+1}$ is a constellation, $s\perp^{+}t$. $~{}~{}\Box$

Let $f:(S,\sigma)\longrightarrow(T,\tau)$ be a map of tree complexes. It gives
rise to maps of faces, for $k\in{\omega}$,

$\vec{f}_{k}=f_{k+1}\lhd f_{k}:S^{*}_{k}=S_{k+1}\lhd S_{k}\longrightarrow
T^{*}_{k}=T_{k+1}\lhd T_{k},$

that, by definition, preserve sups. Note that the maps $\vec{f}_{k}$’s do not
preserve the domains or codomains just defined above, in general. These maps
induce the morphism of p-opetopes $f^{*}:T^{*}\longrightarrow S^{*}$ as
follows. Let $t\in T^{*}_{i}$ and $s\in S^{*}_{j}$, with $0\leq j\leq i\leq
dim(T)$. Then

$f^{*}_{i}(t)=s$

iff

  1. 1.

$j$ is the maximal such number such that there is $s^{\prime}\in S^{*}_{j}$
such that

$\vec{f}_{j}(s^{\prime})\leq^{co}\delta\gamma^{(j+1)}(t);$

  2. 2.

and $s$ is the $\leq^{co}$-maximal $s^{\prime}\in S^{*}_{j}$ satisfying the
above inequality.

######  Lemma 3.6.

Let $f:(S,\sigma)\longrightarrow(T,\tau)$ be a morphism of tree complexes.
Then the set of maps
$f^{*}=\\{f^{*}_{i}\\}_{i\in{\omega}}:(T,\tau)^{*}\rightarrow(S,\sigma)^{*}$
is a $\iota$-epimorphism of positive opetopes.

Proof. Let us fix $i\in{\omega}$ and $s\in S^{*}_{i}=S_{i+1}\lhd S_{i}$. Let
$t=\vec{f}_{i}(s)$. Then $\vec{f}_{i}(s)\leq^{co}t$ and, as $\vec{f}_{i}$ is
one-to-one, it is the largest such $s$. Thus $f^{*}_{i}(t)=s$. Since $s$ was
arbitrary, $f^{*}_{i}$ is onto, for any $i\in{\omega}$ and hence $f^{*}$ epi.

For preservation of both codomains and domains by $f^{*}$, we fix $i>0$ and
$t\in T^{*}_{i}$ and we consider three cases:

  1. 1.

$\vec{f}_{i}(t)\in S^{*}_{i}$;

  2. 2.

$\vec{f}_{i}(t)\in S^{*}_{i-1}$;

  3. 3.

$\vec{f}_{i}(t)\in S^{*}_{j}$, for some $j<i-1$.

Preservation of codomains $\gamma$.

Case $\gamma$.1: $\vec{f}_{i}(t)=s\in S^{*}_{i}$.

First we shall show that $\vec{f}_{i-1}(\gamma(s))\leq^{co}\gamma(t)$. Since
$\vec{f}_{i}$’s are monotone and preserve leaves, we have

$\vec{f}_{i}({\rm lvs^{i}}(s))\subseteq{\rm
lvs^{i}}(\vec{f}_{i}(s))\subseteq{\rm{lvs}}^{i}(t).$

Using the above and the fact that $\vec{f}_{i}$’s preserve sups, we have

$\vec{f}_{i-1}(\gamma(s))=$
$=\vec{f}_{i-1}(\rm{sup}\\!^{i-1}({\rm{lvs}}^{i}(s)^{\circ}))=$ $={\rm
sup}\\!^{i-1}(\vec{f}_{i-1}({\rm{lvs}}^{i}(s)^{\circ}))\leq^{co}$
$\leq^{co}{\rm sup}\\!^{i-1}({\rm{lvs}}^{i}(t)^{\circ}))=$ $=\gamma(t).$

Now, contrary to the claim we want to prove, we assume that there is
$s_{1}^{\circ}\in S^{*}_{i-1}$ such that $\gamma(s)<^{co}s_{1}$ and

$\vec{f}_{i-1}(\gamma(s))<^{co}\vec{f}_{i-1}(s_{1}^{\circ})\leq^{co}\gamma(t)\in{\rm{lvs}}^{i}(t)^{\circ}\subseteq
T^{*}_{i-1}.$

Thus $\vec{f}_{i-1}(s_{1}^{\bullet})\in
T^{*}_{i}-\vec{f}_{i-1}({\rm{lvs}}^{i}(s)^{\circ})$. As
$\vec{f}_{i}(s)\leq^{co}t$, we have
$\vec{f}_{i}({\rm{lvs}}^{i}(s))\subseteq{\rm{lvs}}^{i}(t)$. Hence
$\vec{f}_{i-1}(\gamma(s))\in{\rm{lvs}}^{i}(t)^{\circ}$. Since
${\rm{lvs}}^{i}(t)^{\circ}$ is a convex subtree, we have

$\vec{f}_{i-1}(s_{1}^{\circ})\in{\rm{lvs}}^{i}(t)^{\circ}.$

As $\vec{f}_{i}(s_{1}^{\bullet})^{\circ}=\vec{f}_{i-1}(s_{1}^{\circ})$, we
have $\vec{f}_{i}(s_{1}^{\bullet})\leq^{co}t$. Since we also have
$s_{1}^{\bullet}\not\in{\rm{lvs}}^{i}(s)$, we get that

$s<\rm{sup}\\!^{i}(\\{s_{1}^{\bullet}\\}\cup{\rm{lvs}}^{i}(s))=s_{2}^{\circ}$

and

$\vec{f}_{i}(s)<^{co}\vec{f}_{i}(s_{2}^{\circ})=$
$=\vec{f}_{i}(\rm{sup}\\!^{i}(\\{s_{1}^{\bullet}\\}\cup{\rm{lvs}}^{i}(s)))=$
$={\rm
sup^{i}}(\\{\vec{f}_{i}(s_{1}^{\bullet})\\}\cup\vec{f}_{i}({\rm{lvs}}^{i}(s)))\leq^{co}t.$

This is a contradiction with the fact that $f^{*}_{i}(t)=s$.

Case $\gamma$.2: $f^{*}_{i}(t)=s_{1}\in S^{*}_{i-1}$.

Thus we have a $t_{1}\in\delta(t)$ such that

$\vec{f}_{i-1}(s_{1})\leq^{co}t_{1}<^{co}\gamma(t).$

We need to show that $s_{1}$ is the largest such an element of $S^{*}_{i-1}$
that $\vec{f}_{i-1}(s_{1})\leq^{co}\gamma(t)$. Suppose to the contrary that
there is $s_{2}^{\circ}\in S^{*}_{i-1}$ such that $s_{1}<^{co}s_{2}^{\circ}$
and

$\vec{f}_{i-1}(s_{1})<^{co}\vec{f}_{i-1}(s_{2}^{\circ})\leq^{co}\gamma(t).$

We have $\vec{f}_{i-1}(s_{1})\leq^{co}t_{1}$ and
$\vec{f}_{i-1}(s_{1})<^{co}\vec{f}_{i-1}(s_{2}^{\circ})$, and, as we cannot
have $\vec{f}_{i-1}(s_{2}^{\circ})\leq^{co}t_{1}$, we have

$\vec{f}_{i-1}(s_{1})\leq^{co}t_{1}<^{co}\vec{f}_{i-1}(s_{2}^{\circ})\leq^{co}\gamma(t).$

Since ${\rm{lvs}}^{i}(t)^{\circ}$ is a convex subtree of $T^{*}_{i-1}$, it
follows that $f^{*}_{i-1}(s_{2}^{\circ})\in{\rm{lvs}}^{i}(t)^{\circ}$. Thus
$f^{*}_{i}(s_{2}^{\bullet})\in{\rm{lvs}}^{i}(t)$, i.e.
$f^{*}_{i}(s_{2}^{\bullet})\leq^{co}t$. Hence $f^{*}_{i}(t)\in S^{*}_{i}$,
contrary to the supposition. End of Case 2.

Case $\gamma$.3: $f^{*}_{i}(t)=s_{1}\in S^{*}_{j}$, for some $j<i-1$.

Suppose there is $s_{2}\in S^{*}_{i-1}$ such that
$\vec{f}_{i-1}(s_{2})\leq^{co}\gamma(t)$. Then, as $j<i-1$,
$\vec{f}_{i-1}(s_{2})\not\leq^{co}t^{\prime}$, for all
$t^{\prime}\in\delta(t)$. Thus there is $t_{1}\in\delta(t)$ and $s_{3}\in
S^{*}_{i}$ so that $s_{2}=s_{3}^{\circ}$ and

$t_{1}<^{co}\vec{f}_{i-1}(s_{3}^{\circ})\leq^{co}\gamma(t).$

As ${\rm{lvs}}^{i}(t)$ is a convex subtree, we have
$f^{*}_{i-1}(s_{3}^{\circ})\in{\rm{lvs}}^{i}(t)^{\circ}$ and then
$f^{*}_{i}(s_{3}^{\bullet})\in{\rm{lvs}}^{i}(t)$. This contradicts the fact
that $j<i-1$. Thus $f^{*}_{i-1}(\gamma(t))=s_{4}\in S^{*}_{j^{\prime}}$ such
that $j^{\prime}<i-1$. If $j^{\prime}>j$, then

$\vec{f}_{j^{\prime}}(s_{4})\leq^{co}\delta\gamma^{(j^{\prime}+1)}(\gamma(t))=\delta\gamma^{(j^{\prime}+1)}(t)$

and this contradicts the choice of $s_{1}\in S^{*}_{j}$. If $j>j^{\prime}$,
then

$\vec{f}_{j}(s_{1})\leq^{co}\delta\gamma^{(j+1)}(t)=\delta\gamma^{(j+1)}(\gamma(t))$

contradicting the choice of $s_{4}\in S^{*}_{j^{\prime}}$. Thus $j=j^{\prime}$
and $s_{1}=s_{3}$, as required.

Preservation of domains $\delta$.

Case $\delta$.1: $f^{*}_{i}(t)=s\in S^{*}_{i}$.

We shall show that $f^{*}_{i}$ restricts to a bijection

$f^{*}_{\lceil t}:\delta(t)-\ker(f^{*})\longrightarrow\delta(s).$

Let $t_{1}\in\delta(t)-\ker(f^{*})$. Thus there is $s_{1}\in S^{*}_{i-1}$ such
that $f^{*}_{i-1}(t_{1})=s_{1}$ and hence
$\vec{f}_{i-1}(s_{1})\leq^{co}t_{1}$. Since $f^{*}$ preserves codomains
$\gamma(s)=f^{*}_{i-1}(\gamma(t))$.

Since $\vec{f}_{i-1}(s_{1})\not\in{\rm
lvs^{i}}(t)^{\circ}\supseteq\vec{f}_{i-1}({\rm{lvs}}^{i}(s)^{\circ})$, it
follows that $s_{1}\not\in{\rm{lvs}}^{i}(s)^{\circ}$.

We shall show that $s_{1}<^{co}\gamma(s)$. Suppose not. Then
$\gamma(s)<^{co}s_{1}\vee\gamma(s)$ and

$\vec{f}_{i-1}(s_{1}\vee\gamma(s))=\vec{f}_{i-1}(s_{1})\vee\vec{f}_{i-1}(\gamma(s))\leq^{co}\gamma(t).$

This means that

$\gamma(f^{*}_{i}(t))=\gamma(s)<^{co}s_{1}\vee\gamma(s)\leq^{co}f^{*}_{i-1}(\gamma(t)).$

and that the codomains are not preserved. Thus $s_{1}<^{co}\gamma(s)$ indeed.

Next we show that $s_{1}\in\delta(s)$. Again, we suppose that this is not the
case. Then there is $s_{2}^{\circ}\in\delta(s)$ such that
$s_{1}<^{co}s^{\circ}_{2}$. We have

$\vec{f}_{i-1}(s_{1})<^{co}\vec{f}_{i-1}(s_{2}^{\circ})<^{co}\vec{f}_{i-1}(\gamma(s))\leq^{co}\gamma(t),$

and

$\vec{f}_{i-1}(s_{1})\leq^{co}t_{1}<^{co}\gamma(t).$

As $f^{*}_{i-1}(t_{1})=s_{1}$, we have

$\vec{f}_{i-1}(s_{1})<^{co}t_{1}<^{co}\vec{f}_{i-1}(s_{2}^{\circ})<^{co}\vec{f}_{i-1}(\gamma(s))\leq^{co}\gamma(t).$

As the set ${\rm{lvs}}^{i}(t)^{\circ}$ is a convex subtree of $T^{*}_{i}$, we
have $\vec{f}_{i-1}(s_{2}^{\circ})\in{\rm{lvs}}(t)^{\circ}$. Hence
$s_{2}^{\bullet}\not\in{\rm{lvs}}^{i}(s)$ and
$\vec{f}_{i}(s_{2}^{\bullet})\in{\rm{lvs}}^{i}(t)$. Thus we have

$\vec{f}_{i}(s_{2}^{\bullet}\vee
s)=\vec{f}_{i}(s_{2}^{\bullet}\vee\vec{f}_{i}(s))\leq^{co}t,$

and $s<^{co}s_{2}^{\bullet}\vee s$. This contradicts the fact
$f^{*}_{i}(t)=s$. Thus $s_{1}\in\delta(s)$, as claimed.

So far we have shown that $f^{*}_{i}$ restricts to a well defined function

$f^{*}_{\lceil t}:\delta(t)-\ker(f^{*})\longrightarrow\delta(s).$

We shall show that it is a bijection.

Let $t_{1},t_{2}\in\delta(t)$ and $s\in S^{*}_{i-1}$ and
$f^{*}_{i}(t_{1})=s_{1}=f^{*}_{i}(t_{2})$. Hence
$s_{1}<^{co}f^{*}_{i-1}(t_{1})$ and $s_{1}<^{co}f^{*}_{i-1}(t_{2})$ and then
$t_{1}\perp^{+}t_{2}$ or $t_{1}=t_{2}$. As $\delta(t)$ is an antichain in
$S^{*}_{i-1}$, $t_{1}=t_{2}$. Thus $f^{*}_{\lceil t}$ is one-to-one.

To see that $f^{*}_{\lceil t}$ is onto, let us fix an arbitrary
$s_{1}\in\delta(s)$. Then $s_{1}\not\in{\rm{lvs}}(s)^{\circ}$.

We shall show that $\vec{f}_{i-1}(s_{1})\not\in{\rm{lvs}}^{i}(t)^{\circ}$.
Suppose to the contrary that $s_{1}=s_{3}^{\circ}$, for some $s_{3}\in S_{i}$
and that

${\rm
lvs}^{i}(t)^{\circ}\ni\vec{f}_{i-1}(s_{3}^{\circ})\not\in\vec{f}_{i-1}({\rm{lvs}}^{i}(s)^{\circ}).$

Hence $s_{3}^{\bullet}\not\in{\rm{lvs}}^{i}(s)$ and
$f(s_{3}^{\bullet})\in{\rm{lvs}}^{i}(t)$. Thus

$\vec{f}_{i}(s_{3}^{\bullet}\vee
s)=\vec{f}_{i}(s_{3}^{\bullet})\vee\vec{f}_{i}(s)\leq^{co}t$

and $s<s_{3}^{\bullet}\vee s$. This contradicts the fact that
$f^{*}_{i}(t)=s$. Thus $\vec{f}_{i}(s_{1})\not\in{\rm{lvs}}^{i}(t)^{\circ}$
indeed.

There is $t_{1}\in\delta(t)$ such that $\vec{f}_{i-1}(s_{1})\leq^{co}t_{1}$.
Let $s_{2}\in{\rm{lvs}}^{i}(s)^{\circ}$ such that $s_{1}\prec^{co}s_{2}$. Then

$\vec{f}_{i-1}(s_{2})\in\vec{f}_{i-1}({\rm{lvs}}^{i}(s)^{\circ})\subseteq{\rm{lvs}}^{i}(t)^{\circ}.$

Hence $s_{1}$ is the largest element of $S^{*}_{i-1}$ such that
$\vec{f}_{i}(s_{1})\leq^{co}t$, and hence $f^{*}_{i}(t_{1})=s_{1}$, as
required. End of proof of Case $\delta$.1.

Case $\delta$.2: $f^{*}_{i}(t)=s_{1}\in S^{*}_{i-1}$.

In this case we have a $t_{1}\in\delta(t)$ such that
$\vec{f}_{i-1}(s_{1})\leq^{co}t_{1}$. Clearly
$f^{*}_{i-1}(t_{1})=s_{1}=f^{*}_{i-1}(t)$. It remains to show that
$\delta(t)-\\{t_{1}\\}\subseteq\ker(f^{*})$. Suppose to the contrary that
there is $t_{2}\in\delta(t)$, $t_{2}\neq t_{1}$ such that
$f^{*}_{i-1}(t_{2})\in S^{*}_{i-1}$. Thus there is $s_{2}\in S^{*}_{i-1}$ such
that $\vec{f}_{i-1}(s_{2})\leq^{co}t_{2}\leq\gamma(t)$. Hence $s_{1}<s_{1}\vee
s_{2}$ and

$\vec{f}_{i-1}(s_{1}\vee
s_{2})=\vec{f}_{i-1}(s_{1})\vee\vec{f}_{i-1}(s_{2})\leq^{co}\gamma(t).$

But then

$\gamma^{(i-1)}(f^{*}_{i}(t))=\gamma^{(i-1)}(s_{1})=s_{1}<^{co}s_{1}\vee
s_{2}\leq^{co}f^{*}_{i}(\gamma^{(i-1)}(t))$

and this contradicts the fact that $f^{*}$ preserves codomains.

Case $\delta$.3: $f^{*}_{i}(t)=s_{1}\in S^{*}_{j}$, for some $j<i-1$.

We need to show that $\delta(t)\subseteq\ker(f^{*})$. Suppose not. Then there
is $t_{1}\in\delta(t)$ and $s_{2}\in S^{*}_{i-1}$ such that
$\vec{f}_{i}(s_{2})\leq^{co}t_{1}$. But this means that $f^{*}_{i}(t)\in
S^{*}_{j}$, for some $j\geq i-1$, contrary to the supposition.

$~{}~{}\Box$

For the proof of duality we need the following observations. We use the
notation introduced above.

######  Lemma 3.7.

Let $P$ be a positive opetope, $i\in{\omega}$, $p\in P_{i+1}-\gamma(P_{i+2})$,
$p_{root}=\rm{sup}_{<^{-}}(\pi(\gamma(p)))$. Then

  1. 1.

$\gamma\gamma(p)=\gamma(p_{root})$;

  2. 2.

the map

$\xi_{p}:(\pi(\gamma(p)),<^{-})\longrightarrow({\rm{lvs}}^{i}(p^{\circ})^{\circ},<^{co}),$

such that, for $q\in\pi(\gamma(p))\subseteq P_{i}-\gamma(P_{i+1})$,
$\xi_{p}(q)=q^{\circ}$ is an order isomorphism.

  3. 3.

In particular,
$p_{root}^{\bullet}=\xi_{p}(p_{root})=\rm{sup}^{i}({\rm{lvs}}^{i+1}(p^{\circ})).$

Proof. Exercise (use Path Lemma). $~{}~{}\Box$

###  3.3 The main theorem

In this section we shall prove that the functors defined in previous sections
are essential inverse one to the other.

######  Theorem 3.8.

The functors

${\bf pOpe}_{\iota,epi}\vphantom{{\bf cTree}^{op}}$$\vphantom{{\bf
pOpe}_{\iota,epi}}{\bf cTree}^{op}$ $(-)^{*}$ $(-)^{*}$

defined above, establish a dual equivalence of categories between categories
of positive opetopes with $\iota$-epimorphisms and tree complexes with
embeddings.

Proof of Theorem 3.8.

We shall define two natural isomorphisms $\eta$ and $\varepsilon$.

Let $(S,\sigma)$ be a tree complex. Recall that

$S^{*}_{i}=S_{i+1}\lhd S_{i},\;\;{\rm and}\;\;S^{**}_{i}=(S_{i+1}\lhd
S_{i})-\gamma(S_{i+2}\lhd S_{i-1}).$

For $i\in{\omega}$, the $i$-th component

$\eta_{S,i}:S_{i}\longrightarrow S^{**}_{i}$

of $\eta_{S}:(S,\sigma)\longrightarrow(S^{**},\sigma^{**})$ is defined as

$\eta_{S,i}(s)=s^{\bullet},$

i.e. it is a vertex in $S_{i+1}\lhd S_{i}$. Clearly $\eta_{S,i}$ is one-to-one
and, as all circles in $S_{i+1}\lhd S_{i}$ are of form $\gamma(S_{i+2}\lhd
S_{i-1})$, $\eta_{S,i}(s)$ is onto, as well. To see that $\eta_{S,i}$ is an
order isomorphism, consider $s_{1},s_{2}\in S_{i}$. Then

$s^{\bullet}_{1}\prec^{-}s^{\bullet}_{2}$

iff

$\gamma(s^{\bullet}_{1})\in\delta(s^{\bullet}_{2})$

iff

$s^{\circ}_{1}\in\rm{cvr}^{i-1}(s^{\circ}_{2})$

iff

$s^{\circ}_{1}\prec^{co}s^{\circ}_{2}$

iff

$s_{1}\prec^{S_{i}}s_{2}.$

To see that $\eta_{S}$ is an isomorphism of tree complexes, it is enough to
show that, for $i\in{\omega}$,
$(\eta_{S,i+1},\eta_{S,i}:(S_{i+1},\sigma_{i},S_{i})\longrightarrow(S^{**}_{i+1},\sigma^{**}_{i},S^{**}_{i})$
is an isomorphism of constellations. To this aim, it is enough to show that
the square

$S^{**}_{i+1}\vphantom{St(S^{**}_{i})}$$\vphantom{S^{**}_{i+1}}St(S^{**}_{i})$$\sigma^{**}_{i}$$S_{i+1}\vphantom{St(S_{i})}$$\vphantom{S_{i+1}}St(S_{i})$$\sigma_{i}$$\eta_{S,i+1}$$\vec{\eta}_{S,i}$

commutes, where $\vec{\eta}_{S,i}$ is the image function induced by the
function $\eta_{S,i}$. Let $s\in S^{*}_{i+1}$. We have

$\vec{\eta}_{S,i}(\sigma_{i}(s))=\\{t^{\bullet}:\;t\in
S_{i},\;t\in\sigma_{i}(s)\\}=$ $=\\{t^{\bullet}\in
S^{**}_{i}:t^{\bullet}<^{co}s^{\circ}\\}=$ $=\\{t^{\bullet}\in
S^{**}_{i}:t^{\bullet}<^{+}\gamma(s^{\bullet})\\}=$
$=\sigma^{**}_{i}(s^{\bullet})=\sigma^{**}_{i}(\eta_{S,i}(s).$

The naturality of $\eta$ is clear.

Now we turn to the natural isomorphism $\varepsilon$. Let $P$ be a p-opetope,
$i\in{\omega}$. The maps

$\varepsilon_{P,i}:(P^{*}_{i+1}\lhd
P^{*}_{i},<^{co})\longrightarrow(P_{i},<^{+})$

were already defined in section 3.1. So, for $p_{1}\in P_{i}-\gamma(P_{i+1})$,
$p_{1}^{\bullet}\in P^{*}_{i+1}\lhd P^{*}_{i}$, we have
$\varepsilon_{P,i}(p_{1}^{\bullet})=p_{1}$ and, for $p_{2}\in
P_{i+1}-\gamma(P_{i+2})$, $p_{2}^{\circ}\in P^{*}_{i+1}\lhd P^{*}_{i}$, we
have $\varepsilon_{P,i}(p_{2}^{\circ})=\gamma(p_{2})$.

We need to show that $\varepsilon_{P}$ preserves both codomains $\gamma$ and
domains $\delta$. Nautrality of $\varepsilon$ is again clear.

Preservation of codomains. Let $p_{1}\in P_{i+1}-\gamma(P_{i+2})$. We have

$\varepsilon_{P,i}(\gamma(p_{1}^{\bullet}))=\varepsilon_{P,i}(p_{1}^{\circ})=$
$=\varepsilon_{P,i+1}(p_{1}^{\bullet})^{\circ}=\gamma(\varepsilon_{P,i+1}(p_{1}^{\bullet})).$

Let $p_{2}\in P_{i+2}-\gamma(P_{i+3})$ and $p_{root}\in
P_{i+1}-\gamma(P_{i+2})$ such that
$p_{root}^{\circ}=\rm{sup}^{i}({\rm{lvs}}^{i+1}(p_{2}^{\circ})^{\circ})$.
Using Lemma 3.7, we have

$\varepsilon_{P,i}(\gamma(p_{2}^{\circ}))=\varepsilon_{P,i}(\rm{sup}^{i}({\rm{lvs}}^{i+1}(p_{2}^{\circ})^{\circ}))=$
$=\varepsilon_{P,i}(p^{\circ}_{root})=\gamma(p_{root}^{\bullet})=$
$=\gamma\gamma(p_{2}^{\bullet})=\gamma(\varepsilon_{P,i+1}(p_{2}^{\bullet})).$

Preservation of domains. Let $p\in P_{i+1}^{**}$ and $q\in P^{**}_{i}$. We
need to verify that

$q\in\delta(p)\;\;\;{\rm
iff}\;\;\;\varepsilon_{P,i}(q)\in\delta(\varepsilon_{P,i+1}(p)).$ (1)

We shall prove the above equivalence by cases depending on the form of $p$ and
$q$.

Let $p_{1}\in P_{i+1}-\gamma(P_{i+2})$, $p_{2}\in P_{i+2}-\gamma(P_{i+3})$,
$q_{1}\in P_{i}-\gamma(P_{i+1})$, $q_{2}\in P_{i+1}-\gamma(P_{i+2})$. Then we
shall consider four cases, one by one.

Case 1: $p=p_{1}^{\bullet}$, $q=q_{1}^{\bullet}$. We have

$q_{1}^{\bullet}\in\delta(p_{1}^{\bullet})$

iff

$q_{1}^{\bullet}\in\rm{cvr}^{i-1}(p_{1}^{\circ})$

iff

$q_{1}^{\bullet}\prec^{co}p_{1}^{\circ}$

iff

$q_{1}\in\delta(p_{1})$

iff

$\varepsilon_{P,i}(q_{1}^{\bullet})\in\delta(\varepsilon_{P,i+1}(p_{1}^{\bullet})).$

Case 2: $p=p_{1}^{\bullet}$, $q=q_{2}^{\circ}$.

$q_{2}^{\circ}\in\delta(p_{1}^{\bullet})$

iff

$q_{2}^{\circ}\prec^{co}p_{1}^{\circ}$

iff

$q_{2}\prec^{-}p_{1}$

iff

$\gamma(q_{2})\in\delta(p_{1})$

iff

$\varepsilon_{P,i}(q_{2}^{\circ})\in\delta(\varepsilon_{P,i+1}(p_{1}^{\bullet})).$

Case 3: $p=p_{2}^{\circ}$, $q=q_{1}^{\bullet}$.

$q_{1}^{\bullet}\in\delta(p_{2}^{\circ})$

iff

$q_{1}^{\bullet}\in\rm{cvr}^{i-1}({\rm{lvs}}^{1}(p_{2}^{\circ})^{\circ})$

iff

$q_{1}^{\bullet}\not\in{\rm{lvs}}^{1}(p_{2}^{\circ})^{\circ})\;{\rm and}\;{\rm
there\;is}\;q_{3}\in P_{i}-\gamma(P_{i+1})\;{\rm
such\;that}\;q_{1}^{\bullet}\prec^{co}q_{3}^{\circ}\;{\rm
and}\;q_{3}^{\circ}\in{\rm{lvs}}^{i}(p_{2}^{\circ})^{\circ}$

iff

${\rm there\;is}\;q_{3}\in P_{i}-\gamma(P_{i+1})\;{\rm
such\;that}\;q_{1}\in\delta(q_{3})\;{\rm and}\;q_{3}\leq^{+}\gamma(p_{2})$

iff (Path Lemma)

$q_{1}\in\delta\gamma(p_{2})$

iff

$\varepsilon_{P,i}(q_{1}^{\bullet})\in\delta(\varepsilon_{P,i+1}(p_{2}^{\circ})).$

Case 4: $p=p_{2}^{\circ}$, $q=q_{2}^{\circ}$.

$q_{2}^{\circ}\in\delta(p_{2}^{\circ})$

iff

$q_{2}^{\circ}\in\rm{cvr}^{i-1}({\rm{lvs}}^{1}(p_{2}^{\circ})^{\circ})$

iff

$q_{2}^{\circ}\not\in{\rm{lvs}}^{1}(p_{2}^{\circ})^{\circ})\;{\rm and}\;{\rm
there\;is}\;q_{3}\in P_{i}-\gamma(P_{i+1})\;{\rm
such\;that}\;q_{2}^{\circ}\prec^{co}q_{3}^{\circ}\;{\rm
and}\;q_{3}^{\circ}\in{\rm{lvs}}^{i}(p_{2}^{\circ})^{\circ}$

iff

$q_{2}\not\leq^{+}\gamma(p_{2})\;{\rm and}\;{\rm there\;is}\;q_{3}\in
P_{i}-\gamma(P_{i+1})\;{\rm such\;that}\;q_{2}\prec^{-}q_{3}\;{\rm
and}\;q_{3}\leq^{+}\gamma(p_{2})$

iff (Path Lemma)

$\gamma(q_{2})\in\delta\gamma(p_{2})$

iff

$\varepsilon_{P,i}(q_{2}^{\circ})\in\delta(\varepsilon_{P,i+1}(p_{2}^{\circ})).$

$~{}~{}\Box$

##  4 Thicket complexes

###  4.1 Thicket constellations

A thicket is a finite poset $(P,\leq)$ such that, for any $p\in P$, the
subposet $\\{p^{\prime}\in P:p^{\prime}\leq p\\}$ is a tree. A morphism of
thickets is a one-to-one function that preserves and reflects order. $\bf
Thicket$ is the category of thickets. ${\bf St}(P)$ is the poset of convex
sub-trees of thicket $P$.

A thicket-constellation  (or t-constellation or even constellation for short)
is a triple $(T^{\prime},\tau,T)$ where $\tau$ is a monotone function

$\tau:T^{\prime}\rightarrow{\bf St}(T)$

such that if $t,t^{\prime}\in T^{\prime}$ and
$\sigma(t)\cap\sigma(t^{\prime})\neq\emptyset$, then $t\perp t^{\prime}$.

Let $\tau:T^{\prime}\rightarrow{\bf St}(T)$ be a t-constellation. Then the
thicket $T^{\prime}\lhd_{\tau}T$, the extension of $T^{\prime}$ by $T$ along
$\tau$, is the thicket $T^{\prime}$ with nodes of $T$ added as leaves so that
if $x\in T$ and $y\in T^{\prime}$, then $x<^{co}y$ in $T^{\prime}\lhd_{\tau}T$
iff $x\in\tau(y)$. The order $<^{co}$ is called the constellation order of the
t-constellation $\tau:T^{\prime}\rightarrow{\bf St}(T)$, or just constellation
order if the t-constellation is understood. Any pair of tree maps
$f:S\rightarrow T$, $f^{\prime}:S^{\prime}\rightarrow T^{\prime}$ induces a
monotone map $f^{\prime}\lhd f:S^{\prime}\lhd_{\sigma}S\longrightarrow
T^{\prime}\lhd_{\tau}T$. Such a pair

$(f,f^{\prime}):(S^{\prime},\sigma,S)\longrightarrow(T^{\prime},\tau,T)$

is a morphism of t-constellations if the induced map $f^{\prime}\lhd
f:S^{\prime}\lhd_{\sigma}S\longrightarrow T^{\prime}\lhd_{\tau}T$ preserves
binary sups.

###  4.2 Thicket complexes and duality

A thicket complex $(T,\tau)$ is a sequence of t-constellations:

$\tau_{0}:T_{1}\rightarrow{\bf St}(T_{0}),$ $\tau_{1}:T_{2}\rightarrow{\bf
St}(T_{1}),$ $\ldots$ $\tau_{i}:T_{i+1}\rightarrow{\bf St}(T_{i}),$ $\ldots$

$i\in{\omega}$, with almost all sets $T_{i}$ empty. The dimension $(T,\tau)$
is $n$ iff $T_{n}$ be the last non-empty set. We write $dim(T)$ for dimension
of the tree complex $(T,\tau)$. $T_{0}$ is required to be a singleton.

A morphism of thicket complexes $f:(S,\sigma)\rightarrow(T,\tau)$ is a family
of thicket embeddings $f_{i}:S_{i}\rightarrow T_{i}$, for $i\in{\omega}$, such
that, for $i\in{\omega}$,

$(f_{i+1},f_{i}):(S_{i+1},\sigma_{i},S_{i})\longrightarrow(T_{i+1},\tau_{i},T_{i})$

is a morphism of t-constellations, i.e. the thicket embeddings

$\vec{f}_{i}=f_{i}\lhd f_{i}:S_{i+1}\lhd_{\sigma_{i}}S_{i}\longrightarrow
T_{i+1}\lhd_{\tau_{i}}T_{i}$

that preserve binary sups.

Size of a w-complex $(T,\tau)$ is a sequence of natural numbers
$size(T,\tau)=\\{s_{i}\\}_{i\in{\omega}}$ so that $s_{i}=size(T,\tau)_{i}$ is
the number of maximal elements in the thicket $T_{i}$. A w-complex $(T,\tau)$
i a tree complex iff $size(T,\tau)_{i}\leq 1$, for all $i\in{\omega}$.

The category of thicket complexes and their morphisms will be denoted by ${\bf
cThicket}$. Clearly ${\bf cTree}$ is a full subcategory of ${\bf cThicket}$.

######  Theorem 4.1.

The functors

${\bf pOpeCard}_{\iota,epi}\vphantom{{\bf cThicket}^{op}}$$\vphantom{{\bf
pOpeCard}_{\iota,epi}}{\bf cThicket}^{op}$ $(-)^{*}$ $(-)^{*}$

defined as those for trees, establish a dual equivalence of categories between
categories of opetopic cardinals with $\iota$-epimorphisms and thicket
complexes with embeddings.

Proof This is an easy extension of the corresponding fact concerning tree
complexes and p-opetopes. $~{}~{}\Box$

##  5 Appendix: Positive opetopes and positive opetopic cardinals

In this appendix we recall the notion of positive opetopes, positive opetopic
cardinals, their morphisms: face maps and $\iota$-maps. We also quote without
proofs some facts from [Z1] and [Z4].

###  5.1 Positive hypergraphs

A positive hypergraph $S$ is a family $\\{S_{k}\\}_{k\in{\omega}}$ of finite
sets of faces, a family of functions $\\{\gamma_{k}:S_{k+1}\rightarrow
S_{k}\\}_{k\in{\omega}}$, and a family of total relations
$\\{\delta_{k}:S_{k+1}\rightarrow S_{k}\\}_{k\in{\omega}}$. Moreover,
$\delta_{0}:S_{1}\rightarrow S_{0}$ is a function and only finitely many among
sets $\\{S_{k}\\}_{k\in{\omega}}$ are non-empty. As it is always clear from
the context, we shall never use the indices of the functions $\gamma$ and
$\delta$. We shall ignore the difference between $\gamma(x)$ and
$\\{\gamma(x)\\}$ and in consequence we shall consider iterated applications
of $\gamma$’s and $\delta$’s as sets of faces, e.g.
$\delta\delta(x)=\bigcup_{y\in\delta(x)}\delta(y)$ and
$\gamma\delta(x)=\\{\gamma(y)\,|\,y\in\delta(x)\\}$.

A morphism of positive hypergraphs $f:S\longrightarrow T$ is a family of
functions $f_{k}:S_{k}\longrightarrow T_{k}$, for $k\in{\omega}$, such that,
for $k>0$ and $a\in S_{k}$, we have $\gamma(f(a))=f(\gamma(a))$ and $f_{k-1}$
restricts to a bijection

$f_{a}:\delta(a)\longrightarrow\delta(f(a)).$

The category of positive hypergraphs is denoted by $\bf pHg$.

We define a binary relation of lower order on $<^{S_{k},-}$ for $k>0$ as the
transitive closure of the relation $\lhd^{S_{k},-}$ on $S_{k}$ such that, for
$a,b\in S_{k}$, $a\lhd^{S_{k},-}b$ iff $\gamma(a)\in\delta(b)$. We write
$a\perp^{-}b$ iff either $a<^{-}b$ or $b<^{-}a$, and we write $a\leq^{-}b$ iff
either $a=b$ or $a<^{-}b$.

We also define a binary relation of upper order on $<^{S_{k},+}$ for $k\geq 0$
as the transitive closure of the relation $\lhd^{S_{k},+}$ on $S_{k}$ such
that, for $a,b\in S_{k}$, $a\lhd^{S_{k},+}b$ iff there is $\alpha\in S_{k+1}$
so that $a\in\delta(\alpha)$ and $\gamma(\alpha)=b$. We write $a\perp^{+}b$
iff either $a<^{+}b$ or $b<^{+}a$, and we write $a\leq^{+}b$ iff either $a=b$
or $a<^{+}b$.

###  5.2 Positive opetopic cardinals

A positive hypergraph $S$ is a positive opetopic cardinal if it is non-empty,
i.e. $S_{0}\neq\emptyset$ and it satisfies the following four conditions

  1. 1.

Globularity: for $a\in S_{\geq 2}$:

$\gamma\gamma(a)=\gamma\delta(a)-\delta\delta(a),\hskip
42.67912pt\delta\gamma(a)=\delta\delta(a)-\gamma\delta(a);$

  2. 2.

Strictness: for $k\in{\omega}$, the relation $<^{S_{k},+}$ is a strict order;
$<^{S_{0},+}$ is linear;

  3. 3.

Disjointness: for $k>0$,

$\perp^{S_{k},-}\cap\perp^{S_{k},+}=\emptyset$

  4. 4.

Pencil linearity: for any $k>0$ and $x\in S_{k-1}$, the sets

$\\{a\in S_{k}\;|\;x=\gamma(a)\\}\;\;\;\;{\rm and}\;\;\;\;\\{a\in
S_{k}\;|\;x\in\delta(a)\\}$

are linearly ordered by $<^{S_{k},+}$.

The category of positive opetopic cardinals is the full subcategory of $\bf
pHg$ whose objects are positive opetopic cardinals. It is denoted by ${\bf
pOpeCard}$.

###  5.3 Positive opetopes

The size of positive opetopic cardinal $S$ is a sequence of natural numbers
$size(S)=\\{|S_{n}-\delta(S_{n+1})|\\}_{n\in{\omega}}$, with all elements
above $dim(S)$ being equal $0$. We have an order $<$ on such sequences of
natural numbers so that
$\\{x_{n}\\}_{n\in{\omega}}<\\{y_{n}\\}_{n\in{\omega}}$ iff there is
$k\in{\omega}$ such that $x_{k}<y_{k}$ and, for all $l>k$, $x_{l}=y_{l}$. This
order is well founded and hence facts about positive opetopic cardinals can be
proven by induction on their size.

Let $P$ be an positive opetopic cardinal. We say that $P$ is a positive
opetope iff $size(P)_{l}\leq 1$, for $l\in{\omega}$. By ${\bf pOpe}$ we denote
full subcategory of $\bf pHg$ whose objects are positive opetopes.

Some notions and notation.

  1. 1.

Let $S$ be a positive hypergraph $S$, $x$ a face of $S$. Then $S[x]$ denotes
the least sub-hypergraph of $S$ containing face $x$. The dimension of $S$ is
maximal $k$ such that $S_{k}\neq\emptyset$. We denote by $dim(S)$ the
dimension of $S$.

  2. 2.

Let $P$ be an opetope. If $dim(P)=n$, then the unique face in $P_{n}$ is
denoted by ${\bf m}_{P}$.

  3. 3.

The function $\gamma^{(k)}:P\rightarrow P_{\leq k}$ is an iterated version of
the codomain function $\gamma$ defined as follows. For any $k,l\in{\omega}$
and $p\in P_{l}$,

$\gamma^{(k)}(p)\;=\;\left\\{\begin{array}[]{ll}\gamma\gamma^{(k+1)}(p)&\mbox{
if }l>k\\\ p&\mbox{ if }l\leq k.\end{array}\right.$

  4. 4.

Let $a,b\in P_{k}$. A lower path $a_{0},\ldots,a_{m}$ from $a$ to $b$ in $P$
is a sequence of faces $a_{0},\ldots,a_{m}\in S_{k}$ such that $a=a_{0}$,
$b=a_{m}$ and, for $\gamma(a_{i-1})\in\delta(a_{i})$, $i=1,\ldots,m$.

  5. 5.

Let $x,y\in P_{k}$. An upper path $x,a_{0},\ldots,a_{m},y$ from $x$ to $y$ in
$S$ is a sequence of faces $a_{0},\ldots,a_{m}\in P_{k+1}$ such that
$x\in\delta(a_{0})$, $y=\gamma(a_{m})$ and $\gamma(a_{i-1})\in\delta(a_{i})$,
for $i=1,\ldots,m$.

######  Lemma 5.1.

Let $P$ be a positive opetopic cardinal, $n\in{\omega}$, $a,b\in P_{n}$,
$a<^{+}b$. Then, there is an upper $P_{n+1}-\gamma(P_{n+2})$-path from $a$ to
$b$. $\Box$

######  Lemma 5.2 (Path Lemma).

Let $P$ be an opetope. Let $k\geq 0$, $B=(a_{0},\ldots,a_{k})$ be a maximal
$S_{n}$-lower path in a positive opetopic cardinal $P$, $b\in S_{n}$, $0\leq
s\leq k$, $a_{s}<^{+}b$. Then there are $0\leq l\leq s\leq p\leq k$ such that

  1. 1.

$a_{i}<^{+}b$ for $i=l,\ldots,p$;

  2. 2.

$\gamma(a_{p})=\gamma(b)$;

  3. 3.

either $l=0$ and $\delta(a_{0})\subseteq\delta(b)$ or $l>0$ and
$\gamma(a_{l-1})\in\delta(b)$;

  4. 4.

$\gamma(a_{i})\in\iota(S)$, for $l\leq i<p$. $\Box$

###  5.4 The embedding of ${\bf pOpeCard}$ into ${\bf{\omega}Cat}$

There is an embedding

$(-)^{*}:{\bf pOpeCard}\longrightarrow{\bf{\omega}Cat}$

defined as follows, c.f. [Z1]. Let $T$ be an opetopic cardinal. The
${\omega}$-category $T^{*}$ has as $n$-cells pairs $(S,n)$, where $S$ is a
subopetopic cardinal of $T$, $dim(S)\leq n$, and $n\geq 0$. A cell $(S,n)$ is
called proper iff $n=dim(S)$.

For $k<n$, the domain and codomain operations

${\bf d}^{(k)},{\bf c}^{(k)}:T^{*}_{n}\longrightarrow T^{*}_{k}$

are given, for $(S,n)\in T^{*}_{n}$, by

$({\bf d}^{(k)}(S,n))=({\bf d}^{(k)}(S),k),\hskip 42.67912pt({\bf
c}^{(k)}(S,n))=({\bf c}^{(k)}(S),k)$

where

$({\bf d}^{(k)}(S)_{l}\;=\;\left\\{\begin{array}[]{ll}\emptyset&\mbox{ if
}l>k\\\ S_{k}-\gamma(S_{k+1})&\mbox{ if }l=k\\\ S_{l}&\mbox{ if
}l<k\end{array}\right.$

and

$({\bf c}^{(k)}(S)_{l}\;=\;\left\\{\begin{array}[]{ll}\emptyset&\mbox{ if
}l>k\\\ S_{k}-\delta(S_{k+1})&\mbox{ if }l=k\\\ S_{k-1}-\iota(S_{k+1})&\mbox{
if }l=k-1\geq 0\\\ S_{l}&\mbox{ if }l<k-1\end{array}\right.$

The identity operation

$id:T^{*}_{n}\longrightarrow T^{*}_{n+1}$

is given by

$(S,n)\mapsto(S,n+1).$

The composition operation is defined, for pairs of cells
$(S,n),(S^{\prime},n^{\prime})\in T^{*}$ with $k\leq n,n^{\prime}$ such that
${\bf d}^{(k)}(S,n)={\bf c}^{(k)}(S^{\prime},n^{\prime})$, as the sum

$(S,n)\circ(S^{\prime},n^{\prime})=(S\cup S^{\prime},\max(n,n^{\prime})).$

Now $T^{*}$ together with operations of domain, codomain, identity and
composition is an ${\omega}$-category. If $f:S\rightarrow T$ is a map of
opetopic cardinals and $S^{\prime}$ is a sub-opetopic cardinal of $S$, then
the image $f(S^{\prime})$ is a sub-opetopic cardinal of $T$. This defines the
functor $(-)^{*}$ on morphisms. We recall from [Z1]

######  Theorem 5.3.

The embedding

$(-)^{*}:{\bf pOpeCard}\longrightarrow{\bf{\omega}Cat}$

is well defined and full on isomorphisms and it factorises through ${\bf
Poly}\longrightarrow{\bf{\omega}Cat}$ via full and faithful functor,
$(-)^{*}:{\bf pOpeCard}\longrightarrow{\bf Poly}$, into the category of
polygraphs. $\Box$

###  5.5 $\iota$-maps of positive opetopes

The embedding $(-)^{\ast}:{\bf pOpe}\rightarrow{\bf{\omega}Cat}$ is not full.
The morphisms $P^{\ast}\rightarrow Q^{\ast}$ in ${\bf{\omega}Cat}$ between
images of opetopes are ${\omega}$-functors that send generators to generators.
The category ${\bf pOpe}_{\iota}$ with the same objects as ${\bf pOpe}$ will
be so defined that the embedding $(-)^{\ast}:{\bf
pOpe}_{\iota}\rightarrow{\bf{\omega}Cat}$ (denoted the same way) will be full
on ${\omega}$-functors that send generators to either generators or identities
on generators of a smaller dimension.

Let $P$ and $Q$ be positive opetopes. A contraction morphism of opetopes (or
$\iota$-map, for short), $h:Q\longrightarrow P$, is a function
$h:|Q|\longrightarrow|P|$ between faces of opetopes such that

  1. 1.

$dim(q)\geq dim(h(q))$, for $q\in Q$;

  2. 2.

(preservation of codomains) $h(\gamma^{(k)}(q))=\gamma^{(k)}(h(q))$, for
$k\geq 0$ and $q\in Q_{k+1}$;

  3. 3.

(preservation of domains)

    1. (a)

if $dim(h(q))=dim(q)$, then $h$ restricts to a bijection

$(\delta(q)-\ker(h))\vphantom{\delta(h(q))}$$\vphantom{(\delta(q)-\ker(h))}\delta(h(q))$$h$

for $k\geq 0$ and $q\in Q_{k+1}$, where the kernel of h is defined as

$\ker(h)=\\{q\in Q|dim(q)>dim(h(q))\\};$

    2. (b)

if $dim(h(q))=dim(q)-1$, then $h$ restricts to a bijection

$(\delta(q)-\ker(h))\vphantom{\\{h(q)\\}}$$\vphantom{(\delta(q)-\ker(h))}\\{h(q)\\}$$h$

for $k\geq 0$ and $q\in Q_{k+1}$;

    3. (c)

if $dim(h(q))<dim(q)-1$, then $\delta^{(k)}(q)\subseteq\ker(h)$.

We call a face $q\in Q_{m}$ $k$-collapsing iff $h(q)\in P_{m-k}$. A
$0$-collapsing face is called non-collapsing.

We have an embedding $\kappa:{\bf pOpe}\longrightarrow{\bf pOpe}_{\iota}$ that
induces the usual adjunction $\kappa_{!}\dashv\kappa^{*}$

$\widehat{{\bf pOpe}}\vphantom{\widehat{{\bf
pOpe}_{\iota}}}$$\vphantom{\widehat{{\bf pOpe}}}\widehat{{\bf pOpe}_{\iota}}$
$\kappa_{!}$ $\kappa^{*}$

######  Lemma 5.4.

Let $h:Q\rightarrow P$ be a $\iota$-map, $q_{1},q_{2}\in Q-\ker(h)$ and
$l<k\in{\omega}$ such that

$\begin{array}[]{rcl}\gamma^{(k+1)}(q_{1})&<^{-}&\gamma^{(k+1)}(q_{2})\\\
\gamma^{(k)}(q_{1})&<^{+}&\gamma^{(k)}(q_{2})\\\ \ldots&\ldots&\ldots\\\
\gamma^{(l+1)}(q_{1})&<^{+}&\gamma^{(l+1)}(q_{2})\\\
\gamma^{(l)}(q_{1})&=&\gamma^{(l)}(q_{2}).\\\ \end{array}$

Then there is $l\leq l^{\prime}<k$ such that

$\begin{array}[]{rcl}h(\gamma^{(k+1)}(q_{1}))&<^{-}&h(\gamma^{(k+1)}(q_{2}))\\\
h(\gamma^{(k)}(q_{1}))&<^{+}&h(\gamma^{(k)}(q_{2}))\\\ \ldots&\ldots&\ldots\\\
h(\gamma^{(l^{\prime}+1)}(q_{1}))&<^{+}&h(\gamma^{(l^{\prime}+1)}(q_{2}))\\\
h(\gamma^{(l^{\prime})}(q_{1}))&=&h(\gamma^{(l^{\prime})}(q_{2})).\\\
\end{array}$

Proof. Simple check. $\Box$

From the above we get immediately

######  Corollary 5.5.

Let $h:Q\rightarrow P$ be a $\iota$-map, $q_{1},q_{2}\in Q-\ker(h)$. Then

  1. 1.

$q_{1}<^{-}q_{2}$ iff $h(q_{1})<^{-}h(q_{2})$;

  2. 2.

if $q_{1}<^{+}q_{2}$, then $h(q_{1})\leq^{+}h(q_{2})$;

  3. 3.

if $h(q_{1})<^{+}h(q_{2})$, then $q_{1}<^{+}q_{2}$;

  4. 4.

if $h(q_{1})=h(q_{2})$, then $q_{1}\perp^{+}q_{2}$. $\Box$

A set $X$ of $k$-faces in a positive opetope $P$ is a $<^{+}$-interval (or
interval, for short) if it is either empty or there are two k-faces
$x_{0},x_{1}\in P_{k}$ such that $x_{0}\leq^{+}x_{1}$ and $X=\\{x\in
P_{k}|x_{0}\leq^{+}x\leq^{+}x_{1}\\}$. Any interval in any positive opetope is
linearly ordered by $\leq^{+}$.

######  Corollary 5.6.

Let $h:Q\rightarrow P$ -be a contraction of positive opetopes, $p\in P_{k}$.
Then the fiber of $k$-faces (of non-degenerating faces) $h^{-1}(p)-\ker(h)$ of
$h$ over $p$ is an interval.

Proof. From Corollary 5.5.4 we get that $h^{-1}(p)-\ker(h)$ is linearly
ordered. And from Corollary 5.5.2 that this linear order is an interval.
$\Box$

###  5.6 The embedding of ${\bf pOpe}_{\iota}$ into ${\bf{\omega}Cat}$

We extend the embedding functor $(-)^{\ast}$ to contractions

$(-)^{*}:{\bf pOpe}_{\iota}\longrightarrow{\bf{\omega}Cat}.$

Let $h:Q\rightarrow P$ be a contraction morphism in ${\bf pOpe}_{\iota}$. Then

$h^{\ast}:Q^{\ast}\rightarrow P^{\ast}$

is an ${\omega}$-functor such that

$h^{\ast}(k,A)=(k,\vec{h}(A))$

where $(k,A)\in Q^{*}_{k}$, and $\vec{h}(A)$ is the set-theoretic image of the
opetopic cardinal $A$ under $h$.

######  Theorem 5.7.

The functor

$(-)^{*}:{\bf pOpe}_{\iota}\longrightarrow{\bf{\omega}Cat}$

is well defined. The objects of ${\bf pOpe}_{\iota}$ are sent under $(-)^{*}$
to positive-to-one polygraphs. $(-)^{*}$ is faithful, conservative and full on
those ${\omega}$-functors that send generators to either generators or to
(possibly iterated) identities on generators of smaller dimensions. In
particular, it is full on isomorphisms. $\Box$

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