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# A pasting theorem for iterated Segal spaces

Jaco Ruit  Mathematisch Instituut, Universiteit Utrecht, The Netherlands
<EMAIL_ADDRESS>
###### Abstract.

We introduce a novel notion of pasting shapes for iterated Segal spaces which
classify particular arrangements of composing cells in $d$-uple Segal spaces.
Using this formalism, we then continue to prove a pasting theorem for these
iterated Segal spaces.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Pasting shapes and their nerves
  4. 4 Proof of the pasting theorem
  5. 5 Outlook: an $(\infty,d)$-categorical pasting theorem

##  1\. Introduction

A multifold category is a generalization of a category which has multiple
categorical directions. For instance, a 2-fold category is better known as a
double category, a concept that was originally introduced by Ehresmann
[Ehr63], which has both a horizontal and vertical direction for 1-cells, and
2-cells that interact with both directions. Since its introduction, the theory
of double categories has found a wide range of applications throughout
category theory. For instance, the theory of 2-categorical limits admits a
neat description using double categories [GP99]. Pseudo 2-functors that form a
proarrow equipment are better understood as being double categories with
additional properties [Ver92], [Shu08]. These proarrow equipments give rise to
formal category theories. In [Shu11], we see an application of the theory of
double categories to homotopy theory.

When dealing with double categories, one would like to obtain new 2-cells by
composing compatible 2-cells. For instance, one may start with compatible
2-cells $v_{1},v_{2},v_{3},v_{4},v_{5},v_{6}$ in a double category $D$:

${x_{00}}$${x_{10}}$${x_{20}}$${x_{40}}$${x_{01}}$${x_{11}}$${x_{21}}$${x_{31}}$${x_{41}}$${x_{32}}$${x_{42}}$${x_{03}}$${x_{23}}$${x_{33}}$${x_{43},}$$\scriptstyle{v_{1}}$$\scriptstyle{v_{4}}$$\scriptstyle{v_{3}}$$\scriptstyle{v_{2}}$$\scriptstyle{v_{5}}$$\scriptstyle{v_{6}}$

and wonder: does there exist a unique composite 2-cell $v$ in $D$? It has been
shown by Dawson and Paré [DP93] that (in particular) this arrangement of
2-cells admits such a composite $v$. However, not every compatible arrangement
of 2-cells in a general double category admits a composite. The authors of
loc. cit. established that there is an arrangement that does not have a
composite in a general double category, which is called the pinwheel (we will
see this arrangement again in 3.2), and which is in a particular sense, the
canonical example of such an ‘illegal’ arrangement (see [Daw95]). This
procedure of obtaining new 2-cells by composing compatible arrangements, is
also called pasting.

Pasting problems occur in many categorical structures. Nowadays, a wide range
of a pasting theorems for (strict) $\omega$-categories are available. Forest
[For22] has recently unified the main pasting theorems in this context to a
more general pasting theorem for $\omega$-categories. The emergence of (weak)
$\infty$-category theory has created a need for variants of these pasting
theorems in the weaker setting. In this context, it is no longer natural to
ask for unique composites, but instead ask that the space111A space always
means an $\infty$-groupoid in this paper. of composites is contractible.
Lately, Hackney, Ozornova, Riehl and Rovelli [HORR21] have proven a pasting
theorem for $(\infty,2)$-categories, which generalizes Power’s pasting theorem
[Pow91] in two dimensions, to the $\infty$-setting. There is an
$\infty$-analogue for double categories as well, so called double
$\infty$-categories, introduced by Haugseng in [Hau13]. Thus, one may now ask:
does there exist a pasting theorem for these double $\infty$-categories?

This goal of this paper is to answer this question affirmatively. We will
treat a more general pasting problem for multifold categories in the
$\infty$-categorical setting. Although we have only spoken about double
$(\infty-)$categories so far, we will not restrict to the two dimensional
case, but instead prove a pasting theorem in every dimension. Treating this
general case felt more natural to the author as the demonstration of the
theorem proceeds by an induction on the dimension. Note that this is already
interesting from the classical perspective: the pasting theorem in particular
specializes to a novel pasting theorem for multifold categories.

To model higher variants of multifold categories (without completeness), we
will use iterated Segal spaces. Recall that ordinary Segal spaces can be used
to model $\infty$-categories222Again without completeness, one needs to invert
Dwyer-Kan equivalences between Segal spaces which corresponds to restricting
to the complete Segal spaces.; a model which was originally studied by Rezk in
[Rez01]. Iterated Segal spaces are variants of these which have additional
directions for morphisms. For instance, a $2$-uple Segal space $X$ contains a
space of objects, and between any two objects, a space of vertical and
horizontal arrows. Thus a 2-uple Segal space has two categorical directions
instead of merely one as is the case for an ordinary Segal space. Compatible
arrows of $X$ that have the same direction, can be composed in a coherently
associative fashion. Moreover, $X$ contains 2-cells. A 2-cell may be pictured
as a square

${a}$${b}$${c}$${d}$

in $X$. Here the arrows that point horizontally are horizontal arrows of $X$
and similarly for the vertical ones. That is, 2-cells have a source and target
vertical arrow and a source and target horizontal arrow. Again, $X$ has a
coherently associative composition for these 2-cells, which is compatible with
the composition of 1-cells. In general, $d$-uple Segal spaces contain $d$
categorical directions, which may interact using higher cells in a coherent
fashion.

These $d$-uple Segal spaces are rich structures which play a useful role in
category theory. For instance, by ‘truncating’ all but one of the categorical
directions and imposing a completeness condition, they can be used to model
$(\infty,d)$-categories. This is the model for $(\infty,d)$-categories that is
due to Barwick [Bar05] which we will briefly discuss in 5. It has been
(directly) compared to other models for $(\infty,d)$-categories by Bergner and
Rezk in [BR13] and [BR20], and Loubaton, Ozornova and Rovelli in [Lou22] and
[OR22]. Consequently, $d$-uple Segal spaces may act as an useful intermediate
step towards constructing $(\infty,d)$-categories. To construct the
$(\infty,d)$-category of iterated spans in an $\infty$-category with finite
limits, it is easer to first define the $d$-uple Segal space of iterated spans
[Hau18]. In dimension two, $d=2$, double $\infty$-categories are precisely
those 2-uple Segal spaces which satisfy a certain completeness assumption. In
upcoming work of the author [Rui], we study $\infty$-equipments: a
generalization of proarrow equipments to the $\infty$-categorical world. These
$\infty$-equipments are particular kinds of double $\infty$-categories which
offer a context in which to do formal category theory. There exists suitable
$\infty$-equipments for equivariant, parametrized and ordinary
$\infty$-category theory. Moreover, these $\infty$-equipments may also be used
to construct $(\infty,2)$-functors which are otherwise hard to write down.

In this paper, we will introduce $d$-dimensional pasting shapes whose nerve
classify arrangements of cells in a $d$-uple Segal space. For $d=3$, these
include arrangements of rectangular cuboids, whose faces may be subdivided
into smaller rectangles. The faces of these rectangles may in turn be
subdivided into smaller edges. The main result of this paper, the pasting
theorem (3.2), asserts that the nerve functor preserves certain colimits. In
particular, the pasting theorem can be used to show 3.3 which states that the
spine inclusion associated to a composable pasting shape, is an equivalence.
The composable pasting shapes thus classify arrangements of cells which admit
a composite that is unique up to contractible choice (see 3.3). These
composable pasting shapes form a particular class of well-behaved pasting
shapes. For $d=2$, the aforementioned pinwheel of Dawson and Paré is an
example of a pasting shape which is not composable. Furthermore, 3.3 is a
result of the pasting theorem, which gives another description for $d$-uple
Segal spaces: they can be viewed as presheaves on a certain class of
$d$-dimensional pasting shapes, which satisfy a Segal condition.

As a final remark, we would like to mention that we work within the framework
of $\infty$-categories throughout this article (for definiteness, the reader
may choose the quasi-categories of Joyal and Lurie to model
$\infty$-categories). However, if the reader prefers the language of model
categories, one may interpret the main results of this paper within the
appropiate model categories associated to the $\infty$-categories in question.

### Acknowledgements

I want to thank my PhD-supervisor, Lennart Meier, for the helpful
conversations during the writing of this paper and his useful comments on the
draft versions.

During the writing of this paper, the author was funded by the Dutch Research
Council (NWO) through the grant “The interplay of orientations and symmetry”,
grant no. OCENW.KLEIN.364.

##  2\. Preliminaries

In this section, we recall the general definition of $d$-uple categorical
objects in an $\infty$-category $\mathscr{C}$ which admits all finite limits.

######  Definition .

A categorical object $X$ in $\mathscr{C}$ is a simplicial object
$X:\Delta^{\mathrm{op}}\rightarrow\mathscr{C}$ such that the Segal map

$X([n])\rightarrow X(\\{0\leq
1\\})\times_{X(\\{1\\})}\dotsb\times_{X(\\{n-1\\})}X(\\{n-1\leq n\\})$

is an equivalence for all $n$. The full subcategory of
$\mathrm{Fun}(\Delta^{\mathrm{op}},\mathscr{C})$ spanned by the categorical
objects in $\mathscr{C}$ is denoted by $\textup{Cat}(\mathscr{C})$. We define
the category of $d$-uple categorical objects in $\mathscr{C}$ by

$\textup{Cat}^{d}(\mathscr{C}):=\textup{Cat}(\dotsb\textup{Cat}(\textup{Cat}(\mathscr{C}))\dotsb).$

We will refer to $d$-uple categorical objects in the category of sets Set and
in the $\infty$-category of spaces $\mathscr{S}$ as respectively $d$-uple
Segal sets and spaces.

######  Remark .

By adjunction, the category of $d$-uple categorical objects in $\mathscr{C}$
can be described as a full subcategory of

$\mathrm{Fun}(\Delta^{\mathrm{op},\times d},\mathscr{C}),$

the $d$-uple simplicial objects in $\mathscr{C}$. If $\mathscr{C}$ is
presentable then we can describe $\textup{Cat}^{d}(\mathscr{C})$ as a left
reflective subcategory (i.e. a left localization) of the $d$-uple simplicial
objects in $\mathscr{C}$.

This goes as follows. Let us write

$\Delta[n_{1},\dotsc,n_{d}]:\Delta^{\mathrm{op},\times
d}\rightarrow\textup{Set}\subset\mathscr{S}$

for the discrete presheaf represented by
$([n_{1}],\dotsc,[n_{d}])\in\Delta^{\times d}$. We consider the category
$\Lambda^{\times d}/([n_{1}],\dotsc,[n_{d}])$ defined by the pullback square

${\Lambda^{\times d}/([n_{1}],\dotsc,[n_{d}])}$${\Delta^{\times
d}_{\mathrm{in}}/([n_{1}],\dotsc,[n_{d}])}$${\Lambda^{\times
d}}$${\Delta_{\mathrm{in}}^{\times d}.}$

Here $\Delta_{\mathrm{in}}$ is the subcategory of $\Delta$ spanned by the
inert maps: i.e. maps that are injective and have convex image. The category
$\Lambda$ is in turn the full subcategory of $\Delta_{\mathrm{in}}$ spanned by
$[0]$ and $[1]$. The spine inclusion for $([n_{1}],\dotsc,[n_{d}])$ is now
defined to be the map of presheaves

$i_{n_{1},\dotsc,n_{d}}:\operatorname{\mathrm{colim}}_{([k_{1}],\dotsc,[k_{d}])\in\Lambda^{\times
d}/([n_{1}],\dotsc,[n_{d}])}\Delta[k_{1},\dotsc,k_{d}]\rightarrow\Delta[n_{1},\dotsc,n_{d}].$

We will see a more geometrically flavored definition of this map in 3.

If $I$ is a (small) set of compact generators of $\mathscr{C}$, we obtain a
set $S$ of maps of $d$-uple simplicial $\mathscr{C}$-objects whose objects are
given by

$i_{n_{1},\dotsc,n_{m}}\otimes c,\quad n_{1},\dotsc,n_{m}\in\mathbb{N},c\in
I.$

The $S$-local objects are now precisely the $d$-uple categorical objects in
$\mathscr{C}$. Here $\otimes$ denotes the canonical tensor product
$\mathscr{S}\times\mathscr{C}\rightarrow\mathscr{C}$. Using the theory of
Bousfield localizations, we obtain a functor

$L:\mathrm{Fun}(\Delta^{\mathrm{op},\times
d},\mathscr{C})\rightarrow\textup{Cat}^{d}(\mathscr{C}),$

which witnesses $\textup{Cat}^{d}(\mathscr{C})$ as the localization of
$\mathrm{Fun}(\Delta^{\mathrm{op},\times d},\mathscr{C})$ at the set of
morphisms $S$. It is right adjoint to the inclusion
$\textup{Cat}^{d}(\mathscr{C})\subset\mathrm{Fun}(\Delta^{\mathrm{op},\times
d},\mathscr{C})$.

######  Definition .

We say that a map between $d$-uple simplicial spaces is a Segal equivalence if
it is inverted by the functor $L:\mathrm{Fun}(\Delta^{\mathrm{op},\times
d},\mathscr{S})\rightarrow\textup{Cat}^{d}(\mathscr{S})$.

The definition of $d$-uple categorical objects directly implies the following:

######  Proposition .

Suppose that $f:\mathscr{C}\rightarrow\mathscr{D}$ is a finite limit
preserving functor between $\infty$-categories that admit all finite limits.
Then the induced functor $\mathrm{Fun}(\Delta^{\mathrm{op},\times
d},\mathscr{C})\rightarrow\mathrm{Fun}(\Delta^{\mathrm{op},\times
d},\mathscr{D})$ restricts to a functor
$\textup{Cat}^{d}(\mathscr{C})\rightarrow\textup{Cat}^{d}(\mathscr{D})$.

We will apply the above in particular to the inclusion
$\textup{Set}\rightarrow\mathscr{S}$. This inclusion preserves limits, hence
we obtain an induced inclusion

$\textup{Cat}^{d}(\textup{Set})\rightarrow\textup{Cat}^{d}(\mathscr{S}).$

This is an example of a nerve functor. We leave this inclusion implicit, and
view every $d$-uple Segal set as a (discrete) $d$-uple Segal space. We will
make a lot of use of the following observation:

######  Proposition .

Suppose that we have a pushout diagram

${A}$${X}$${B}$${Y}$

of $d$-uple simplicial sets such that $A\rightarrow B$ is injective and a
Segal equivalence. Then $X\rightarrow Y$ is injective and a Segal equivalence
as well.

###### Proof.

Since Segal equivalences are closed under pushouts in
$\mathrm{Fun}(\Delta^{\mathrm{op},\times d},\mathscr{S})$, it suffices to show
that the pushout square is preserved under the functor

$\mathrm{Fun}(\Delta^{\mathrm{op},\times
d},\textup{Set})\rightarrow\mathrm{Fun}(\Delta^{\mathrm{op},\times
d},\mathscr{S}).$

This can be checked levelwise. The inclusion
$\textup{Set}\rightarrow\mathscr{S}$ factors as

$\textup{Set}\rightarrow\textup{sSet}\rightarrow\mathscr{S},$

where the functor $\textup{sSet}\rightarrow\mathscr{S}$ witnesses the
$\infty$-category of spaces $\mathscr{S}$ as the localization of the category
of simplicial sets sSet at the weak homotopy equivalences, i.e. it witnesses
$\mathscr{S}$ as the underlying $\infty$-category of sSet equipped with the
Kan-Quillen model structure. The levelwise pushout squares are colimiting
cones of a Reedy cofibrant diagram
$\\{\cdot\leftarrow\cdot\rightarrow\cdot\\}\rightarrow\textup{Set}\rightarrow\textup{sSet}$
as all objects are cofibrant, and one arrow is a monomorphism. Hence these are
carried to colimiting cones under the localization functor
$\textup{sSet}\rightarrow\mathscr{S}$. That is precisely to say that the
levelwise pushout squares are preserved. ∎

######  Remark .

The $\infty$-category of double $\infty$-categories is given by
$\textup{Cat}(\textup{Cat}_{\infty})$. Joyal and Tierney have shown that
$\infty$-categories may be identified with the complete Segal spaces (see 5
for a definition of completeness and precise statement of this theorem). In
turn, one may use this result to identify
$\textup{Cat}(\textup{Cat}_{\infty})$ with the (reflective) full subcategory
of $\textup{Cat}^{2}(\mathscr{S})$ spanned by the 2-uple Segal spaces $X$ such
that $X_{0,\bullet}$ and $X_{1,\bullet}$ are complete.

##  3\. Pasting shapes and their nerves

We commence this section by introducing pasting shapes and their nerves. The
remaining of this section will be devoted to studying basic properties of
these pasting shapes and introducing the pasting theorem.

######  Definition .

A $(d,k)$-box (resp. non-degenerate $(d,k)$-box) is a pair

$(x,y)\in\mathbb{N}^{\times d}\times\mathbb{N}^{\times d}$

such that $x_{a}=y_{a}$ for $d-k$ indices $a$, and $x_{a}\leq y_{a}$ (resp.
$x_{a}<y_{a}$) for the remaining $k$ indices $a$. We denote the set of
$(d,k)$-boxes by $B^{d,k}$.

We say that two (non-degenerate) $(d,k)$-boxes $(x,y)$,
$(x^{\prime},y^{\prime})$ are adjacent if
$x_{a}=y_{a}=x^{\prime}_{a}=y^{\prime}_{a}$ for $d-k$ indices $a$, and out of
the remaining $k$ indices $a_{1},\dotsc,a_{k}$, there is at most one index $i$
such that $x_{a_{i}}<y_{a_{i}}=x^{\prime}_{a_{i}}<y^{\prime}_{a_{i}}$ and for
the other indices $j$, we have
$x_{a_{j}}=x^{\prime}_{a_{j}}<y^{\prime}_{a_{j}}=y_{a_{j}}$. In this case
$(x,y^{\prime})$ is a non-degenerate $(d,k)$-box, which we will call the join
of $(x,y)$ and $(x^{\prime},y^{\prime})$.

######  Definition .

A $d$-dimensional pasting shape $I$ is a subset

$B^{d}(I)\subset B^{d,d}$

of boxes in $I$ giving rise to a chain

$B^{0}(I)\subset B^{1}(I)\subset\dotsb\subset B^{d}(I),$

where $B^{k}(I):=B^{d,k}\cap B^{d}(I)$ are called the $k$-boxes in $I$,
satisfying the following two properties:

  1. (1)

closure under faces: for every box $(x,y)$ in $I$, the boxes
$(x^{\prime},y^{\prime})$ such that
$x^{\prime}_{a},y^{\prime}_{a}\in\\{x_{a},y_{a}\\}$ for each index $a$, are in
$I$

  2. (2)

closure under joins: if $(x,y)$ and $(x^{\prime},y^{\prime})$ are adjacent
$k$-boxes in $I$, then its join $(x,y^{\prime})$ must be contained in $I$.

Note that $B^{0}(I)$ is the diagonal of a subset
$V(I)\subset\mathbb{N}^{\times d}$, which we will call the vertices of $I$.
Any box $(x,y)$ in $I$ has $x,y\in V(I)$.

A map of pasting shapes $f:I\rightarrow J$ between $d$-dimensional pasting
shapes $I$ and $J$ is a map $f:V(I)\rightarrow V(J)$ between their underlying
sets of vertices, such that for any box $(x,y)$ in $I$, $((f(x),f(y))$ is a
box in $J$ with the property that $f(x)_{a}=f(y)_{a}$ whenever $x_{a}=y_{a}$.
With these maps, the $d$-dimensional pasting shapes form a category which we
will denote by $\textup{Shape}^{d}$.

######  Example .

The $1$-dimensional pasting shapes may be identified with subposets of
$\mathbb{N}$.

######  Example .

We may view $d$-dimensional pasting shapes as being particular graphs equipped
with extra markings. For instance, for $d=2$, the following graph

${(0,0)}$${(1,0)}$${(2,0)}$${(4,0)}$${(0,1)}$${(1,1)}$${(2,1)}$${(3,1)}$${(4,1)}$${(3,2)}$${(4,2)}$${(0,3)}$${(2,3)}$${(3,3)}$${(4,3)}$

depicts the smallest $2$-dimensional pasting shape $I^{\prime}$ whose vertices
are given by the vertices in the graph, and contain those 1-boxes $(x,y)$ so
that there exists a (directed) path of edges between $x$ and $y$. Note that
the resulting pasting shape has no non-degenerate 2-boxes. To remedy this, our
picture needs to reflect which 2-boxes needs to be included in the associated
pasting shape.

In this article, we will use the convention to color the backgrounds of the
2-boxes we would like to include. Thus the following picture

${(0,0)}$${(1,0)}$${(2,0)}$${(4,0)}$${(0,1)}$${(1,1)}$${(2,1)}$${(3,1)}$${(4,1)}$${(3,2)}$${(4,2)}$${(0,3)}$${(2,3)}$${(3,3)}$${(4,3)}$

depicts the $2$-dimensional pasting shape $I$ which contains $I^{\prime}$ and
all the non-degenerate 2-boxes whose faces are in $I$ and enclose a colored
region in the picture. In this case, this are precisely all non-degenerate
2-boxes whose faces are in $I^{\prime}$. In 4.1, we see an example of a
colored graph where this is not the case.

For $d=3$, we can draw similar pictures. Accompanied with the data of
$2$-boxes and $3$-boxes, the following graph

${(0,0,0)}$${(1,0,0)}$${(2,0,0)}$${(0,0,1)}$${(1,0,1)}$${(2,0,1)}$${(0,0,2)}$${(1,0,2)}$${(2,0,2)}$${(1,1,0)}$${(2,1,0)}$${(0,1,1)}$${(1,1,1)}$${(2,1,1)}$${(0,1,2)}$${(1,1,2)}$${(2,1,2)}$${(0,2,0)}$${(1,2,0)}$${(2,2,0)}$${(0,2,1)}$${(1,2,1)}$${(0,2,2)}$${(1,2,2)}$${(2,2,2)}$
would define a pasting shape $J$. The graph above determines $B^{1}(J)$, but
it now becomes visually intractable to color 2- and 3-boxes in the graph.
Instead, we will describe the sets of 2- and 3-boxes in this case. For
instance, we may take

$\displaystyle B^{3}(J)\setminus B^{2}(J)=N^{3}:=\\{$
$\displaystyle((0,0,0),(1,2,2)),((1,0,0),(2,1,1)),((1,0,1),(2,1,2)),$
$\displaystyle((1,0,0),(2,1,2)),((1,1,0),(2,2,2)),((1,0,0),(2,2,2)),$
$\displaystyle((0,0,0),(2,2,2))\\},$

for the non-degenerate 3-boxes of $J$. For the set

$B^{2}(J)\setminus B^{1}(J)$

of non-degenerate 2-boxes of $J$, we can pick all those non-degenerate 2-boxes
$(x,y)$ with the property that the faces of $(x,y)$ are in the graph above,
and there exist a face $(\alpha,\omega)\in B^{3,2}$ of a box in $N^{3}$ such
that $\alpha_{a}\leq x_{a},y_{a}\leq\omega_{a}$ for all $a$.

Note that pasting shapes have a notion of subshapes, unions and intersections:

######  Definition .

Let $I$ be a pasting shape. A subshape $I^{\prime}\subset I$ of $I$ is a
pasting shape $I^{\prime}$ such that $B^{d}(I^{\prime})\subset B^{d}(I)$.

A subshape $E\subset I$ is called a $k$-entire subshape of $I$ if there there
exists a non-degenerate $k$-box $(\alpha,\omega)$ whose faces are in $I$, with
the property that any box $(x,y)$ of $I$ is contained in $E$ if and only if
$\alpha_{a}\leq x_{a},y_{a}\leq\omega_{a}$ for all $a$. In this case, the pair
$(\alpha,\omega)$ is called the pair of bounding box of $E$. We say that $E$
is closed if $(\alpha,\omega)\in I$, in which case also $(\alpha,\omega)$ is
also contained in $E$. If $E$ is $k$-entire and every $k$-box of $E$ is
degenerate, then $E$ is called open.

A $k$-entire subshape $V\subset I$ is called a $k$-vertebra of $I$ if for any
$k$-entire subshape $E\subset V$, we have $E=V$.

If $k=d$, we will drop $d$ from the notation, and consequently call $d$-entire
subshapes and $d$-vertebrae of $I$, respectively, entire subshapes and
vertebrae.

######  Remark .

For a $d$-dimensional pasting shape belong to a certain good class of pasting
shapes, its vertebrae should be viewed as certain ‘indecomposable’
$d$-dimensional subshapes so that taking their union will always recover the
original pasting shape. We will make this precise in 3.2 and 3.3.

######  Definition .

Let $I$ and $J$ be pasting shapes. Then we define the union of $I$ and $J$,
$I\cup J$, to be the smallest pasting shape that contains both $I$ and $J$.
Moreover, we define the intersection of $I$ and $J$, $I\cap J$, to be the
largest pasting shape that is contained in $I$ and $J$. Concretely, $I\cap J$
has boxes given by $B^{d}(I\cap J)=B^{d}(I)\cap B^{d}(J)$.

We have a few important pasting shapes. For natural numbers
$n_{1},\dotsc,n_{d}$, we define the $d$-dimensional pasting shape

$\square[n_{1},\dotsc,n_{d}]$

to be the largest pasting shape whose vertices are the $d$-tuples contained in
$\\{0,\dotsc,n_{1}\\}\times\dotsb\times\\{0,\dotsc,n_{d}\\}$, i.e. its boxes
are given by

$B^{d}(\square[n_{1},\dotsc,n_{d}])=\\{(x,y)\in B^{d,d}\mid 0\leq
x_{a},y_{a}\leq n_{a}\text{ for all $a$}\\}.$

With the obvious structure maps, these pasting shapes assemble to a $d$-uple
cosimplicial object in $\textup{Shape}^{d}$

$\Delta^{\times
d}\rightarrow\textup{Shape}^{d}:(n_{1},\dotsc,n_{d})\mapsto\square[n_{1},\dotsc,n_{d}].$

We will show that this cosimplicial object satisfies the co-Segal property,
allowing us to define a nerve functor for pasting shapes.

######  Proposition .

There is a bijection

$\textstyle\textup{Shape}^{d}(\square[n_{1},\dotsc,n_{d}],I)\rightarrow\mathrm{Hom}_{I}(\coprod_{1\leq
a\leq d}[n_{a}],\mathbb{N}),$

natural in $([n_{1}],\dotsc,[n_{d}])\in\Delta^{\times d}$, where the right-
hand side is the subset

$\textstyle\mathrm{Hom}_{I}(\coprod_{1\leq a\leq
d}[n_{a}],\mathbb{N})\subset\textup{Poset}(\coprod_{1\leq a\leq
d}[n_{a}],\mathbb{N})=\prod_{1\leq a\leq d}\textup{Poset}([n_{a}],\mathbb{N})$

of those tuples $(f_{1},\dotsc,f_{d})$ which have the property that each box
$(x,y)$ with $x_{a},y_{a}\in\mathrm{im}(f_{a})$ for all $a$, is contained in
$I$.

###### Proof.

The key observation is the following. Let
$f:\square[n_{1},\dotsc,n_{d}]\rightarrow I$ be a map of pasting shapes, and
define

$f_{a}(i):=f(0,\dotsc,0,i,0,\dotsc,0)_{a}.$

Then we claim that the image of a vertex $x\in\square[n_{1},\dotsc,n_{d}]$ is
given by $f(x)=(f_{1}(x_{1}),\dotsc,f_{d}(x_{d})).$ Indeed, for any index
$1\leq a\leq d$, we may consider the projection
$p^{a}:=(0,\dotsc,0,x_{a},0,\dotsc,0)$ of $x$. Then $(p^{a},x)$ is a
$(d-1)$-box of $\square[n_{1},\dotsc,n_{d}]$. By the definition of pasting
maps, we must then have that $f_{a}(x_{a})=f(p^{a})_{a}=f(x)_{a}$.

The correspondence is now given as follows: we carry a map of pasting shapes
$f:\square[n_{1},\dotsc,n_{d}]\rightarrow I$ to the tuple
$(f_{1},\dotsc,f_{d})$ defined above. It is clear that each $f_{a}$ is a map
of posets $[n_{a}]\rightarrow\mathbb{N}$. There is an inverse to this
correspondence: we may carry a tuple
$(f_{1},\dotsc,f_{d})\in\mathrm{Hom}_{I}(\coprod[n_{a}],\mathbb{N})$, to the
unique map of pasting shapes $f:\square[n_{1},\dotsc,n_{d}]\rightarrow I$
which is given on vertices by $f(x):=(f_{1}(x_{1}),\dotsc,f_{d}(x_{d})).$ ∎

######  Corollary .

The structure maps of $\square[\bullet,\dotsc,\bullet]$ witness the colimit

$\operatorname{\mathrm{colim}}_{[k_{a}]\in\Lambda/[n_{a}]}\square[n_{1},\dotsc,n_{a-1},k_{a},n_{a+1},\dotsc,n_{d}]=\square[n_{1},\dotsc,n_{d}].$

in $\textup{Shape}^{d}$.

###### Proof.

This can be readily deduced from 3 and the fact that boxes in a pasting shape
are closed under joins. ∎

This observation now leads to the following construction:

######  Construction .

Given a $d$-dimensional pasting shape $I$, we define a $d$-uple Segal set
$[I]$, the nerve of $I$, by setting

$[I]_{n_{1},\dotsc,n_{d}}:=\textup{Shape}^{d}(\square[n_{1},\dotsc,n_{d}],I).$

This construction gives rise to a nerve functor

$[-]:\textup{Shape}^{d}\rightarrow\textup{Cat}^{d}(\textup{Set}).$

Note that the simplices of the nerve may be described using 3. In particular,
one can use this description to readily verify that $[-]$ is fully faithful.

######  Corollary .

The $d$-uple cosimplicial object $[\square[\bullet,\dotsc,\bullet]]$ concides
with $\Delta[\bullet,\dotsc,\bullet]$.

###### Proof.

In light of 3, we see that

$[\square[m_{1},\dotsc,m_{d}]]_{n_{1},\dotsc,n_{d}}\cong\prod_{1\leq a\leq
d}\Delta([n_{a}],[m_{a}]),$

natural in all variables. ∎

###  3.1. Truncations of pasting shapes and grids

In order to define a good class of pasting shapes, we need to introduce the
building blocks of these shapes: so-called grids. These are pasting shapes
that are in some sense shaped by the standard grids
$\square[n_{1},\dotsc,n_{d}]$. Let us start with an illustrative example.
Consider the obvious injective map of pasting shapes

$\square[2,1]\xrightarrow{\makebox[45.0pt]{}}\leavevmode\hbox to143.28pt{\vbox
to88.06pt{\pgfpicture\makeatletter\hbox{\hskip
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{}{}{}}{}{{}}{}{ {}{}{}} {}{}{ {}{}{}} {}{}{ {}{}{}}
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Note that this map does not create any new 2-cells, but factors one of the
boundary 1-cells. The codomain is in this sense shaped by the domain (modulo
boxes of dimension $\leq 1$, they agree), and would be an example of a grid.
We want to make this precise and capture this notion. First, we observe that
$d$-uple simplicial sets support a notion of dimension:

######  Definition .

Suppose that $S$ is a $d$-uple simplicial set. Then a $d$-uple simplex
$\sigma:\Delta[n_{1},\dotsc,n_{d}]\rightarrow S$ is called of dimension $\leq
k$ if there exists a factorization

$\Delta[n_{1},\dotsc,n_{d}]\rightarrow\Delta[m_{1},\dotsc,m_{d}]\rightarrow S$

of $\sigma$ such that there are at most $k$ indices $i$ such that $m_{i}\neq
0$. Moreover, we say that $\sigma$ is of dimension $k$ if it is of dimension
$\leq k$ but not of dimension $\leq(k-1)$. The $d$-uple simplicial subset of
$S$ containing those $d$-uple simplices of dimension $\leq k$ is called the
$k$-truncation of $S$ and denoted by

$\mathrm{tr}_{\leq k}S\subset S.$

Whenever $S$ is the nerve of a pasting shape, the truncation
$\mathrm{tr}_{\leq k}S$ can be described as follows.

######  Definition .

Let $I$ be a $d$-dimensional pasting shape. Then for $0\leq k\leq d$, we
define its $k$-truncation $\mathrm{tr}_{\leq k}I\subset I$ to be the pasting
shape with boxes

$B^{i}(\mathrm{tr}_{\leq k}I)=\begin{cases}B^{i}(I)&\text{if $i\leq k$},\\\
B^{k}(I)&\text{if $i\geq k$}.\end{cases}$

We say that $I$ is k-truncated if $\mathrm{tr}_{\leq k}I=I$, i.e., whenever
every $l$-box of $I$ is degenerate for $l>k$.

######  Example .

The following is an important class of examples of $k$-truncated
$d$-dimensional pasting shapes. For $d-k$ different indices $1\leq
a_{1},\dotsc,a_{d-k}\leq d$ and integers $c_{1},\dotsc,c_{d-k}$, we can
consider the $k$-dimensional hyperplane $H$ whose boxes are given by

$B^{d}(H)=\\{(x,y)\in B^{d,d}\mid x_{a_{i}}=y_{a_{i}}=c_{i}\text{ for all
$i$}\\}.$

We will denote the set of $k$-dimensional hyperplanes by $\mathscr{H}_{k}$.

The following can readily be deduced from the two definitions of truncations
that we have seen above:

######  Proposition .

Suppose that $I$ is a $d$-dimensional pasting shape, and let $0\leq k\leq d$.
Then

$\mathrm{tr}_{\leq k}I=\bigcup_{H\in\mathscr{H}_{k}}I\cap H,$

and

$\mathrm{tr}_{\leq k}[I]=[\mathrm{tr}_{\leq
k}I]=\bigcup_{H\in\mathscr{H}_{k}}[I\cap H].$

Using this notion of truncations, we can give a description of what it means
for a $d$-dimensional pasting shape $I$ to shape a $d$-dimensional pasting
shape $J$ modulo $(d-1)$-boxes.

######  Proposition .

Let $f:I\rightarrow J$ be an injective map of $d$-dimensional pasting shapes.
Then the following are equivalent:

  1. (1)

every non-degenerate $d$-box of $J$ is the image of a non-degenerate $d$-box
in $I$,

  2. (2)

$J$ can be written as

$J=f(I)\cup\mathrm{tr}_{\leq d-1}J,$

  3. (3)

every $d$-dimensional simplex of $[J]$ lies in $f([I])$,

  4. (4)

the nerve of $J$ can be written as

$[J]=f([I])\cup\mathrm{tr}_{\leq d-1}[J].$

###### Proof.

We only prove that (1) implies (3). The remaining implications are readily
verified. Suppose that we have a $d$-dimensional simplex of $[J]$
corresponding to a map

$g:\square[n_{1},\dotsc,n_{d}]\rightarrow J,$

then we must show that $f$ factors through $I$. Since $f$ is injective, it
suffices to show that $g$ has image in $f(I)$. We may assume that each map
$g_{a}:[n_{a}]\rightarrow\mathbb{N}$ is injective and $n_{a}\neq 0$. Now, any
box $(x,y)$ of $\square[n_{1},\dotsc,n_{d}]$ lies in the image of an injective
map $i:\square[1,\dotsc,1]\rightarrow\square[n_{1},\dotsc,n_{d}]$. Thus it
suffices to show that $gi$ has image in $f(I$). But $gi$ classifies a non-
degenerate $d$-box of $J$, hence has image in $f(I)$ by assumption. ∎

######  Definition .

We call a map of $d$-dimensional pasting shapes $I\rightarrow J$ $d$-shaping
if it is injective and satisfies any of the equivalent conditions above.

With this terminology now in place, we can introduce the following notion:

######  Definition .

A $d$-dimensional pasting shape $A$ is called an open grid, respectively, a
closed grid, if there exists a map

$f:\mathrm{tr}_{\leq d-1}\square[n_{1},\dotsc,n_{d}]\rightarrow A,\text{
resp.\ a $d$-shaping map }f:\square[n_{1},\dotsc,n_{d}]\rightarrow A,$

with $n_{1},\dotsc,n_{d}\neq 0$, having the property that any $(d-1)$-box
$(x,y)$ in $A$ satisfies $0\leq x_{a},y_{a}\leq f_{a}(n_{a})$ for all indices
$a$, and there exists an index $b$ such that
$x_{b}=y_{b}\in\mathrm{im}(f_{b})$.

######  Proposition .

Let $A$ be a grid witnessed by a map $f$. Then this map $f$ is unique: every
other map that witnesses $A$ to be a grid, coincides with $f$.

###### Proof.

Note that $A$ cannot be open and closed simultaneously. We restrict to the
case that $A$ is open as the closed case is handled analogously. Let

$g:\mathrm{tr}_{\leq d-1}\square[m_{1},\dotsc,m_{d}]\rightarrow A$

be another map that witnesses $A$ to be a grid. Similarly to the analysis of
3, we can identify $f$ and $g$ respectively with tuples
$(f_{1},\dotsc,f_{d})$, $(g_{1},\dotsc,g_{d})$ such that each $(d-1)$-box
$(x,y)$ with $x_{a},y_{a}\in\mathrm{im}(f_{a})$, resp.
$x_{a},y_{a}\in\mathrm{im}(g_{a})$, is in $A$. Since $f$ and $g$ are
injective, it suffices to show that $\mathrm{im}(f_{b})=\mathrm{im}(g_{b})$
for each index $1\leq b\leq d$. Let $x_{b}\in\mathrm{im}(f_{b})$. Then for
$a\neq b$, we may find integers $x_{a}<y_{a}$ in $\mathrm{im}(f_{a})$ since
$n_{a}\neq 0$. Setting $y_{b}:=x_{b}$, we obtain a non-degenerate $(d-1)$-box
$(x,y)$ in $A$. Since $g$ witnesses $A$ to be a grid, we must then have that
$x_{b}=y_{b}\in\mathrm{im}(g_{b})$. Hence
$\mathrm{im}(f_{b})\subset\mathrm{im}(g_{b})$. Interchanging the roles of $f$
and $g$, we also obtain the reverse inclusion. ∎

The above observation shows that the following definition is well-defined.

######  Definition .

Let $A$ be a grid witnessed by a map $f$. We call the $d$-tuple of points

$((f_{1}(0),f_{1}(n_{1})),\dotsc,(f_{d}(0),f_{d}(n_{d})))$

the corners of $A$. We define the boundary

$\partial A\subset A$

to be the subshape of $A$ that consists of the boxes $(x,y)$ in $A$ such that
$x_{a}=y_{a}=f_{a}(0),f_{a}(n_{a})$ for some index $a$. If
$n_{1}=\dotsb=n_{d}=1$, then $A$ is called a cell.

An inclusion $A\subset B$ between grids whose corners coincide, is called a
subdivision inclusion.

######  Remark .

Note that the following 2-dimensional pasting shape fails to be a grid:

${(0,0)}$${(1,0)}$${(2,0)}$${(1,1)}$${(0,2)}$${(2,2).}$

######  Example .

The pasting shape $\square[n_{1},\dotsc,n_{d}]$ with $n_{1},\dotsc,n_{d}\neq
0$ is of course a grid. Note that its vertebrae are given by the cells

$V_{(j_{1},\dotsc,j_{d})},\quad 0\leq j_{a}\leq n_{a}-1,$

which is the largest $d$-dimensional pasting shape whose vertices $x$ satisfy
$x_{a}=j_{a},j_{a}+1$ for all $a$.

From the definition of grids, it may be readily deduced that the vertebrae of
a grid $A$ witnessed by a map

$f:\mathrm{tr}_{\leq d-1}\square[n_{1},\dotsc,n_{d}]\rightarrow
A,\quad\text{or},\quad f:\square[n_{1},\dotsc,n_{d}]\rightarrow A,$

are given by cells whose corners are given by the corners of $f(V_{j})$, for
some tuple $j$ as above.

###  3.2. Admittable and composable pasting shapes

Suppose that $I$ is a pasting shape. Then we would like to show that $I$ is
the union of its vertebrae

$I=\bigcup_{V\subset I\text{ vertebra}}V,$

and more importantly, that this colimit is preserved by the nerve functor
$[-]:\textup{Shape}^{d}\rightarrow\textup{Cat}^{d}(\mathscr{S})$. We must
however restrict to a good class of pasting shapes: called the composable
pasting shapes. For instance, if $d=2$, then we can consider the famous
pinwheel introduced by Dawson and Paré [DP93]. This is the 2-uple Segal set
associated to the pasting shape

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One can show that mapping out of $[PW]$ into a 2-uple Segal space, is not
equivalent to mapping out of the nerves of its vertebrae in a compatible
fashion. In fact, at the level of pasting shapes, the pinwheel cannot be
written as the union of its vertebrae. We will demonstrate this in 3.2 and
3.3.

Shortly, we will introduce the notion of admittable and composable pasting
shapes. The main goal of this subsection is to study properties of these
particular pasting shapes and see that they are considerably better behaved
than arbitrary pasting shapes. In particular, we will show that any composable
shape can be written as the union of its vertebrae. This can be considered as
an early version of the pasting theorem introduced in 3.3, and the same ideas
will be used as input to prove the full strength of the pasting theorem.

######  Definition .

A $d$-dimensional pasting shape $I$ is called admittable if there exists a
filtration

$I_{0}\subset I_{1}\dotsb\subset I_{n}=I,$

where $I_{0}$ is a grid and $I_{k}=I_{k-1}\cup A_{k}$ for a grid $A_{k}\subset
I$ such that $I_{k-1}\cap A_{k}=\partial A_{k}$ is an open vertebra of
$I_{k-1}$. The corners of $I$ are given by the corners of $I_{0}$ (it is
readily verified that this is independent of the filtration). Moreover, we
define the boundary of $I$

$\partial I\subset I$

to be the smallest subshape of $I$ that contains all boxes $(x,y)$ such that
$x_{a}=y_{a}$ is equal to a coordinate of the $a$’th corner for some index
$a$, or equivalently $\partial I=\partial I_{0}$.

######  Definition .

A $d$-dimensional pasting shape $I$ is called composable if it meets the
following conditions:

  1. (1)

$I$ is closed admittable (i.e. admittable and closed $d$-entire),

  2. (2)

any closed $k$-entire subshape of $I$, with $k<d$, is a $k$-dimensional
admittable pasting shape.

######  Example .

Here are some examples:

  * •

The 1-dimensional admittable pasting shapes correspond to those subposets of
$\mathbb{N}$ which are disjoint unions of intervals. The composable ones
correspond to the intervals.

  * •

An example of a 2-dimensional composable pasting shape is the pasting shape

${(0,0)}$${(2,0)}$${(3,0)}$${(0,1)}$${(1,1)}$${(2,1)}$${(3,1)}$${(0,2)}$${(1,2)}$${(3,2),}$

which classifies one of the relevant composites of the triangle identity in a
2-fold Segal space (see 5).

  * •

The pinwheel $PW$ is not admittable, thus in particular not composable.

  * •

The 2-dimensional pasting shape $I$ and the 3-dimensional pasting shape $J$ of
3 are composable.

We would now like to study these admittable and composable pasting shapes. In
particular, we will show at the end of this subsection the following theorem
which may be considered as a stepping stone to 3.2, the general pasting
theorem:

######  Theorem 3.1.

Let $I$ be a composable pasting shape. Then each vertebra of $I$ is
composable, and $I$ can be written as the union of its vertebrae.

In order to prove structural properties of admittable pasting shapes (it is
also used in the proof of the general pasting theorem), the following notion
of division pairs turns out to be useful. It is precisely the datum of a stage
in a filtration that witnesses admittability (cf. 3.2):

######  Definition .

A division pair for an admittable pasting shape $I$ is a tuple $(K,J)$ of
admittable subshapes of $I$ such that $K\cap J=\partial J$ is an open vertebra
of $K$, and $I=K\cup J$.

######  Proposition .

Let $I$ be an admittable pasting shape. Then the following is true:

  1. (1)

For any open vertebra $V$ of $I$ with corners
$((\alpha_{1},\omega_{1}),\dotsc,(\alpha_{d},\omega_{d}))$, every box $(x,y)$
in $I$ has $x_{a},y_{a}\leq\alpha_{a}$ or $x_{a},y_{a}\geq\omega_{a}$ for some
index $a$.

  2. (2)

If $(K,J)$ is a division pair for an admittable pasting shape $I$, where $J$
has corners given by
$((\alpha_{1},\omega_{1}),\dotsc,(\alpha_{d},\omega_{d}))$, then $J$ is
entire, and for any box $(x,y)$ in $I$ we have that $(x,y)$ is in $K$ if and
only if there exists an index $a$ such that $x_{a},y_{a}\leq\alpha_{a}$ or
$x_{a},y_{a}\geq\omega_{a}$.

###### Proof.

Suppose that $(K,J)$ is an division pair for an admittable pasting shape $I$
such that (1) holds for $K$. Denote the corners of $J$ by
$((\alpha_{1},\omega_{1}),\dotsc,(\alpha_{d},\omega_{d}))$. We inductively
define a subset $N^{k}$ of non-degenerate $k$-boxes of $I$ as follows. We set
$N^{0}:=B^{0}(K)\cup B^{0}(J)$. If $N^{k-1}$ is defined, then we define
$N^{k}$ to be the set of non-degenerate $k$-boxes $(x,y)$ in $I$ such that the
faces of $(x,y)$ are in $N^{k-1}$ and one of the following properties is met:

  1. (i)

$(x,y)\in B^{k}(K)\cup B^{k}(J)$,

  2. (ii)

$\max(x_{a},\alpha_{a})<\min(y_{a},\omega_{a})$ if $x_{a}<y_{a}$ and
$\alpha_{a}\leq x_{a},y_{a}\leq\omega_{a}$ if $x_{a}=y_{a}$.

One now readily checks that the boxes in the subset

$\bigcup_{k=0}^{d}N^{k}\subset B^{d}(I)$

are closed under faces and joins (by induction on $k$), and hence, constitute
a subshape $I^{\prime}\subset I$. By construction, $K$ and $J$ are contained
in $I$. Since $I=K\cup J$, we must have that $I=I^{\prime}$. Thus the non-
degenerate $k$-boxes of $I$ are given by $N^{k}$, and from this description
and the fact that (1) holds for $K$, it can be deduced that (2) holds.

It remains to show that (1) holds for any admittable pasting shape $I$. From
what we have shown so far, it can be deduced that that if $(K,J)$ is a
division pair for $I$ such that (1) holds for $K$ and $J$, then (1) also holds
for $I$. Thus, we may reduce to showing that (1) holds for any grid, which is
clear. ∎

In the demonstration of 3.2 above, we have used division pairs to induct our
way up to each admittable pasting shape. Another convenient way of giving
similarly flavored inductive proofs, is using the notion of heights:

######  Definition .

A decomposition of an admittable pasting shape $I$ is a pair
$(A,\\{J_{i}\\})$, where $A\subset I$ is an open subgrid of $I$, $\\{J_{i}\\}$
is a collection of admittable subshapes of $I$, such the intersections $A\cap
J_{i}=\partial J_{i}$ are all mutually distinct open vertebrae of $A$, and $I$
can be written as

$I=A\cup\bigcup J_{i}.$

We say that an admittable pasting shape $I$ has height 0 if it is a cell.
Inductively, we say that a pasting shape has height $h$ if there exists a
decomposition $(A,\\{J_{i}\\})$ of $I$ where each $J_{i}$ height $h-1$.

The proof of the following proposition shows how one can obtain a
decomposition for an admittable pasting shape:

######  Proposition .

Any admittable pasting shape admits a decomposition and has a height.

###### Proof.

Suppose that $I$ is an admittable pasting shape with a filtration as in 3.2.
Then for $1\leq k\leq n$, we consider the pair

$D_{k}=(\mathrm{tr}_{\leq d-1}I_{0},S_{k}),$

where $S_{k}$ is the set of entire subshapes of $I_{k}$ whose bounding boxes
are given by the corners of vertebrae of $I_{0}$. We show by induction that
$D_{k}$ is a decomposition for $I_{k}$. For $k=0$, this is clear. Suppose that
$D_{k-1}$ is a decomposition for $I_{k-1}$. Now, $I_{k}=I_{k-1}\cup A_{k}$,
where $A_{k}$ is a grid and $A_{k}\cap I_{k-1}=\partial A_{k}$ is a vertebra
of $I_{k-1}$. 3.2 implies that $\partial A_{k}\subset E$ of some subshape
$E\in S_{k-1}$. Let $E^{\prime}$ be the entire subshape in $S_{k}$ with the
same bounding box. Then $E^{\prime}=(I_{k-1}\cup A_{k})\cap E^{\prime}=E\cup
A_{k}$ and $E\cap A_{k}=\partial A_{k}$ must be a vertebra of $E$. Thus
$E^{\prime}$ is again admittable. Moreover, we see that
$S_{k}=S_{k-1}\setminus\\{E\\}\cup\\{E^{\prime}\\}$. All in all, this implies
that $D_{k}$ is a decomposition for $I_{k}$. It is readily verified that $I$
has a height. ∎

######  Remark .

Note that the height of an admittable pasting shape $I$ is not unique. If $I$
has height $h$ then $I$ has all heights greater than $h$ as well.

######  Example .

The 2-dimensional admittable pasting shape $I$ of 3 has (minimal) height $4$.

The following factorization property is an important property of admittable
pasting shapes:

######  Lemma .

Suppose that $I$ is an admittable pasting shape with decomposition
$(A,\\{J_{i}\\})$, where the open grid $A$ is witnessed by the map

$f:\mathrm{tr}_{\leq d-1}\square[n_{1},\dotsc,n_{d}]\rightarrow A.$

Then for any box $(x,y)\in I$, we have $(x,y)\in A$ if and only if
$x_{b}=y_{b}\in\mathrm{im}(f_{b})$ for some index $b$. If $(x,y)\notin A$,
then there exists an injective map

$g:\square[m_{1},\dotsc,m_{d}]\rightarrow I,$

such that

$\mathrm{im}(g_{a})=\\{t_{a}\in\mathrm{im}(f_{a})\mid x_{a}\leq t_{a}\leq
y_{a}\\}\cup\\{x_{a},y_{a}\\}.$

In particular, it follows that any injective map

$h:\square[p_{1},\dotsc,p_{d}]\rightarrow I$

with $p_{1},\dotsc,p_{d}\neq 0$, admits an extension

$h^{\prime}:\square[p_{1}^{\prime},\dotsc,p_{d}^{\prime}]\rightarrow I$

such that $\mathrm{im}(h^{\prime}_{a})=\\{t_{a}\in\mathrm{im}(f_{a})\mid
h_{a}(0)\leq t_{a}\leq h_{a}(p_{a})\\}\cup\mathrm{im}(h_{a})$.

###### Proof.

Similarly as in the proof of 3.2, we inductively define sets $N^{k}$ of non-
degenerate $k$-boxes of $I$. We set
$N^{0}:=B^{0}(A)\cup\bigcup_{i}B^{0}(J_{i})$. If $N^{k-1}$ is defined, we
define $N^{k}$ to be the set of non-degenerate $k$-boxes $(x,y)$ of $I$ whose
faces are in $N^{k-1}$, and which have one of the following properties:

  1. (1)

$(x,y)\in B^{k}(A)$,

  2. (2)

$(x,y)\notin B^{k}(A)$, $x_{a},y_{a}\notin\mathrm{im}(f_{a})$ if
$x_{a}=y_{a}$, and there exists an injective map

$g:\square[m_{1},\dotsc,m_{d}]\rightarrow I,$

such that

$\mathrm{im}(g_{a})=\\{t_{a}\in\mathrm{im}(f_{a})\mid x_{a}\leq t_{a}\leq
y_{a}\\}\cup\\{x_{a},y_{a}\\}.$

It follows from the definition that

$\bigcup_{k=0}^{d}N^{k}\subset B^{d}(I)$

is closed under joins. We will show that it is also closed under faces, and
hence defines a subshape $I^{\prime}\subset I$.

Clearly $N^{0}$ is closed under joins. Suppose that $N^{k-1}$ is closed under
joins. Then we show that the join of adjacent boxes $(x,y)$ and
$(x^{\prime},y^{\prime})$ in $N^{k}$ is again in $N^{k}$. Clearly, the faces
of the join $(x,y^{\prime})$ must be in $N^{k-1}$, since $N^{k-1}$ is assumed
to be closed under joins. Thus we must check that the join $(x,y^{\prime})$
has property (1) or (2). If $k=d$, $(x,y)$ and $(x^{\prime},y^{\prime})$ must
have property (2) since $A$ has no non-degenerate $d$-boxes. If $k<d$, then
$(x,y)$ and $(x^{\prime},y^{\prime})$ either must have both property (1) or
both property (2), since any box $(s,t)$ in $A$ has the property that
$s_{b}=t_{b}\in\mathrm{im}(f_{b})$ for some index $b$. In both cases, it is
readily verified that the join of $(x,y)$ and $(x^{\prime},y^{\prime})$ again
has the same property, hence is in $N^{k}$.

We now claim that the resulting subshape $I^{\prime}$ coincides with $I$. We
then obtain a description of the boxes of $I$, from which the first statement
of the lemma follows. By assumption, we may write $I$ as the union

$I=A\cup J_{1}\cup\dotsb\cup J_{m},$

Thus it suffices that $A$ and the $J_{i}$’s are contained in $I^{\prime}$. The
first assertion is clear. Suppose that $(x,y)$ is a box in $J_{i}$, then
clearly the desired injective map $g$ exists (with $n_{1},\dotsc,n_{d}\leq
1)$. If $x_{b}=y_{b}\in\mathrm{im}(f_{b})$ for some index $b$, then we must
have that $(x,y)$ is contained in the boundary $\partial J_{i}\subset A$. Thus
in either case, we have $(x,y)\in I^{\prime}$.

The latter statement of the lemma follows from the preceding assertion and the
fact that $A$ has no non-degenerate $d$-boxes. ∎

######  Proposition .

Any closed entire subshape $E$ of a $d$-dimensional admittable pasting shape
$I$ is again admittable.

###### Proof.

We proceed by induction on the height $h$ of $I$. If $h=0$, the statement is
clear. Suppose that the statement holds for any admittable pasting shape of
height $h$.

Let $I$ be a composable pasting shape of height $h+1$ with decomposition
$(A,\\{J_{i}\\})$. Write $f:\mathrm{tr}_{\leq
d-1}\square[n_{1},\dotsc,n_{d}]\rightarrow A$ for the map that witnesses $A$
to be an open grid. Denote the bounding box of $E$ by $(x,y)$. Since $E$ is
entire, $(x,y)$ must be a non-degenerate $d$-box contained in $I$.
Consequently, there exists an extension

$g:\square[m_{1},\dotsc,m_{d}]\rightarrow I,$

with $\mathrm{im}(g_{a})=\\{t_{a}\in\mathrm{im}(f_{a})\mid x_{a}\leq t_{a}\leq
y_{a}\\}\cup\\{x_{a},y_{a}\\}$ on account of 3.2. Let $B$ be the open subgrid
of $I$ determined by $g$. We note that for any inert map

$j:\square[1,\dotsc,1]\rightarrow\square[m_{1},\dotsc,m_{d}],$

the restriction $gj$ classifies the corners of a closed entire,
$d$-dimensional subshape $E_{j}\subset E$, which has the property that
$E\subset J_{i}$ for some $i$. Thus $E_{j}$ is admittable by the induction
hypothesis. Note that $E_{j}\cap B=\partial E_{j}$. Using 3.2, one may again
show that

$E=B\cup\bigcup_{j}E_{j}.$

From this it follows that $E$ is admittable with decomposition
$(B,\\{E_{j}\\})$. ∎

Using a similar strategy as employed in the proof of 3.2, we can show a
stronger factorization property for composable pasting shapes:

######  Lemma .

Suppose that $I$ is a $d$-dimensional composable pasting shape and let

$f:\square[n_{1},\dotsc,n_{d}]\rightarrow I$

be an injective map that carries corners to corners. Then for any box
$(x,y)\in I$, there exists an injective map

$g:\square[m_{1},\dotsc,m_{d}]\rightarrow I,$

such that

$\mathrm{im}(g_{a})=\\{t_{a}\in\mathrm{im}(f_{a})\mid x_{a}\leq t_{a}\leq
y_{a}\\}\cup\\{x_{a},y_{a}\\}.$

###### Proof.

We proceed by induction on the dimension $d$. For $d=0$, the statement is
clear. Suppose the statement holds for $(d-1)$-dimensional composable pasting
shapes. If $I$ has height $0$, then $I$ is a cell, and there is nothing to
prove. Suppose that the statement holds for all $d$-dimensional composable
pasting shapes that have height $h$.

Let $I$ be a composable pasting shape of height $h+1$ with decomposition
$(A,\\{J_{i}\\})$. During this proof, we write $N^{k}$ for the set of non-
degenerate $k$-boxes $(x,y)$ of $I$ with the property that there exists an
injective map

$g:\square[m_{1},\dotsc,m_{d}]\rightarrow I,$

such that

$\mathrm{im}(g_{a})=\\{t_{a}\in\mathrm{im}(f_{a})\mid x_{a}\leq t_{a}\leq
y_{a}\\}\cup\\{x_{a},y_{a}\\}.$

Then the subset of boxes given by

$\bigcup_{k=0}^{d}N^{k}\subset B^{d}(I)$

is clearly closed under joins and faces, hence constitutes a subshape
$I^{\prime}\subset I$. It now suffices to show that $I^{\prime}$ contains
$J_{1},\dotsc,J_{n}$ and $A$.

Suppose that $(x,y)$ is a box of $A$. Then there exists an index $b$ so that
$x_{b}=y_{b}\in\mathrm{im}(f_{b})$. The map $f$ restricts to a
$(d-1)$-truncated map

$f^{\prime}:\square[n_{1},\dotsc,n_{b-1},0,n_{b+1},\dotsc,n_{d}]\rightarrow
A\cap H.$

This can be viewed as a map between $(d-1)$-dimensional pasting shapes. As the
slice $A\cap H$ must be $(d-1)$-dimensional composable pasting shape, for
which the statement already holds, it can now readily be deduced that
$(x,y)\in I^{\prime}$.

It remains to show that each box $(x,y)$ of $J_{i}$ is contained in
$I^{\prime}$. We will write

$\phi:\mathrm{tr}_{\leq d-1}\square[p_{1},\dotsc,p_{d}]\rightarrow A,$

for the injective map that witnesses $A$ to be an open grid. On account of
3.2, there exists an injective map

$f^{\prime}:\square[n_{1}^{\prime},\dotsc,n_{d}^{\prime}]\rightarrow I$

such that
$\mathrm{im}(f^{\prime}_{a})=\mathrm{im}(\phi_{a})\cup\mathrm{im}(f_{a})$.
Note that the corners of $J_{i}$ are determined by an inert map

$j:\square[1,\dotsc,1]\rightarrow\square[p_{1},\dotsc,p_{d}],$

which determines a restriction $r$ of $f^{\prime}$ determined by having image

$\mathrm{im}(r_{a})=\\{t_{a}\in\mathrm{im}(f_{a})\mid j_{a}(0)\leq t_{a}\leq
j_{a}(1)\\}\cup\\{j_{a}(0),j_{a}(1)\\}\subset\mathrm{im}(f^{\prime}_{a})$

for all $a$. The lemma holds for $J_{i}$ by assumption as it is composable and
has height $h$, hence we may apply the lemma with respect to $(x,y)$ and $r$,
yielding the desired factorization $g$ for $(x,y)$. ∎

###### Proof of 3.1.

We proceed by induction on the height $h$ of $I$. If $I$ has height $0$, then
there is nothing to show. Suppose now that the theorem holds for any
$d$-dimensional composable pasting shape $I$ of height $h$. Let $I$ be a
composable pasting shape of height $h+1$ with decomposition $(A,\\{J_{i}\\})$.
Then there exists an injective map

$f:\square[n_{1},\dotsc,n_{d}]\rightarrow I$

such that $\mathrm{tr}_{\leq d-1}f$ witnesses $A$ to be an open grid. Then the
vertebrae of $I$ are given by the vertebrae of the $J_{i}$’s. Thus the
induction hypothesis asserts that the vertebrae of $I$ are composable, and
shows that we must demonstrate that every box $(x,y)$ of $I$ is contained in
the union

$\bigcup_{V\subset I\textit{ vertebra }}V=\bigcup_{i}J_{i}\subset I,$

and this readily follows from the factorization property of 3.2. ∎

We may slightly strengthen 3.1 by introducing so-called locally composable
pasting shapes. This is an auxiliary notion, albeit pragmatic notion to set up
the theory.

######  Definition .

A $d$-dimensional pasting shape $I$ with finitely many boxes is called locally
composable if every closed $k$-entire subshape of $I$, with $1\leq k\leq d$,
is a $k$-dimensional admittable pasting shape.

The following is a direct result of 3.2:

######  Proposition .

A $d$-dimensional pasting shape $I$ is composable if and only if $I$ is
locally composable and closed $d$-entire.

######  Corollary .

Suppose that $I$ is a locally composable, then $I$ can be written as the union

$I=\bigcup_{V\subset I\text{ closed k-vertebra, }0\leq k\leq d}V.$

###### Proof.

Suppose that $(x,y)$ is a $k$-box of $I$. Then $(x,y)$ determines a $k$-entire
subshape $E$ of $I$ which is a $k$-dimensional composable pasting shape by
assumption. In view of 3.1, $E$ may be written as the union of its
(necessarily closed) $k$-vertebrae. Hence $(x,y)$ is contained in the union
that is displayed in the statement of the corollary. ∎

######  Example .

Note that 3.1 fails to be true in the setting of non-composable pasting
shapes. Consider the pinwheel $PW$ that was defined at the start of 3.2. Then
there is a subshape

$PW^{\circ}\subset PW$

whose boxes are given by
$B^{2}(PW^{\circ})=B^{2}(PW)\setminus\\{((0,0),(3,3))\\}$. It is readily
verified that $PW^{\circ}$ is locally composable. Since every $1$-vertebra of
$PW^{\circ}$ is contained in a $2$-vertebra of $PW^{\circ}$, 3.2 asserts that

$PW^{\circ}=\bigcup_{V\subset PW^{\circ}\text{ $k$-vertebra, }0\leq k\leq
2}V=\bigcup_{V\subset PW^{\circ}\text{ vertebra}}V.$

But the vertebrae of $PW^{\circ}$ are precisely those of $PW$. Since $PW\neq
PW^{\circ}$, this shows that 3.1 fails for the pinwheel.

###  3.3. The pasting theorem

Now that we have singled out a certain class of pasting shapes, we are able to
formulate the pasting theorem which will be proven in the next section.

######  Definition .

Let $I$ be a $d$-dimensional pasting shape. Then a covering of $I$ is a
collection $I_{1},\dotsc,I_{n}$ of subshapes of $I$, such that

  1. (1)

every closed $k$-entire subshape of $I_{i}$, with $0\leq k\leq d$, is entire
in $I$,

  2. (2)

every closed $k$-vertebra of $I$, with $0\leq k\leq d$, is contained in some
$I_{i}$.

######  Example .

We have the following canonical examples of coverings:

  1. (1)

Every composable pasting shape is covered by its vertebrae.

  2. (2)

In general, if $I$ is a locally composable pasting shape, then the collection
of closed $k$-vertebrae of $I$, where $k$ ranges over all dimensions
$0,\dotsc,d$, is a covering of $I$.

  3. (3)

If $I$ is an admittable pasting shape with decomposition $(A,\\{J_{i}\\})$
then $A$ together with the $J_{i}$’s form a covering for $I$.

######  Theorem 3.2 (The pasting theorem).

Suppose that $I_{1},\dotsc,I_{n}$ is a covering of a $d$-dimensional locally
composable pasting shape $I$. Then $I$ can be written as

$I=\bigcup_{i=1}^{n}I_{i},$

and this union is preserved by the nerve functor so that the canonical map

$\operatorname{\mathrm{colim}}_{[k]\in\Delta^{\mathrm{op}}}\left(\coprod_{\sigma:\\{0,\dotsc,k\\}\rightarrow\\{1,\dotsc,n\\}}[I_{\sigma(0)}\cap\dotsb\cap
I_{\sigma(k)}]\right)\rightarrow[I]$

is an equivalence of $d$-uple Segal spaces.

######  Remark .

The pasting theorem can also be rephrased as follows. Suppose that
$I_{1},\dotsc,I_{n}$ is a covering of a locally composable pasting shape $I$.
Since $\textup{Cat}^{d}(\mathscr{S})$ is a reflective subcategory of
$\mathrm{Fun}(\Delta^{\mathrm{op},\times d},\mathscr{S})$, we may compute and
reflect the colimit appearing in the statement in the latter category so that
the statement of the theorem is equivalent to saying that the map

$\operatorname{\mathrm{colim}}_{[k]\in\Delta^{\mathrm{op}}}\left(\coprod_{\sigma:\\{0,\dotsc,k\\}\rightarrow\\{1,\dotsc,n\\}}[I_{\sigma(0)}\cap\dotsb\cap
I_{\sigma(k)}]\right)\rightarrow[I]$

of $d$-uple simplicial spaces is a Segal equivalence. More is true: since
colimits in $\mathrm{Fun}(\Delta^{\mathrm{op},\times d},\mathscr{S})$ are
computed pointwise, and the inclusion $\textup{Set}\rightarrow\mathscr{S}$
commutes with geometric realization, i.e. commutes with colimits of
$\Delta^{\mathrm{op}}$-shaped diagrams (this follows from similar reasoning as
in the proof 2: using that any set-valued simplicial diagram in
$\mathrm{Fun}(\Delta^{\mathrm{op}},\mathrm{sSet})$ is Reedy cofibrant), we can
even compute the colimit in $\mathrm{Fun}(\Delta^{\mathrm{op},\times
d},\textup{Set})$, where it is given by the ordinary union. Consequently, the
theorem equivalently asserts that the inclusion

$\bigcup_{i=1}^{n}[I_{i}]\rightarrow[I]$

of $d$-uple simplicial sets is a Segal equivalence.

If $I$ is a composable pasting shape, then the vertebrae of $I$ give a
canonical covering of $I$ to which the above theorem applies.

######  Definition .

Let $I$ be a $d$-dimensional composable pasting shape. The spine of $[I]$ is
the $d$-uple simplicial subset of $[I]$ given by the union

$\mathrm{Sp}[I]:=\bigcup_{V\subset I\textit{ vertebra}}[V]\subset[I].$

We will call the resulting inclusion $\mathrm{Sp}[I]\rightarrow[I]$ the spine
inclusion for $I$.

######  Corollary .

The spine inclusion for a composable pasting shape $I$ is a Segal equivalence.

######  Remark .

The spine inclusions defined above generalize the spine inclusions defined in
2. Namely, consider the grid

$I=\square[n_{1},\dotsc,n_{d}],$

so that $n_{1},\dotsc,n_{d}\neq 0$. As seen in 3.1, the vertebrae of $I$ are
given by the $V_{j}$’s. Note that
$[V_{j}]\cong[\square[1,\dotsc,1]]=\Delta[1,\dotsc,1]$ and
$[I]\cong\Delta[n_{1},\dotsc,n_{d}]$, and under these identifications, the
spine inclusion $\mathrm{Sp}[I]\rightarrow[I]$ can be identified with the map

$\operatorname{\mathrm{colim}}_{([k_{1}],\dotsc,[k_{d}])\in\Lambda^{\times
d}/([n_{1}],\dotsc,[n_{d}])}\Delta[k_{1},\dotsc,k_{d}]\rightarrow\Delta[n_{1},\dotsc,n_{d}],$

which is indeed a Segal equivalence by definition. To verify this, one puts a
Reedy structure on the slice categories of $\Lambda$ (one can take the one
inherited from the Reedy structure on $\Delta$, concretely, every morphism in
$\Lambda$ raises degree, and the identities are the only morphism that lower
degree) to compute the colimit on the left. One then readily sees that the
diagram over which the colimit is taken, is Reedy cofibrant. Hence this
colimit may be computed in set-valued presheaves (similarly as in 3.3), and is
hence given by the union of the images of the $\Delta[k_{1},\dotsc,k_{d}]$’s
in $\Delta[n_{1},\dotsc,n_{d}]$. This is precisely the spine $\mathrm{Sp}[I]$.

######  Example .

Using a similar argument as in 3.2, one sees that the spine inclusion for $PW$
is not a Segal equivalence. Namely, it factors as

$\mathrm{Sp}[PW]=\mathrm{Sp}[PW^{\circ}]\rightarrow[PW^{\circ}]\rightarrow[PW].$

The pasting theorem asserts that the first inclusion is a Segal equivalence.
By the 2-out-of-3 principle, the spine inclusion for $PW$ is a Segal
equivalence precisely when the inclusion $[PW^{\circ}]\rightarrow[PW]$ is a
Segal equivalence. But this cannot be the case since the nerve functor is
fully faithful and $PW\neq PW^{\circ}$.

######  Corollary .

Consider a $d$-dimensional pasting shape $I$ and a $d$-uple Segal space $X$.
Suppose that $I$ is a composable shape and that we have family of maps

$f_{V}:[V]\rightarrow X,\quad V\subset I\text{ vertebra},$

which are compatible in the sense that it defines a map
$\mathrm{Sp}[I]\rightarrow X$ of $d$-uple simplicial spaces. Then the space
$C$ of composites defined by the pullback square

${C}$${\mathrm{Map}([I],X)}$${\\{(f_{V})\\}}$${\mathrm{Map}(\mathrm{Sp}[I],X),}$

is contractible.

Consider the full subcategory

$\textup{CShape}^{\leq d}\subset\textup{Shape}^{d}$

of $\textup{Shape}^{d}$ spanned by the locally composable pasting shapes. Then
the following corollary of the pasting theorem gives a new description of the
$\infty$-category of $d$-uple Segal spaces:

######  Corollary .

The functor

$\textup{Cat}^{d}(\mathscr{S})\rightarrow\mathrm{Fun}(\textup{CShape}^{\leq
d,\mathrm{op}},\mathscr{S}):X\mapsto(I\mapsto\mathrm{Map}([I],X))$

is an equivalence onto the full subcategory of presheaves
$Y:\textup{CShape}^{\leq d,\mathrm{op}}\rightarrow\mathscr{S}$ that satisfy
the following Segal conditions:

  1. (1)

for every locally composable pasting shape $I$ and covering
$I_{1},\dotsc,I_{n}$ of $I$, the map

$\textstyle
Y(I)\rightarrow\lim_{[k]\in\Delta}\prod_{\sigma:\\{0,\dotsc,k\\}\rightarrow\\{1,\dotsc,n\\}}Y(I_{\sigma(0)}\cap\dotsb\cap
I_{\sigma(k)})$

is an equivalence,

  2. (2)

for any $k$-shaping map $f:I\rightarrow J$ between $k$-dimensional locally
composable pasting shapes, the map

$Y(J)\rightarrow Y(I)\times_{Y({\mathrm{tr}_{\leq k-1}}I)}Y(\mathrm{tr}_{\leq
k-1}J),$

is an equivalence.

###### Proof.

Recall that we have a cosimplicial object
$\square[\bullet,\dots,\bullet]:\Delta^{\times
d}\rightarrow\textup{CShape}^{\leq d}$, which we will temporarily denote by
$\square$ for brevity. By right Kan extension, we obtain an adjunction

$\square^{*}:\mathrm{Fun}(\textup{CShape}^{\leq
d,\mathrm{op}},\mathscr{S})\rightleftarrows\mathrm{Fun}(\Delta^{\times
d,\mathrm{op}},\mathscr{S}):\square_{*}.$

We claim that this restricts to an adjunction

$\mathrm{L}\square^{*}:\mathrm{Fun}_{\mathrm{Seg}}(\textup{CShape}^{\leq
d,\mathrm{op}},\mathscr{S})\rightleftarrows\textup{Cat}^{d}(\mathscr{S}):\mathrm{R}\square_{*},$

where the left $\infty$-category is the full subcategory
$\mathrm{Fun}(\textup{CShape}^{\leq d,\mathrm{op}},\mathscr{S})$ spanned by
those presheaves that meet the Segal condition as in the statement of the
corollary. This subcategory admits the following equivalent description. Let

$y:\textup{CShape}^{\leq d}\rightarrow\mathrm{Fun}(\textup{CShape}^{\leq
d,\mathrm{op}},\mathscr{S})$

denote the Yoneda embedding. Then
$\mathrm{Fun}_{\mathrm{Seg}}(\textup{CShape}^{\leq
d,\mathrm{op}},\mathscr{S})$ can be described as the left Bousfield
localization of $\mathrm{Fun}(\textup{CShape}^{\leq
d,\mathrm{op}},\mathscr{S})$ at the set $S$ consistings of the following maps:

  1. (i)

for any covering $I_{1},\dotsc,I_{n}$ of a locally composable pasting shape
$I$, the map
$\operatorname{\mathrm{colim}}_{[k]\in\Delta^{\mathrm{op}}}\coprod_{\sigma}y(I_{\sigma(0)}\cap\dotsb\cap
I_{\sigma(k)})\rightarrow y(I)$,

  2. (ii)

for any $k$-shaping map $f:I\rightarrow J$ between $k$-dimensional locally
composable pasting shapes, the map $y(I)\cup_{y(\mathrm{tr}_{\leq
k-1}I)}y(\mathrm{tr}_{\leq k-1}J)\rightarrow y(J)$.

It suffices to check $\square^{*}Y$ is a $d$-uple Segal space for every
$S$-local presheaf $Y$, and that $\square^{*}$ carries maps in $S$ to Segal
equivalences. The latter follows from the fact that we may identify the
composite $\square^{*}y$ with the nerve functor, the pasting theorem, and the
fact that any map $f$ as in (ii) gives rise to an equivalence
$[I]\cup_{\mathrm{tr}_{\leq k-1}[J]}\mathrm{tr}_{\leq k-1}[J]\rightarrow[J]$
in $\textup{Cat}^{d}(\mathscr{S})$. It remains to show that $\square^{*}Y$ is
a $d$-uple Segal space. To this end, consider a $k$-dimensional grid
$I:=\square[n_{1},\dotsc,n_{d}]$. Then we may cover $I$ by its $k$-vertebrae
to obtain a map in $S$ of type (i). Using similar arguments as in 3.3 and 3.3
to rewrite the colimit appearing in the domain of this map, this can be
written as the canonical map

$\operatorname{\mathrm{colim}}_{i:{([k_{1}],\dotsc,[k_{d}])\rightarrow([n_{1}],\dotsc,[n_{d}])}\in\Lambda^{\times
d}/([n_{1}],\dotsc,[n_{d}])}y(\mathrm{im}(i))\rightarrow y(I).$

Using the maps of type (ii) in $S$, one now deduces that the inclusion

$\operatorname{\mathrm{colim}}_{i:{([k_{1}],\dotsc,[k_{d}])\rightarrow([n_{1}],\dotsc,[n_{d}])}\in\Lambda^{\times
d}/([n_{1}],\dotsc,[n_{d}])}y(\square[k_{1},\dotsc,k_{d}])\rightarrow y(I).$

is $S$-local. Since $Y$ is $S$-local, this implies that $\square^{*}Y$ is a
$d$-uple Segal space.

Finally, we will show that $(\mathrm{L}\square^{*},\mathrm{R}\square_{*})$ is
an adjoint equivalence of $\infty$-categories. Since $\square$ is fully
faithful, the functor $\square_{*}$ is again fully faithful and hence its
restriction $\mathrm{R}\square_{*}$ as well. It thus suffices to check that
$\mathrm{L}\square^{*}$ is conservative. This entails showing that for any map
$f:X\rightarrow Y$ in $\mathrm{Fun}_{\mathrm{Seg}}(\textup{CShape}^{\leq
d,\mathrm{op}},\mathscr{S})$ so that

$f_{n_{1},\dotsc,n_{d}}:X(\square[n_{1},\dotsc,n_{d}])\rightarrow
Y(\square[n_{1},\dotsc,n_{d}])$

is an equivalence for all $n_{1},\dotsc,n_{d}$, then

$f_{I}:X(I)\rightarrow Y(I)$

is an equivalence for all locally composable pasting shapes $I$. We leave this
to the reader to check; it can be shown using an induction on the truncation
degree of $I$, covering locally composable pasting shapes by their
$k$-vertebrae, and the Segal conditions for $X$ and $Y$. ∎

######  Remark .

One may deduce from the proof of the corollary above that a presheaf
$Y:\textup{CShape}^{\leq d,\mathrm{op}}\rightarrow\mathscr{S}$ already
satisfies the Segal conditions if:

  1. (1)

for every locally composable pasting shape $I$ with $k$-vertebrae ($0\leq
k\leq d$) given by $I_{1},\dotsc,I_{n}$, the map

$\textstyle
Y(I)\rightarrow\lim_{[k]\in\Delta}\prod_{\sigma:\\{0,\dotsc,k\\}\rightarrow\\{1,\dotsc,n\\}}Y(I_{\sigma(0)}\cap\dotsb\cap
I_{\sigma(k)})$

is an equivalence,

  2. (2)

for any $k$-shaping map $f:\square[n_{1},\dotsc,n_{d}]\rightarrow C$ between
$k$-dimensional composable cells, the map

$Y(C)\rightarrow
Y(\square[n_{1},\dotsc,n_{d}])\times_{Y(\partial\square[n_{1},\dotsc,n_{d}])}Y(\partial
C),$

is an equivalence.

##  4\. Proof of the pasting theorem

Our strategy for proving the pasting theorem is by proceeding inductively on
the dimension of the pasting shapes. The following theorem is the crucial
ingredient that makes the induction work:

######  Theorem 4.1.

Suppose that $I$ is a $d$-dimensional admittable pasting shape. Then the
inclusion

$\mathrm{Sp}[I]\cup\mathrm{tr}_{\leq d-1}[I]\rightarrow[I]$

is a Segal equivalence.

The proof of this theorem is deferred to the end of this section. Instead, we
will now demonstrate how the pasting theorem follows from this fact. We first
introduce the following auxiliary notion (cf. 3.2):

######  Definition .

A $d$-dimensional pasting shape $I$ with finitely many boxes is called locally
admittable if every closed entire subshape of $I$ is admittable.

######  Remark .

Any admittable pasting shape is locally admittable on account of 3.2.
Furthermore, for any $d$-dimensional locally composable pasting shape $I$,
each slice $I\cap H$, with $H\in\mathscr{H}_{k}$, is a locally admittable
$k$-dimensional pasting shape.

######  Lemma .

Suppose that $I$ is a locally admittable pasting shape, and let
$I_{1},\dotsc,I_{n}$ be a collection of subshapes of $I$ with the property
that any closed entire subshape of $I_{i}$ is an entire subshape of $I$. Then
$I_{1},\dotsc,I_{n}$ are locally admittable and the inclusion

$\textstyle\bigcup_{i}\mathrm{Sp}[I_{i}]\cup\mathrm{tr}_{\leq
d-1}[I_{i}]\rightarrow\bigcup_{i}[I_{i}]$

is a Segal equivalence.

###### Proof.

During this proof, we will use the notion of the maximal box size which is
defined for any pasting shape $J$ as follows

$M(J):=\max_{(x,y)\in B^{d}(I)\setminus
B^{d-1}(I)}(y_{1}-x_{1})+\dotsb+(y_{d}-x_{d}),$

with the convention that $M(J)=0$ if $J$ is $(d-1)$-truncated. If
$J_{1},\dotsc,J_{m}$ are $d$-dimensional pasting shapes, then we define

$M(J_{1},\dotsc,J_{m}):=\max_{1\leq i\leq m}M(J_{i}).$

We now shift our attention to the proof. It is readily verified that
$I_{1},\dotsc,I_{n}$ are again locally admittable. To prove the main content
of the lemma, we proceed by induction on the maximal box size of
$I_{1},\dotsc,I_{n}$. If $M(I_{1},\dotsc,I_{n})=0$, there is nothing to show.
Suppose that the statement holds whenever $M(I_{1},\dotsc,I_{n})\leq r-1$.
Then we have to show that for any collection $I_{1},\dotsc,I_{n}\subset I$ as
in the statement, with $M(I_{1},\dotsc,I_{n})\leq r$, the lemma holds.

In turn, we proceed by induction on $n$. If $n=0$, there is nothing to show.
Suppose that the statement holds for all collections $I_{1},\dotsc,I_{n-1}$ of
length $n-1$ with $M(I_{1},\dotsc,I_{n-1})\leq r$. Consider a collection
$I_{1},\dotsc,I_{n}$ of length $n$ with $M(I_{1},\dotsc,I_{n})\leq r$. Then we
have a factorization

$\textstyle\bigcup_{i}\mathrm{Sp}[I_{i}]\cup\mathrm{tr}_{\leq
d-1}[I_{i}]\rightarrow\bigcup_{i=2}^{n}[I_{i}]\cup\mathrm{Sp}[I_{1}]\cup\mathrm{tr}_{\leq
d-1}[I_{1}]\rightarrow\bigcup_{i}[I_{i}]$

The left map is a Segal equivalence, as it is a pushout along the map

$\textstyle\bigcup_{i=2}^{n}\mathrm{Sp}[I_{i}]\cup\mathrm{tr}_{\leq
d-1}[I_{i}]\rightarrow\bigcup_{i=2}^{n}[I_{i}],$

which is a Segal equivalence by the induction hypothesis. The map on the right
is a pushout along the map

$\textstyle\bigcup_{j=2}^{n}[I_{j}\cap
I_{1}]\cup\mathrm{Sp}[I_{1}]\cup\mathrm{tr}_{\leq
d-1}[I_{1}]\rightarrow[I_{1}].$

From this, we deduce that we may reduce the proof to showing the following:
for $I$ with $M(I)\leq r$ and $I_{1},\dotsc,I_{n}\subset I$ as in the
statement, the inclusion

$\textstyle\bigcup_{j=1}^{n}[I_{j}]\cup\mathrm{Sp}[I]\cup\mathrm{tr}_{\leq
d-1}[I]\rightarrow[I]$

is a Segal equivalence. Using an analogous inductive argument, this can in
turn be reduced to the case that $n=0$.

Thus we have reduced the proof to showing that the map

$\mathrm{Sp}[I]\cup\mathrm{tr}_{\leq d-1}[I]\rightarrow[I],$

is a Segal equivalence for all locally admittable pasting shapes $I$ with
maximal size at most $r$. Let $E_{1},\dotsc,E_{m}$ be all distinct, maximally
closed entire subshapes of $I$ (i.e. for any closed entire $E\supset E_{i}$,
we must have that $E=E_{i}$). Then we have to show that the inclusion

${\bigcup_{i=1}^{k}\mathrm{Sp}[E_{i}]\cup\mathrm{tr}_{\leq
d-1}[I]\rightarrow\bigcup_{i=1}^{k}[E_{i}]\cup\mathrm{tr}_{\leq d-1}[I]}$

is a Segal equivalence for $k=m$. We proceed by induction on $k$. If $k=0$,
there is nothing to show. Suppose that the map is a Segal equivalence for
$k-1$. Then we have a factorization

$\textstyle\bigcup_{i=1}^{k}\mathrm{Sp}[E_{i}]\cup\mathrm{tr}_{\leq
d-1}[I]\rightarrow\bigcup_{i=1}^{k-1}[E_{i}]\cup\mathrm{Sp}[E_{k}]\cup\mathrm{tr}_{\leq
d-1}[I]\rightarrow\bigcup_{i=1}^{k}[E_{i}]\cup\mathrm{tr}_{\leq d-1}[I].$

The left map is a Segal equivalence by the induction hypothesis indexed by
$k$. The map on the right is a pushout along the map

$\textstyle\bigcup_{i=1}^{k-1}[E_{i}\cap
E_{k}]\cup\mathrm{Sp}[E_{k}]\cup\mathrm{tr}_{\leq
d-1}[E_{k}]\rightarrow[E_{k}].$

Since $E_{k}$ is admittable, 4.1 and the 2-out-of-3 principle imply that it
suffices to show that the inclusion

$\textstyle\mathrm{Sp}[E_{k}]\cup\mathrm{tr}_{\leq
d-1}[E_{k}]\rightarrow\bigcup_{i=1}^{k-1}[E_{i}\cap
E_{k}]\cup\mathrm{Sp}[E_{k}]\cup\mathrm{tr}_{\leq d-1}[E_{k}]$

is a Segal equivalence. This map is a pushout along

$\textstyle\bigcup_{i=1}^{k-1}\mathrm{Sp}[E_{i}\cap
E_{k}]\cup\mathrm{tr}_{\leq d-1}[E_{i}\cap
E_{k}]\rightarrow\bigcup_{i=1}^{k-1}[E_{i}\cap E_{k}].$

Now, we note that $E_{1}\cap E_{k},\dotsc,E_{k-1}\cap E_{k}$ are subshapes of
$E_{k}$ that meet the conditions of the lemma. Moreover, since $E_{k}$ is
entire and $E_{i}\cap E_{k}\neq E_{k}$, we must have that $M(E_{i}\cap
E_{k})<M(E_{k})\leq r$. In view of the induction assumption made at the start
of this proof, this implies that that the above map is a Segal equivalence. ∎

###### Proof of 3.2.

The fact that $I=\bigcup_{i}I_{i}$ follows directly from 3.1 and the
definition of a covering. Consider the inclusion

$\textstyle j:\bigcup_{i}[I_{i}]\rightarrow[I].$

We will show that each truncated inclusion $\mathrm{tr}_{\leq k}j$ is a Segal
equivalence by induction on $0\leq k\leq d$. As explained in 3.3, this then
proves the theorem. For $k=0$, the statement is clear. Suppose that the
statement holds for $k-1$. Then we may consider the factorization

$\textstyle\mathrm{tr}_{\leq k}\bigcup_{i}[I_{i}]\rightarrow\mathrm{tr}_{\leq
k}\bigcup_{i}[I_{i}]\cup\mathrm{tr}_{\leq
k-1}[I]\xrightarrow{j_{k}}\mathrm{tr}_{\leq k}[I]$

of the truncated inclusion $\mathrm{tr}_{\leq k}j$. Note that the map on the
left is a pushout along inclusion map $\mathrm{tr}_{\leq k-1}j$, which is a
Segal equivalence by assumption. Hence, this this map is again a Segal
equivalence. In light of the 2-out-of-3 principle, the truncation
$\mathrm{tr}_{\leq k}j$ is a Segal equivalence precisely if the inclusion
$j_{k}$ is a Segal equivalence.

To show that $j_{k}$ is a Segal equivalence, we will make use of 3.1. Let
$H\in\mathscr{H}_{k}$. Then we may view the slices $I_{i}\cap H$ and $I\cap H$
as $k$-dimensional pasting shapes, and these again form a covering of $I\cap
H$. Hence, we have an inclusion

$\mathrm{Sp}[I_{i}\cap H]_{k}\subset\mathrm{Sp}[I\cap H]_{k}.$

Here, we decorated the spines with a $k$ to emphasize the fact that these are
unions of $k$-vertebrae. In fact, we have that

$\textstyle\bigcup_{i}\mathrm{Sp}[I_{i}\cap H]_{k}=\mathrm{Sp}[I\cap H]_{k}.$

Consider now the commutative square of inclusions

${\bigcup_{i}{\mathrm{Sp}[I_{i}\cap H]}_{k}\cup\mathrm{tr}_{\leq
k-1}[I_{i}\cap H]_{k}}$${\bigcup_{i}{[I_{i}\cap H]_{k}}}$${\mathrm{Sp}[I\cap
H]_{k}\cup\mathrm{tr}_{\leq k-1}[I\cap H]_{k}}$${{[I\cap H]_{k}}.}$

We again decorated the nerves with a $k$, as we are taking the nerves of
$k$-dimensional pasting shapes. As explained in 4, the slice $I\cap H$ is
locally $k$-admittable. Thus 4 asserts that the top and bottom horizontal maps
are Segal equivalences between $k$-uple Segal sets. From this, we may now
deduce that the inclusion

$\textstyle j_{k}^{H}:\bigcup_{i}{[I_{i}\cap H]}\cup\mathrm{tr}_{\leq
k-1}[I]\rightarrow[I\cap H]\cup\mathrm{tr}_{\leq k-1}[I],$

is a Segal equivalence (between $d$-uple Segal sets). Here we also used that
$\mathrm{tr}_{\leq k-1}j$ is a Segal equivalence. In view of 3.1, $j_{k}$ is
given by the inclusion

$\textstyle j_{k}:\bigcup_{H\in\mathscr{H}_{k}}\bigcup_{i}{[I_{i}\cap
H]}\cup\mathrm{tr}_{\leq k-1}[I]\rightarrow\bigcup_{H\in\mathscr{H}_{k}}[I\cap
H],$

and this can be written as a composition of maps obtain by pushing out along a
$j_{k}^{H}$. As we established that each $j_{k}^{H}$ is a Segal equivalence,
it follows that the inclusion $j_{k}$ must be a Segal equivalence as well. ∎

###  4.1. Good pasting shapes

It remains to prove 4.1. We will make the following definition:

######  Definition .

We call a $d$-dimensional pasting shape $I$ good if the map

$\mathrm{Sp}[I]\cup\mathrm{tr}_{\leq d-1}[I]\rightarrow[I]$

is a Segal equivalence.

######  Remark .

We may describe the $d$-uple simplicial subset of $[I]$ appearing in the
domain of the map above, alternatively by

$\mathrm{Sp}[I]\cup\mathrm{tr}_{\leq
d-1}[I]=\bigcup_{\begin{subarray}{c}i:\square[1,\dotsc,1]\rightarrow I\text{
injective},\\\ \mathrm{im}(i)\text{ is contained in a (closed) vertebra of
$I$}\end{subarray}}i[\square[1,\dotsc,1]]\cup\mathrm{tr}_{\leq d-1}[I].$

######  Proposition .

Any grid is good.

###### Proof.

Let $A$ be a $d$-dimensional grid. If $A$ is open, then 4.1 implies that there
is nothing to show. Hence, we may assume that there exists a map

$f:\square[n_{1},\dotsc,n_{d}]\rightarrow A,$

with $n_{1},\dotsc,n_{d}\neq 0$, that witnesses $A$ to be a grid. For the sake
of brevity, we suppress the notation of $n_{1},\dotsc,n_{d}$. When we combine
the description given in 4.1, the description of the vertebrae of $A$ in 3.1,
and the fact that $f$ is $d$-shaping, we deduce that we have a pushout square

${\mathrm{Sp}[\square]\cup\mathrm{tr}_{\leq
d-1}[\square]}$${\mathrm{Sp}[A]\cup\mathrm{tr}_{\leq
d-1}[A]}$${{[\square]}}$${{[A]}.}$

Hence, it suffices to show that the map on the left is a Segal equivalence.

Note that the spine inclusion for $\square$ factors as

$\mathrm{Sp}[\square]\xrightarrow{j}\mathrm{Sp}[\square]\cup\mathrm{tr}_{\leq
d-1}[\square]\rightarrow[\square].$

We have already demonstrated in 3.3 that this map is a Segal equivalence.
Using this fact repeatedly (now in lower dimensions), a straightforward
induction similar to the one employed in the proof of 3.2, shows that

$\mathrm{tr}_{\leq
k}\mathrm{Sp}[\square[n_{1},\dotsc,n_{d}]]\rightarrow\mathrm{tr}_{\leq
k}[\square[n_{1},\dotsc,n_{d}]]$

is a Segal equivalence for $k<d$ as well. Note that $j$ is a pushout along
this map for $k=d-1$. This proves the statement in light of the 2-out-of-3
principle. ∎

The following is also an important example of a good pasting shape:

######  Definition .

Suppose that

$i:\square[1,\dotsc,1]\rightarrow\square[n_{1},\dotsc,n_{d}]$

is an injective map of $d$-dimensional pasting shapes. Then we define the
(admittable) pasting shape

$\boxdot[i]\subset\square[n_{1},\dotsc,n_{d}]$

to be the subshape of $\square[n_{1},\dotsc,n_{d}]$ that consists of all those
boxes $(x,y)$ such that their exists an index $a$ with $x_{a}\leq y_{a}\leq
i_{a}(0)$ or $i_{a}(1)\leq x_{a}\leq y_{a}$ (one readily checks that these
boxes are closed under faces and joins).

######  Example .

The following is a picture for $\boxdot[i]$, with maps
$i_{1}:[1]\rightarrow[6],i_{2}:[1]\rightarrow[4]$ starting and ending at $2,5$
and $1,3$ respectively:

${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$${\cdot}$

######  Proposition .

Suppose that $i:\square[1,\dotsc,1]\rightarrow\square[n_{1},\dotsc,n_{d}]$ is
an injective map, then the pasting shape $\boxdot[i]$ is good.

Let us first make the necessary preparations for its proof.

######  Construction .

Let $i:\square[1,\dotsc,1]\rightarrow\square[n_{1},\dotsc,n_{d}]$ be an
injective map, and consider the set of regions $R:=\\{-,0,+\\}^{\times d}$.
Then for $\rho\in R$, we define the subshape

$A^{\rho}[i]\subset\square[n_{1},\dotsc,n_{d}]$

to be the largest subshape of $\square[n_{1},\dotsc,n_{d}]$ whose vertices $x$
satisfy

$\begin{cases}x_{a}\leq i_{a}(0)&\text{if $\rho_{a}=-$},\\\ i_{a}(0)\leq
x_{a}\leq i_{a}(1)&\text{if $\rho_{a}=0$},\\\ i_{a}(1)\leq x_{a}&\text{if
$\rho_{a}=+$},\end{cases}$

for each index $a$. Note that $A^{\rho}[i]$ is contained in $\boxdot[i]$
whenever $\rho\neq(0,\dotsc,0)$. We will suppress $\rho$ from the notation if
$\rho=(0,\dotsc,0)$ and write $A[i]$ for $A^{\rho}[i]$.

######  Lemma .

Let $i:\square[1,\dotsc,1]\rightarrow\square[n_{1},\dotsc,n_{d}]$ be an
injective map. Then any map of pasting shapes
$\square[k_{1},\dotsc,k_{d}]\rightarrow\boxdot[i]$ factors through one of the
subshapes

$M^{\sigma}_{a}:=\bigcup_{\rho\in F^{\sigma}_{a}}A^{\rho}[i],\quad 1\leq a\leq
d,\sigma\in\\{-,+\\},$

where $F^{\sigma}_{a}\subset R$ is the collection of regions $\rho$ such that
$\rho_{a}=\sigma$.

###### Proof.

Firstly, we note that we may describe $M^{\sigma}_{a}$ as the subshape of
$\square[n_{1},\dotsc,n_{d}]$ that contains all boxes whose vertices $x$
satisfy

$\begin{cases}x_{a}\leq i_{a}(0)&\text{if $\sigma=-$},\\\ x_{a}\geq
i_{a}(1)&\text{if $\sigma=+$}.\end{cases}$

Consider a map of pasting shapes
$f:\square[k_{1},\dotsc,k_{d}]\rightarrow\boxdot[i]$ and suppose to the
contrary that $f$ does not factor through a single $M^{\sigma}_{a}$. Then for
any $a$, there exists a $t_{a}$ and $t^{\prime}_{a}$ in the image of $f_{a}$
such that $t_{a}>i_{a}(0)$ and $t^{\prime}_{a}<i_{a}(1)$. Define
$x_{a}:=\min(t_{a},t^{\prime}_{a})$ and $y_{a}:=\max(t_{a},t^{\prime}_{a})$.
Then $(x,y)$ must be a box in $\boxdot[i]$, hence, there is an index $a$ such
that $x_{a}\leq y_{a}\leq i_{a}(0)$ or $i_{a}(1)\leq x_{a}\leq y_{a}$. This is
a contradiction as $i_{a}(1)>t^{\prime}_{a}\geq x_{a}$ and $y_{a}\geq
t_{a}>i_{a}(0)$. ∎

###### Proof of 4.1.

For brevity, we drop the notation of the map $i$ in what follows. We commence
by observing that the inclusion

$\textstyle\bigcup_{\rho\in
R\setminus\\{(0,\dotsc,0)\\}}\mathrm{Sp}[A^{\rho}]\cup\mathrm{tr}_{\leq
d-1}[A^{\rho}]\rightarrow\bigcup_{\rho\in
R\setminus\\{(0,\dotsc,0)\\}}[A^{\rho}]$

is a Segal equivalence. This follows from 4.1 and a similar argument as given
in 4. As the first map in the following factorization is a pushout along this
Segal equivalence

$\mathrm{Sp}[\boxdot]\cup\mathrm{tr}_{\leq
d-1}[\boxdot]\rightarrow\bigcup_{\rho\in
R\setminus\\{(0,\dotsc,0)\\}}[A^{\rho}]\cup\mathrm{tr}_{\leq
d-1}[\boxdot]\rightarrow{[\boxdot]},$

it is a Segal equivalence as well. Thus it suffices to show that the right
inclusion is a Segal equivalence by 2-out-of-3. Since 4.1 precisely asserts
that the nerve of $\boxdot$ decomposes as

$[\boxdot]=\bigcup_{\sigma\in\\{+,-\\},1\leq a\leq d}[M^{\sigma}_{a}],$

this inclusion is a pushout along the map

$\textstyle\bigcup_{\sigma\in\\{+,-\\},1\leq a\leq d}\bigcup_{\rho\in
F^{\sigma}_{a}}[A^{\rho}]\cup\mathrm{tr}_{\leq
d-1}[M_{a}^{\sigma}]\rightarrow\bigcup_{\sigma\in\\{+,-\\},1\leq a\leq
d}[M^{\sigma}_{a}].$

Again, as we already established that grids are good, this must be a Segal
equivalence. ∎

###  4.2. Fillable shapes

Until now we have dealt with concrete admittable pasting shapes and shown that
they are good. In order to deal with arbitrary admittable pasting shapes and
show that they are good as well, we need some machinery to bootstrap the
results we have obtained so far. We will develop this in the following
subsection.

######  Definition .

A division pair $(K,J)$ for an admittable pasting shape $I$ is called good if
$K$ and $J$ are both good.

The following is a result of the main theorem, 4.2, of this final subsection:

######  Corollary .

Suppose that $(K,J)$ is a good division pair for $I$ so that the corners of
$J$ are given by $(\alpha_{1},\omega_{1}),\dotsc,(\alpha_{d},\omega_{d})$.
Then $I$ is good if any injective map

$f:\square[m_{1},\dotsc,m_{d}]\rightarrow I,$

with $m_{1},\dotsc,m_{d}\neq 0$, admits an extension

$g:\square[n_{1},\dotsc,n_{d}]\rightarrow I$

such that for each index $a$, $\mathrm{im}(f_{a})\subset\mathrm{im}(g_{a})$,
and if $g_{a}(0)\leq\alpha_{a}\leq g_{a}(n_{a})$ or
$g_{a}(0)\leq\omega_{a}\leq g_{a}(n_{a})$, then, respectively,
$\alpha_{a}\in\mathrm{im}(g_{a})$ or $\omega_{a}\in\mathrm{im}(g_{a})$.

Let us first show how we can finish the proof of the pasting theorem using
this result.

###### Proof of 4.1.

Let $I$ be a $d$-dimensional admittable pasting shape. If $I$ is height $0$,
then $I$ is a cell and there is nothing to prove. Suppose that the statement
holds for any admittable pasting shape $I$ of height $h$.

Let $I$ now be an admittable pasting shape of height $h+1$ with decomposition
$(A,\\{J_{i}\\}_{i=1}^{m})$, where $A$ is determined by a map
$f:\mathrm{tr}_{\leq d-1}\square[n_{1},\dotsc,n_{d}]\rightarrow A$. For $0\leq
k\leq m$, consider the subshape

$I_{k}:=A\cup\bigcup_{i=1}^{k}J_{i}\subset I.$

Then it suffices to show that $I_{k}$ is good for all $I$. We proceed by
induction on $k$. Trivally, the pasting shape $I_{0}=A$ is good. Suppose that
$I_{k-1}$ is good. Then the pair $(I_{k-1},J_{k})$ is a good division pair for
$I_{k}$. Note that $I_{k}$ has a decomposition given by
$(A,\\{J_{0},\dotsc,J_{k},\partial J_{k+1},\dotsc,\partial J_{m}\\})$. Thus,
in light of the factorization property of 3.2, any map
$h:\square[m_{1},\dotsc,m_{d}]\rightarrow I_{k}$, with $m_{1},\dotsc,m_{d}\neq
0$, admits an extension $g:\square[p_{1},\dotsc,p_{d}]\rightarrow I_{k}$ such
that

$\mathrm{im}(g_{a})=\\{t_{a}\in\mathrm{im}(f_{a})\mid h_{a}(0)\leq t_{a}\leq
h_{a}(m_{a})\\}\cup\mathrm{im}(h_{a}).$

This is a desired extension $g$ of $h$ as in 4.2, hence it follows from this
result that $I_{k}$ is good. ∎

######  Proposition .

Suppose that $(K,J)$ is a division pair for $I$. Then the spine inclusion for
$I$ factors as the compositum of inclusions

$\mathrm{Sp}[I]\cup\mathrm{tr}_{\leq
d-1}[I]\rightarrow[K]\cup[J]\cup\mathrm{tr}_{\leq d-1}[I]\rightarrow[I],$

and the left map is a Segal equivalence if $(K,J)$ is good.

###### Proof.

Suppose that all the distinct vertebrae of $K$ are given by
$V_{1},\dotsc,V_{k},\partial J$, and the vertebrae of $J$ are given by
$W_{1},\dotsc,W_{l}$. Then the vertebrae of $I$ are given by
$V_{1},\dotsc,V_{k},W_{1},\dotsc,W_{l}$. Hence, the spine inclusion factors
through the inclusion $[K]\cup[J]\cup\mathrm{tr}_{\leq d-1}[I]\subset[I]$.
Note that we have pushout diagrams

${\mathrm{Sp}[K]\cup\mathrm{tr}_{\leq
d-1}[K]}$${\mathrm{Sp}[I]\cup\mathrm{tr}_{\leq
d-1}[I]}$${{[K]}}$${\mathrm{Sp}[I]\cup{[K]}\cup\mathrm{tr}_{\leq d-1}[I],}$

and

${\mathrm{Sp}[J]\cup\mathrm{tr}_{\leq
d-1}[J]}$${\mathrm{Sp}[I]\cup[K]\cup\mathrm{tr}_{\leq
d-1}[I]}$${{[J]}}$${{[K]}\cup{[J]}\cup\mathrm{tr}_{\leq d-1}[I].}$

If $(K,J)$ is good, the left vertical maps in these diagrams are Segal
equivalences, hence, the composite of the right vertical maps must then be a
Segal equivalence as well. ∎

######  Example .

The division pair $(\boxdot[i],A[i])$ for $\square[n_{1},\dotsc,n_{d}]$
associated to an injective map

$i:\square[1,\dotsc,1]\rightarrow\square[n_{1},\dotsc,n_{d}],$

is good. Since $\square[n_{1},\dotsc,n_{d}]$ is good, 4.2 implies that the
inclusion

$[\boxdot[i]]\cup[A[i]]\cup\mathrm{tr}_{\leq
d-1}[\square[n_{1},\dotsc,n_{d}]]\rightarrow[\square[n_{1},\dotsc,n_{d}]]$

is a Segal equivalence.

######  Definition .

Suppose that $(K,J)$ and $(K^{\prime},J^{\prime})$ are division pairs for,
respectively, $d$-admittable pasting shapes $I$ and $I^{\prime}$. Then a map
of division pairs

$f:(K,J)\rightarrow(K^{\prime},J^{\prime})$

is a map of pasting shapes $f:I\rightarrow I^{\prime}$ such that

$f^{-1}(J^{\prime})=J,\text{ and, }\quad
f^{-1}(K^{\prime})\cup\mathrm{tr}_{\leq d-1}I=K\cup\mathrm{tr}_{\leq d-1}I.$

In view of 4.2, we have to show that the map appearing on the right is a Segal
equivalence. This amounts to filling simplices (or rather: nerves of
subshapes) that do not appear in the domain of this map, using pushouts along
Segal equivalences. We single out a class of pasting shapes that we will be
able to fill:

######  Definition .

Suppose that $(K,J)$ is a division pair for $I$. We call a subshape $G\subset
I$ $(K,J)$-fillable if $G$ is $(d-1)$-truncated or there exists a $d$-shaping
map

$g:\square[n_{1},\dotsc,n_{d}]\rightarrow G$

such that one of the two following conditions is met:

  1. (1)

$g$ has image in $K$,

  2. (2)

the composite

$\square[n_{1},\dotsc,n_{d}]\xrightarrow{g}G\rightarrow I$

underlies a map of division pairs

$(\boxdot[i],A[i])\rightarrow(K,J)$

for some injective map
$i:\square[1,\dotsc,1]\rightarrow\square[n_{1},\dotsc,n_{d}]$.

The goal of this section is to prove the following theorem:

######  Theorem 4.2.

Let $(K,J)$ be a division pair for $I$. Suppose that $G_{1},\dotsc,G_{n}$ are
$(K,J)$-fillable subshapes of $I$, then the inclusion

$[K]\cup[J]\cup\mathrm{tr}_{\leq
d-1}[I]\rightarrow\bigcup_{i=1}^{n}[G_{i}]\cup[K]\cup[J]\cup\mathrm{tr}_{\leq
d-1}[I]$

is a Segal equivalence.

The proof of this theorem requires some preparation. The following is one of
the crucial observations:

######  Proposition .

Suppose that $(K,J)$ is a division pair. Then for an $(K,J)$-fillable subshape
$G$, the inclusion

$[G\cap K]\cup[G\cap J]\cup\mathrm{tr}_{\leq d-1}[G]\rightarrow[G]$

is a Segal equivalence.

###### Proof.

If $G$ is contained in $I$ or it satisfies condition 4.2(1), this is clear,
since then the left-hand side is the whole $[G]$ and the inclusion in question
is the identity. Thus we may focus on the case that there exists a $d$-shaping
map $f:\square[n_{1},\dotsc,n_{d}]\rightarrow G$ (in particular, $f$ is
injective) that underlies a map of division pairs

$(\boxdot[i],A[i])\rightarrow(K,J),$

for a suitable injective map $i$. It then directly follows from the
definitions that we obtain a pushout square

${{[\boxdot]}\cup[A]\cup\mathrm{tr}_{\leq d-1}[\square]}$${{[G\cap
K]}\cup[G\cap J]\cup\mathrm{tr}_{\leq d-1}[G]}$${{[\square]}}$${{[G]}.}$

Since the map on the left is a Segal equivalence on account of 4.2, the map on
the right must also be a Segal equivalence. ∎

The final ingredient for the proof of 4.2, is the following:

######  Proposition .

The $(K,J)$-fillables for a division pair $(K,J)$ are closed under taking
intersections.

######  Lemma .

Suppose that $I$ is a pasting shape and that we have injective maps
$f:\square[n_{1},\dotsc,n_{d}]\rightarrow I$ and
$g:\square[m_{1},\dotsc,m_{d}]\rightarrow I$ in $I$ with intersecting images,
then one can construct a pullback square

${\square[l_{1},\dotsc,l_{d}]}$${\square[n_{1},\dotsc,n_{d}]}$${\square[m_{1},\dotsc,m_{d}]}$${I,}$

where all arrows appearing in the diagram are inclusions.

###### Proof.

Since the images of $f$ and $g$ intersect, we get that
$\mathrm{im}(f_{a})\cap\mathrm{im}(g_{a})$ is non-empty for each index $a$.
Let $h_{a}:[l_{a}]\rightarrow\mathrm{im}(f_{a})\cap\mathrm{im}(g_{a})$ be the
unique isomorphism for each $a$. These give rise to maps
$[l_{a}]\rightarrow\mathrm{im}(f_{a})\cap\mathrm{im}(g_{a})\rightarrow\mathrm{im}(f_{a})\cong[n_{a}]$,
inducing a map
$i:\square[l_{1},\dotsc,l_{d}]\rightarrow\square[n_{1},\dotsc,n_{d}]$.
Similarly, we obtain a map
$j:\square[l_{1},\dotsc,l_{d}]\rightarrow\square[n_{1},\dotsc,n_{d}]$. It can
be readily checked that this gives a commutative square

${\square[l_{1},\dotsc,l_{d}]}$${\square[n_{1},\dotsc,n_{d}]}$${\square[m_{1},\dotsc,m_{d}]}$${I.}$

To check that this square is a pullback square, it suffices to check this on
level of nerves as the nerve functor is fully faithful. Thus we must verify
that for any $([k_{1}],\dotsc,[k_{d}])\in\Delta^{\times d}$, the following
square is a pullback square

${\prod_{1\leq a\leq d}\Delta([k_{a}],[l_{a}])}$${\prod_{1\leq a\leq
d}\Delta([k_{a}],[n_{a}])}$${\prod_{1\leq a\leq
d}\Delta([k_{a}],[m_{a}])}$${\mathrm{Hom}_{I}(\coprod_{1\leq a\leq
d}[k_{a}],\mathbb{N}),}$

and this readily follows from the construction of the maps $i$ and $j$. ∎

######  Lemma .

Let $(K,J)$ be a division pair for $I$. Suppose that the corners of $J$ are
given by $(\alpha_{1},\omega_{1}),\dotsc,(\alpha_{d},\omega_{d})$. Then for an
injective map $f:\square[n_{1},\dotsc,n_{d}]\rightarrow J$, the following are
equivalent:

  1. (1)

$f$ underlies a map of division pairs

$(\boxdot[i],A[i])\rightarrow(K,J),$

  2. (2)

for all $1\leq a\leq d$, we have
$\max(\min(f_{a}),\alpha_{a})<\min(\max(f_{a}),\omega_{a})\in\mathrm{im}(f_{a})$.

###### Proof.

First, suppose that (1) is satisfied so that we have an injective map

$i:\square[1,\dotsc,1]\rightarrow\square[n_{1},\dotsc,n_{d}]$

with (in particular) the property that $fi$ has image in $J$ and $f$ carries
every non-degenerate $d$-box of $\boxdot[i]$ to a non-degenerate $d$-box of
$K$. Define $\alpha_{a}^{\prime}:=f_{a}(i_{a}(0))$ and
$\omega_{a}^{\prime}:=f_{a}(i_{a}(1))$. From 3.2, it follows that
$\alpha_{a}\leq\alpha_{a}^{\prime}<\omega_{a}^{\prime}\leq\omega_{a}$. If
$f_{a}(0)<\alpha_{a}$, then the box $(x,y)$ given by $x_{b}=0$ for all $b$ and
$y_{b}=n_{b}$ if $a\neq b$ and $y_{a}=i_{a}(0)$, is a non-degenerate $d$-box
in $M^{-}_{a}\subset\boxdot$. Hence $(f(x),f(y))$ is contained in $K$. Again
by 3.2, we necessarily must have that
$f(y)_{a}=\alpha^{\prime}_{a}\leq\alpha_{a}$. Hence
$\alpha_{a}=\alpha_{a}^{\prime}$. Thus we deduce that
$\max(f_{a}(0),\alpha_{a})\in\mathrm{im}(f_{a})$. Similarly, one deduces that
$\min(f_{a}(n_{a}),\omega_{a})\in\mathrm{im}(f_{a})$.

Conversely, suppose that (2) is satisfied. Let

$i:\square[1,\dotsc,1]\rightarrow\square[n_{1},\dotsc,n_{d}]$

be the injective inclusion that satisfies

$f_{a}(i_{a}(0))=\max(f_{a}(0),\alpha_{a})<\min(f_{a}(n_{a}),\omega_{a})=f_{a}(i_{a}(1))$

for all $a$. Suppose that $(x,y)$ is a box in $I$. Then clearly, $(f(x),f(y))$
is contained in $J$ if and only if $x_{a},y_{a}\in\mathrm{im}(f_{a}i_{a})$ for
all $a$. Hence, $f^{-1}(J)=A[i]$. Let $(x,y)$ be a non-degenerate $d$-box of
$\square[n_{1},\dotsc,n_{d}]$. Then we have to show that $(f(x),f(y))$ is in
$K$ if and only if $(x,y)$ is in $\boxdot$. Thus it suffices to show that
$f_{a}(x_{a}),f_{a}(y_{a})\leq\alpha_{a}$ if and only if $x_{a},y_{a}\leq
i_{a}(0)$ (and a similar statement for $\omega_{a}$, $i_{a}(n_{a})$, which is
handled in the same manner). In either case, we must have that
$f_{a}(i_{a}(0))=\alpha_{a}$ since $x_{a}<y_{a}$, and the remaining claim
follows. ∎

######  Corollary .

Let $(K,J)$ be a division pair for $I$. Suppose that the corners of $J$ are
given by $(\alpha_{1},\omega_{1}),\dotsc,(\alpha_{d},\omega_{d})$. Let
$G\subset I$ be a subshape that comes with a $d$-shaping map
$g:\square[n_{1},\dotsc,n_{d}]\rightarrow G$. Then $G$ is $(K,J)$-fillable if
for each index $1\leq a\leq d$, we have that $\max(\min(g_{a}),\alpha_{a})$
and $\min(\max(g_{a}),\omega_{a})$ are in the image of $g_{a}$.

###### Proof.

Let us write $\alpha^{\prime}_{a}:=\max(\min(g_{a}),\alpha_{a})$ and
$\omega_{a}^{\prime}:=\min(\max(g_{a}),\omega_{a})$. If
$\alpha^{\prime}_{a}\geq\omega^{\prime}_{a}$ for some $a$, then we must have
that the image of $g$ is either in $K$, or $g_{a}(0)=g_{a}(n_{a})$, in which
case $G$ is $(d-1)$-truncated. Otherwise, we deduce that $g$ meets condition
4.2(2). ∎

###### Proof of 4.2.

If $G$ and $G^{\prime}$ have empty intersection, or one of these subshapes is
$(d-1)$-truncated, then we are done. Consequently, we may assume there exists
$d$-shaping maps $f:\square[n_{1},\dotsc,n_{d}]\rightarrow I$ and
$g:\square[m_{1},\dotsc,m_{d}]\rightarrow I$ that witness, respectively, $G$
and $G^{\prime}$ to be $(K,J)$-fillable, and which have intersecting images.
On account of 4.2, there is a pullback square

${\square[l_{1},\dotsc,l_{d}]}$${\square[n_{1},\dotsc,n_{d}]}$${\square[m_{1},\dotsc,m_{d}]}$${I,}$