$2$-Segal maps associated to a category with cofibrations

Tanner Carawan

###### Contents

  1. 1 Abstract
  2. 2 Introduction
  3. 3 Waldhausen Categories
    1. 3.1 Categories with cofibrations
    2. 3.2 Waldhausen categories
  4. 4 The $S_{\bullet}$-construction and 2-Segal objects
    1. 4.1 The $S_{\bullet}$-construction
    2. 4.2 2-Segal objects
  5. 5 Homotopy limits and projective 2-limits
  6. 6 2-Segality of discrete variants of the $S_{\bullet}$-construction
  7. 7 2-Segality of topological versions of the $S_{\bullet}$-construction
  8. 8 2-Segality of categorical versions of the $S_{\bullet}$-construction
  9. 9 A sufficient condition for $S_{\bullet}\mathcal{C}$ to be 2-Segal
  10. A Upper and lower 2-Segal

##  1\. Abstract

Waldhausen’s $S_{\bullet}$-construction gives a way to define the algebraic
$K$-theory space of a category with cofibrations. Specifically, the $K$-theory
space of a category with cofibrations $\mathcal{C}$ can be defined as the loop
space of the realization of the simplicial topological space
$|iS_{\bullet}\mathcal{C}|$. Dyckerhoff and Kapranov observed that if
$\mathcal{C}$ is chosen to be a proto-exact category, then this simplicial
topological space is 2-Segal. A natural question is then what variants of this
$S_{\bullet}$-construction give 2-Segal spaces. We find that for
$|iS_{\bullet}\mathcal{C}|$, $S_{\bullet}\mathcal{C}$,
$wS_{\bullet}\mathcal{C}$, and the simplicial set whose $n$th level is the set
of isomorphism classes of $S_{\bullet}\mathcal{C}$, there are certain
$2$-Segal maps which are always equivalences. However for all of these
simplicial objects, none of the rest of the $2$-Segal maps have to be
equivalences. We also reduce the question of whether
$|wS_{\bullet}\mathcal{C}|$ is $2$-Segal in nice cases to the question of
whether a simpler simplicial space is $2$-Segal. Additionally, we give a
sufficient condition for $S_{\bullet}\mathcal{C}$ to be $2$-Segal. Along the
way we introduce the notion of a generated category with cofibrations and
provide an example where the levelwise realization of a simplicial category
which is not $2$-Segal is $2$-Segal.

##  2\. Introduction

Algebraic $K$-theory is an area of math where groups indexed by integers are
assigned to rings, exact categories, and other objects. Algebraic $K$-theory
groups often enjoy theorems like Additivity, Localization, and Approximation
and this area of math has connections to number theory and topology.

In the process of extending the collection of objects for which $K$-groups are
defined to have an algebraic $K$-theory of spaces, Waldhausen gave the
$S_{\bullet}$-construction which can be applied to an exact category or more
generally to a category with cofibrations.

Let $\mathcal{E}$ be an exact category. The $S_{\bullet}$-construction
produces a sequence of spaces, the $n$th one of which is the realization of a
category $iS_{n}\mathcal{E}$ of diagrams built from the exact sequences of
$\mathcal{E}$. This sequence of spaces determines a space called the
$K$-theory space, and the homotopy groups of this space are the $K$-theory
groups.

Although the homotopy-theoretic properties of Waldhausen’s
$S_{\bullet}$-construction are still being investigated, two independent
groups of authors, one consisting of Dyckerhoff and Kapranov [3] and another
consisting of Gálvez-Carrillo, Kock, and Tonks [5], noticed that if
$\mathcal{E}$ is an exact category, then the sequence of spaces
$|iS_{n}\mathcal{E}|$ has the structure of a $2$-Segal space.

A sequence of spaces $X_{n}$, $n\geq 0$, form a $2$-Segal space if for every
order preserving map $\alpha\colon\\{0\leq 1\leq\cdots\leq m\\}\to\\{0\leq
1\leq\cdots\leq n\\}$ (written $\alpha\colon[m]\to[n]$), there are compatibly
chosen maps $X_{n}\to X_{m}$ such that certain induced maps $X_{k}\to
X_{2}\times^{h}_{X_{1}}X_{2}\times^{h}_{X_{1}}\cdots\times^{h}_{X_{1}}X_{2}$,
called $2$-Segal maps, are weak equivalences. We think of a $2$-Segal space as
having a space of objects $X_{0}$, a space of morphisms $X_{1}$, and an up-to-
homotopy sometimes-defined multivalued composition given by a span
$X_{1}\times^{h}_{X_{0}}X_{1}\leftarrow X_{2}\to X_{1}$; notice that the
indices on the left are $0$ and $1$ and not $1$ and $2$. The $2$-Segal maps
being weak equivalences says that this composition is associative up-to-
homotopy.

One benefit of the $S_{\bullet}$-construction is that it applies to a huge
class of categories, categories with cofibrations. For $\mathcal{C}$ a
category with cofibrations this work addresses whether the $2$-Segal maps from
the $|iS_{n}\mathcal{C}|$ to certain iterated homotopy pullbacks are also weak
homotopy equivalences. We find that a certain collection of them are, but none
of the rest have to be. We also give a sufficient condition for
$S_{\bullet}\mathcal{C}$ to be a $2$-Segal space. Finally, we do the
preceeding with several variants of the $S_{\bullet}$-construction. Along the
way, we introduce the notion of a generated category with cofibrations and
provide an example where the levelwise realization of a non $2$-Segal space is
$2$-Segal.

The work is structured as follows. We begin with the inputs to variants of the
$S_{\bullet}$-construction in Section 3. Then in Section 4 we review the
$S_{\bullet}$-construction itself. Section 5 is a technical aside on homotopy
limits. In Section 6 we prove our main result for a discrete version
$\text{iso}(s_{\bullet}\mathcal{C})$ of the $S_{\bullet}$-construction. Then
in Section 7 we prove our main result for a topological version
$|iS_{\bullet}\mathcal{C}|$ of the $S_{\bullet}$-construction and we give our
progress on analyzing the $2$-Segal maps of $|wS_{\bullet}\mathcal{C}|$ when
$\mathcal{C}$ is a Waldhausen category. In Section 8 we prove our main result
for the categorical $S_{\bullet}$-constructions $S_{\bullet}\mathcal{C}$ and
$wS_{\bullet}\mathcal{C}$. In the final non-appendix section we provide a
sufficient condition for $S_{\bullet}\mathcal{C}$ to be $2$-Segal. In the
appendix we show that one of our central definitions is equivalent to a
preexisting one, namely that left and right $2$-Segal are the same as lower
and upper $2$-Segal.

##  3\. Waldhausen Categories

In this section, we review categories with cofibrations and Waldhausen
categories because they are the inputs to the $S_{\bullet}$-construction. We
also introduce a dual notion to that of a category with cofibrations, a
category with fibrations.

###  3.1. Categories with cofibrations

We begin by defining a category with cofibrations. In this section we include
relevant terminology and facts about categories with cofibrations as well.

######  Definition 3.1.

[12, 1.1] A category with cofibrations is a category $\mathcal{C}$ with a zero
object $0$ and a distinguished class of morphisms called the _cofibrations_.
These cofibrations and zero object are such that for any object $A$ of
$\mathcal{C}$, the unique map $0\rightarrow A$ is a cofibration, every
isomorphism is a cofibration, the composite of cofibrations is a cofibration,
and the pushout of a cofibration always exists and is a cofibration.

We indicate that a morphism is a cofibration by drawing our arrows with a tail
($\rightarrowtail$).

######  Example 3.2.

A motivating, and for our purpose central, example is $\mathcal{R}_{f}$, the
category of finite based CW-complexes with specified CW structure and based
cellular maps [14, 9.1.4]. In $\mathcal{R}_{f}$ the cofibrations are the
cellular inclusions and the $0$ object is the singleton space.

One reason for having categories with cofibrations is to be able to consider
cofibration sequences.

######  Definition 3.3.

A _cofibration sequence_ is a cofibration followed by its cokernel. We denote
the cokernel object of $i\colon A\rightarrowtail B$ by
$\operatorname{coker}(i)$ and the cokernel morphism by
$B\twoheadrightarrow\operatorname{coker}(i).$ We sometimes denote the cokernel
object by $B/A$.

######  Definition 3.4.

A functor $F:\mathcal{C}\rightarrow\mathcal{D}$ between categories with
cofibrations is called _exact_ when it preserves zero objects, cofibrations,
and pushouts of cofibrations.

######  Definition 3.5.

We say $\mathcal{A}$ is a _subcategory with cofibrations_ of a category with
cofibrations $\mathcal{C}$ if the inclusion of $\mathcal{A}$ into
$\mathcal{C}$ is an exact functor, and the cofibrations in $\mathcal{A}$ are
the cofibrations of $\mathcal{C}$ with source and target in $\mathcal{A}$ and
with cokernel in $\mathcal{A}$.

The category whose objects are small categories with cofibrations and whose
morphisms are exact functors, is called the category of categories with
cofibrations and we denote it by Cof.

Now we present the dual notion to a category with cofibrations. By dual
notion, we mean that the opposite category of a category with cofibrations is
a category with fibrations and vice versa. Everything that follows in later
sections about categories with cofibrations has a dual version for categories
with fibrations.

######  Definition 3.6.

A _category with fibrations_ is a category $\mathcal{C}$ with a zero object
and a distinguished class of morphisms called the _fibrations_. These
fibrations and zero object are such that for any object $A$ of $\mathcal{C}$,
the unique map $A\rightarrow 0$ is a fibration, every isomorphism is a
fibration, the composite of fibrations is a fibration, and the pullback of a
fibration always exists and is a fibration.

We indicate that a morphism is a fibration by drawing our arrows with two
heads ($\twoheadrightarrow$). This notation does not mean however that the
morphism is the cokernel of a cofibration. Similarly, in a category with
fibrations we denote that a morphism is the kernel of a fibration by drawing
our arrow with a tail ($\rightarrowtail$). We call a fibration preceded by its
kernel a fibration sequence.

We denote the category whose objects are categories with fibrations by Fib.

######  Example 3.7.

Consider the category $\text{Vect}_{k}$ of $k$-vector spaces and linear maps.
The $0$-dimensional $k$-vector space together with the collection of
surjective linear maps give this category the structure of a category with
fibrations.

###  3.2. Waldhausen categories

Waldhausen categories are categories with cofibrations, together with another
class of morphisms meant to behave like weak homotopy equivalences.

######  Definition 3.8.

[12, 1.2] A _Waldhausen category_ is a category with cofibrations that has a
class $w$ of morphisms called _weak equivalences_ , such that all isomorphisms
are weak equivalences, $w$ is closed under composition, and pushouts along
cofibrations are unique up to weak equivalence. To be precise, this last
condition means that if in the commutative diagram

${B}$${A}$${C}$${{B^{\prime}}}$${{A^{\prime}}}$${{C^{\prime}}}$

the vertical arrows arrows are in $w$, then the induced map $B\cup_{A}C\to
B^{\prime}\cup_{A^{\prime}}C^{\prime}$ is in $w$. We write $w\mathcal{C}$ for
the subcategory of $\mathcal{C}$ whose collection of morphisms is $w$ and co
$w\mathcal{C}$ for the subcategory of $\mathcal{C}$ whose collection of
morphisms is the cofibrations in $w$. The maps in co $w\mathcal{C}$ are called
_acyclic cofibrations_.

######  Example 3.9.

Let $\mathcal{C}$ be a category with cofibrations. Then we can make
$\mathcal{C}$ a Waldhausen category by taking the weak equivalences to be the
isomorphisms.

######  Example 3.10.

Another example is given by $\mathcal{R}_{f}$. Here the weak equivalences are
the weak homotopy equivalences.

There are some commonly ascribed additional properties a Waldhausen category
might have. To define these properties it is convenient to give names to some
categories one can construct given a category with cofibrations.

######  Definition 3.11.

Let $\mathcal{C}$ be a category. Its _arrow category_
$\operatorname{Ar}(\mathcal{C})$ is a category whose objects are morphisms in
$\mathcal{C}$ and whose morphisms are commutative squares in $\mathcal{C}$.

We denote a morphism from $f\colon X\to Y$ to $f^{\prime}\colon X^{\prime}\to
Y^{\prime}$, depicted as

${X}$${{X^{\prime}}}$${Y}$${{Y^{\prime}}}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$

by $(\phi,\psi)\colon f\to f^{\prime}$. Two functors that one always has are
the source and target functors $S\colon\text{Ar}(\mathcal{C})\to\mathcal{C}$
and $T\colon\text{Ar}(\mathcal{C})\to\mathcal{C}$ which take morphisms in
$\mathcal{C}$ to their sources and targets, respectively. If $\mathcal{C}$ is
a category with cofibrations, then $\operatorname{Ar}(\mathcal{C})$ is a
category with cofibrations, where a cofibration is given by a pair of
cofibrations in $\mathcal{C}$. The category $\operatorname{Ar}(\mathcal{C})$
is Waldhausen with weak equivalences given by pairs of weak equivalences.

######  Definition 3.12.

The full subcategory of $\operatorname{Ar}(\mathcal{C})$ on the cofibrations
of $\mathcal{C}$, which we denote by $F_{1}\mathcal{C}$, is a category with
cofibrations the maps $(A\rightarrowtail B)\to(A^{\prime}\rightarrowtail
B^{\prime})$ such that $A\rightarrowtail A^{\prime}$, $B\rightarrowtail
B^{\prime}$, and $A^{\prime}\cup_{A}B\rightarrowtail B^{\prime}$ are
cofibrations in $\mathcal{C}$. The category $F_{1}\mathcal{C}$ is Waldhausen
with weak equivalences the component-wise weak equivalences.

######  Definition 3.13.

A Waldhausen category is _saturated_ if whenever $gf$ is a weak equivalence,
then $f$ is a weak equivalence if and only if $g$ is a weak equivalence.

Motivated by mapping cylinders of spaces, one has the following definition.

######  Definition 3.14.

Let $\mathcal{C}$ be a Waldhausen category. A _mapping cylinder functor_ on
$\mathcal{C}$ is a functor from $\text{Ar}(\mathcal{C})$ to the category of
diagrams in $\mathcal{C}$ taking $f\colon A\to B$ to a diagram

${{S(f)}}$${{M(f)}}$${{T(f)}}$${{T(f)}}$$\scriptstyle{i_{f}}$$\scriptstyle{j_{f}}$$\scriptstyle{f}$$\scriptstyle{r_{f}}$$\scriptstyle{=}$

where $M(f)$ is called the cylinder of $f$. The induced natural
transformations, $i\colon S\to M$, $j\colon T\to M$, and $r\colon M\to T$ are
called the _top inclusion_ , _bottom inclusion_ , and _retract_ ,
respectively. The functor $M$ and the natural transformations must satisfy the
following axioms.

  * •

(MapCyl 1) The functor $i\sqcup j\colon\operatorname{Ar}(\mathcal{C})\to
F_{1}C$ is exact.

  * •

(MapCyl 2) For every object $Y$ in $\mathcal{C}$, $M(0\to Y)=Y$ with $r_{0\to
Y}$ and $j_{0\to Y}$ the identity.

We say that $\mathcal{C}$ satisfies the mapping cylinder axiom if it has a
mapping cylinder such that for all $f\colon X\to Y$ in $\mathcal{C}$,
$r_{f}\colon M(f)\to Y$ is a weak equivalence.

Notice that these axioms imply the following.

  * •

(MapCyl 3) For every morphism $(\phi,\psi)\colon f\to f^{\prime}$ in
$\text{Ar}(\mathcal{C})$, $M((\phi,\psi))\colon M(f)\to M(f^{\prime})$ is a
weak equivalence if $\phi$ and $\psi$ are weak equivalences.

  * •

(MapCyl 4) For every morphism $(\phi,\psi)\colon f\to f^{\prime}$ in
$\text{Ar}(\mathcal{C})$, $M((\phi,\psi))\colon M(f)\to M(f^{\prime})$ is a
cofibration if $\phi$ and $\psi$ are cofibrations.

  * •

(MapCyl 5) For every morphism $(\phi,\psi)\colon f\to f^{\prime}$ in
$\text{Ar}(\mathcal{C})$, the diagram

${{X\sqcup Y}}$${{M(f)}}$${Y}$${{X^{\prime}\sqcup
Y^{\prime}}}$${{M(f^{\prime})}}$${{Y^{\prime}}}$$\scriptstyle{\phi\sqcup\psi}$$\scriptstyle{i_{f}\sqcup
j_{f}}$$\scriptstyle{i_{f^{\prime}}\sqcup
j_{f^{\prime}}}$$\scriptstyle{M((\phi,\psi))}$$\scriptstyle{r_{f^{\prime}}}$$\scriptstyle{\psi}$$\scriptstyle{r_{f}}$

commutes. Furthermore, if $\phi$ and $\psi$ are cofibrations, then the induced
map $(X^{\prime}\sqcup Y^{\prime})\bigsqcup_{X\sqcup Y}M(f)\to M(f^{\prime})$
is a cofibration.

  * •

(MapCyl 6) Given a diagram of the form

${A}$${B}$${{A^{\prime}}}$${{B^{\prime}}}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$

in $\mathcal{C}$, there is a commutative diagram

${{A\sqcup B}}$${{M(f)}}$${{A^{\prime}\sqcup
B^{\prime}}}$${{M(f^{\prime})}}$${{A^{\prime}/A\sqcup
B^{\prime}/B}}$${{M(f^{\prime}/f)}.}$

where $f^{\prime}/f$ is the map from $A^{\prime}/A$ to $B^{\prime}/B$ induced
by $f$ and $f^{\prime}$.

######  Proposition 3.15.

Let $\mathcal{C}$ be a saturated Waldhausen category together with a functor
$M\colon\textrm{Ar}(\mathcal{C})\to\mathcal{C}$ and natural transformations
$r\colon M\to T$, $i\colon S\to M$, and $j\colon T\to M$, where $T$ is the
target functor and $S$ is the source functor. Suppose that the $f$ component
of $r$, $r_{f}$, is a weak equivalence when $f$ is a weak equivalence and that
$M$ satisfies (MapCyl1), (MapCyl3) and (MapCyl4). Then the inclusion
$\iota\colon$co $w\mathcal{C}\to w\mathcal{C}$ is a homotopy equivalence
[14][IV, Exercise 8.15].

###### Proof.

This proof comes from [13][Section 18, Proposition 3]. By Quillen’s Theorem A
[9, 1], it suffices to show that $\iota$ is initial. We must show that for
every $Y\in ob(w\mathcal{C})$, $\iota/Y$ is contractible. An object of
$\iota/Y$ is an object $X$ in $\mathcal{C}$ together with a weak equivalence
$f$ from $X$ to $Y$. A morphism in $\iota/Y$ from $(X,f)$ to $(Z,g)$ is an
acyclic cofibration $h\colon X\to Z$ such that $gh=f$. We define a functor
$m\colon\iota/Y\to\iota/Y$ that sends $(X,f)\to(M(f),r_{f})$. The fact that
$m$ is a functor follows from $r$ being a natural transformation, $r_{f}$
being a weak equivalence, and $M$ satisfying (MapCyl3) and (MapCyl4). Now $i$
defines a natural transformation $id_{\iota/Y}\to m$ using (MapCyl1), the
assumption that $\mathcal{C}$ is saturated, and that $i$ is a natural
transformation from $S$ to $M$. Similarly, $j$ defines a natural
transformation from the constant functor with image object $(id_{y},Y)$ to
$m$. We therefore have homotopies from the identity on $|\iota/Y|$ to $|m|$
and from a constant map from $\iota/Y\to\iota/Y$ to $|m|$. We conclude that
$\iota/Y$ is contractible, and since $Y$ was arbitrary, $|\iota|$ is a
homotopy equivalence. ∎

##  4\. The $S_{\bullet}$-construction and 2-Segal objects

###  4.1. The $S_{\bullet}$-construction

A sequence of cofibrations in $\mathcal{C}$ together with choices of cokernels
gives an object in the category $S_{n}\mathcal{C}$. The categories
$S_{n}\mathcal{C}$ taken together with certain functors between them form the
simplicial category $S_{\bullet}\mathcal{C}$. This construction was given in
Waldhausen’s paper [12, §1] en route to creating an algebraic $K$-theory of
spaces. We now review this construction and $2$-Segal objects.

######  Definition 4.1.

The category $\Delta$, called the _simplex category_ , has objects the ordered
sets $[n]\colon=\\{0\leq 1\leq\cdots\leq n\\}$ and morphisms order-preserving
maps.

We note that if we view $[n]$ as a poset category, then $\Delta$ can be viewed
as the full subcategory of the category of small categories $\mathcal{C}at$,
on those categories of the form $[n]$ for some nonnegative integer $n$. It is
often helpful to know that every morphism in $\Delta$ is the composite of face
and degeneracy maps. The face maps $\delta^{n}_{i}\colon[n-1]\to[n]$ are the
injective maps "skipping $i$", and the degeneracy maps
$\sigma^{n}_{i}\colon[n+1]\to[n]$ are the surjective maps "hitting $i$ twice".
Given a functor $X\colon\Delta^{op}\to\mathcal{D}$, a simplicial object in
$\mathcal{D}$, we denote $X([n])$ by $X_{n}$ and we denote the image of
$\delta^{n}_{i}$ by $d_{i}$ and the image of $\sigma^{n}_{i}$ by $s_{i}$.

######  Definition 4.2.

Given categories $\mathcal{A}$ and $\mathcal{B}$, the _category of functors_
$\operatorname{Fun}(\mathcal{A},\mathcal{B})$ has functors from $\mathcal{A}$
to $\mathcal{B}$ as its objects and natural transformations as its morphisms.

######  Example 4.3.

The arrow category of $\mathcal{C}$ (Definition 3.11) may be identified with
$\operatorname{Fun}([1],\mathcal{C})$.

######  Example 4.4.

Fix $n\in\mathbb{N}$. Viewing $[n]$ as a poset category we consider
$\operatorname{Ar}[n]$. Any morphism $i\to j$ in $[n]$ may be identified with
the pair $(i,j)$ where $i\leq j$. Any commutative square

${i}$${j}$${{i^{\prime}}}$${{j^{\prime}}}$

in $[n]$ may be identified with two pairs, $(i,j)$ and
$(i^{\prime},j^{\prime})$, where $i\leq j$ and $i^{\prime}\leq j^{\prime}$,
and with $i\leq i^{\prime}$ and $j\leq j^{\prime}$. In other words,
$\operatorname{Ar}[n]$ may be identified with the poset category whose objects
are pairs $(i,j)$ with $0\leq i\leq j\leq n$, and where
$(i,j)\leq(i^{\prime},j^{\prime})$ if and only if $i\leq i^{\prime}$ and
$j\leq j^{\prime}$. The diagram

${{(n,n)}}$${\vdots}$${{(2,2)}}$${\cdots}$${{(2.n)}}$${{(1,1)}}$${{(1,2)}}$${\cdots}$${{(1,n)}}$${{(0,0)}}$${{(0,1)}}$${{(0,2)}}$${\cdots}$${{(0,n)}}$

gives a picture of this category.

######  Definition 4.5.

For $\mathcal{C}$ a category with cofibrations, $S_{n}\mathcal{C}$ is the full
subcategory of the category of functors
$A\colon\operatorname{Ar}[n]\rightarrow\mathcal{C},(i,j)\mapsto A_{i,j}$ on
those functors with $A_{j,j}=0$ for every $j$, and $A_{i,j}\rightarrowtail
A_{i,k}\twoheadrightarrow A_{j,k}$ a cofibration sequence for every $i<j<k$.
For $\mathcal{C}$ a category with fibrations $S_{n}\mathcal{C}$ is the full
subcategory of $\operatorname{Fun}(\operatorname{Ar}[n],\mathcal{C}$) on
functors $A$ with $A_{j,j}=0$ for every $j$ and $A_{i,j}\rightarrowtail
A_{i,k}\twoheadrightarrow A_{j,k}$ a fibration sequence for every $i<j<k$.

Waldhausen [12, 1.3] showed that $S_{n}\mathcal{C}$ is a category with
cofibrations whenever $\mathcal{C}$ is.

######  Proposition 4.6.

The assignment $[n]\mapsto S_{n}\mathcal{C}$ defines a functor
$S_{\bullet}\mathcal{C}\colon\Delta^{\operatorname{op}}\rightarrow\textit{Cof}.$

###### Proof.

Objects of $S_{m}\mathcal{C}$ are also functors
$\operatorname{Ar}[m]\to\mathcal{C}$. The map $[n]\to[m]$ in $\Delta$ induces
a map $\operatorname{Ar}[n]\to\operatorname{Ar}[m]$ which in turn induces a
map $S_{m}\mathcal{C}\to S_{n}\mathcal{C}$ given by precomposition. Therefore
our assignment is a functor from $\Delta^{op}\to\mathcal{C}at$. We have
mentioned that $S_{n}\mathcal{C}$ is a category with cofibrations so it
remains to check that the images of morphisms in $\Delta^{op}$ are exact. See
[12, 1.1].

∎

Similarly, if $\mathcal{C}$ is a category with fibrations then $[n]\mapsto
S_{n}\mathcal{C}$ defines a functor $\Delta^{op}\to\textit{Fib}$.

######  Definition 4.7.

There is a forgetful functor $\mathcal{C}at\rightarrow\mathcal{S}et$ that
sends a small category to its set of objects and a functor to the
corresponding function on sets of objects. We denote the composite
$\Delta^{\operatorname{op}}\xrightarrow{S_{\bullet}\mathcal{C}}\mathcal{C}at\rightarrow\mathcal{S}et$
by $s_{\bullet}\mathcal{C}$.

Each simplicial map $\alpha\colon[m]\to[n]$ induces a map between the set of
isomorphism classes of the objects of $S_{n}\mathcal{C}$ and the set of
isomorphism classes of the objects of $S_{m}\mathcal{C}$

######  Definition 4.8.

We denote by $\text{iso}(s_{\bullet}\mathcal{C})$ the simplicial set that
takes $[n]$ to $\text{iso}(s_{n}\mathcal{C})$, the set of isomorphism classes
of objects in $S_{n}\mathcal{C}$.

Since a Waldhausen category is also a category with cofibrations, we may apply
the $\text{iso}(s_{\bullet}-$),$s_{\bullet}$-, and
$S_{\bullet}$-constructions. For each $n$, $S_{n}\mathcal{C}$ is a Waldhausen
category where a weak equivalence is a component-wise weak equivalence. The
subcategory of weak equivalences in $S_{n}\mathcal{C}$ is denoted by
$wS_{n}\mathcal{C}$. Since $\mathcal{C}$ is also a Waldhausen category with
weak equivalences the isomorphisms, we can consider $S_{n}\mathcal{C}$ as a
Waldhausen category whenever $\mathcal{C}$ is a category with cofibrations.

######  Proposition 4.9.

[12, 1.6.1] If $\mathcal{C}$ is a saturated Waldhausen category satisfying the
mapping cylinder axiom, then $S_{n}\mathcal{C}$ satisfies the hypotheses of
Proposition 3.15.

###  4.2. 2-Segal objects

The goal of this paper broadly is to see what structure
$S_{\bullet}\mathcal{C}$ and its variants have, given the properties we assume
for $\mathcal{C}$. Some structure that $S_{\bullet}\mathcal{C}$ might have is
that of a $2$-Segal category. We review $2$-Segal objects here. The definition
of a $2$-Segal object was inspired by the definition of a $1$-Segal object so
we start there.

######  Definition 4.10.

A simplicial set $X$ is a _$1$ -Segal set_ if the map $X_{n}\rightarrow
X_{1}\times_{X_{0}}X_{1}\times_{X_{0}}X_{1}\cdots\times_{X_{0}}X_{1}$ induced
by $\\{i,i+1\\}\hookrightarrow[n]$ for all $i\in[n-1]$ is a bijection for all
$n\geq 2$ [2, 1.1].

A $1$-Segal set is always isomorphic to the nerve of a category, so we think
of a $1$-Segal set as corresponding to a category with objects the
$0$-simplices and $1$-simplices the morphisms [3, 2.2.2]. Therefore in some
sense, one can study categories by studying $1$-Segal sets. If, instead of
strict composition, we wanted to model composition up to homotopy we could
replace pullbacks with homotopy pullbacks and change the requirement of having
a bijection to having a weak equivalence. One framework where we have homotopy
pullbacks and weak equivalences is that of model categories.

######  Definition 4.11.

For a model category $\mathcal{D}$, a simplicial object in $\mathcal{D}$ is
_$1$ -Segal_ if the map $X_{n}\rightarrow
X_{1}\times^{h}_{X_{0}}X_{1}\times^{h}_{X_{0}}X_{1}\cdots\times^{h}_{X_{0}}X_{1}$
induced by $\\{i,i+1\\}\hookrightarrow[n]$ for $i\in[n-1]$ is a weak
equivalence where the right-hand side is an iterated homotopy pullback.

######  Remark 4.12.

We could instead take $\mathcal{D}$ to be an $(\infty,1)$-category or any
other category with a notion of homotopy limits and weak equivalences.

The maps in the above definitions are called _$1$ -Segal maps_. We can
describe them in a different way that better suggests the generalization to
$2$-Segal maps. In a sufficiently nice model category $\mathcal{D}$ there are
derived mapping objects $\text{Map}^{h}(K,X)$ for all simplicial sets $K$ and
simplicial objects $X$ in $\mathcal{D}$ [2, 1.1]. The derived mapping objects
are such that if $W\colon A\to\mathcal{SS}ets$ is a diagram, then
$\text{Map}^{h}(\operatorname{colim}_{A}W_{a},X)$ is weakly equivalent to
$\text{holim}_{A}\text{Map}^{h}(W_{a},X)$ and $\text{Map}^{h}(\Delta[n],X)$ is
weakly equivalent to $X_{n}$ for all $n$. Now fix $n$ and consider the diagram

${{\Delta[0]}}$${{\Delta[1]}}$${{\Delta[0]}}$${{\Delta[1]}}$${{\Delta[0]}}$${\cdots}$${{\Delta[0]}}$${{\Delta[n].}}$$\scriptstyle{0\to
0}$$\scriptstyle{0\to 1}$$\scriptstyle{0\to 0}$$\scriptstyle{0\to
1}$$\scriptstyle{0\to 0}$$\scriptstyle{0\to 1}$$\scriptstyle{0\to
0}$$\scriptstyle{0\to 0,1\to 1}$$\scriptstyle{0\to 1,1\to
2}$$\scriptstyle{0\to 2}$$\scriptstyle{0\to n}$

From the diagram above there is an induced map
$\Delta[1]\sqcup_{\Delta[0]}\Delta[1]\sqcup_{\Delta[0]}\cdots\sqcup_{\Delta[0]}\Delta[1]\to\Delta[n]$
that we think of as the inclusion of the spine of $\Delta[n]$ into
$\Delta[n]$. It turns out that applying $\text{Map}^{h}(-,X)$ yields a map
fitting into the commutative square

${{\text{Map}^{h}(\Delta[1]\sqcup_{\Delta[0]}\Delta[1]\sqcup_{\Delta[0]}\cdots\sqcup_{\Delta[0]}\Delta[1],X)}}$${{\text{Map}^{h}(\Delta[n],X)}}$${{X_{1}\times_{X_{0}}^{h}X_{1}\times_{X_{0}}^{h}\cdots\times_{X_{0}}^{h}X_{1}}}$${{X_{n}.}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Therefore the $1$-Segal maps are induced by the inclusion of a $1$-dimensional
subsimplicial set of $\Delta[n]$ made up of ($n-1$) $1$-simplices glued to
each other along $0$-simplices. To obtain $2$-Segal maps we can start with
subsimplicial sets of $\Delta[n]$ consisting of ($n-2$) standard $2$-simplices
glued to each other along $1$-simplices. These $2$-dimensional subsimplicial
sets can be unfolded and stretched into a triangulation of a regular $n$-gon.
We number the vertices of polygons to give a way of specifying triangulations.

######  Definition 4.13.

We say that a polygon has its vertices _cyclically labeled_ if it is possible
to read the vertices in counterclockwise order as $0,1,2\ldots,n$ for some
$n$. We write $P_{n}$ for the regular $(n+1)$-gon with cyclically labeled
vertices.

123012340123ExampleNon-exampleNon-exampleFigure 1

Figure 1 shows some examples and non-examples of cyclically labeled polygons.
The middle polygon in that figure is $P_{4}$.

######  Definition 4.14.

A _polygonal subdivision_ of a cyclically labeled polygon $P$ is a collection
$\mathcal{P}$ of polygons contained in $P$ such that members of $\mathcal{P}$
have their vertices among the vertices of $P$, and such that any two members
of $\mathcal{P}$ are either disjoint or intersect on a common edge. A
_triangulation_ is a polygonal subdivision consisting of triangles. A
polygonal subdivision inherits labelings on the vertices of its members from
the labeling of the polygon it subdivides. A _diagonal_ of a polygonal
subdivision is an edge of a member of the subdivision that is not an edge of
the polygon being subdivided.

120123402ExampleNon-exampleNon-exampleFigure 2013

Figure 2 above shows some examples and non-examples of polygonal subdivisions
of cyclically labeled polygons. In the right picture, the triangle with
vertices 0, 2, and 3 intersects the triangle with vertices 0, 1, 3, on more
than a common edge. The picture on the left has subdividing polygons with a
vertex not among the vertices of the polygon being subdivided. In the center
picture, the edge from vertex 1 to vertex 3 is a diagonal since it is not an
edge of the five-sided polygon being subdivided.

We now describe how we can define $2$-Segal maps without derived mapping
objects using the middle triangulation in Figure 2 as an example. The
triangulation corresponds to the category

$\\{1,2,3\\}\leftarrow\\{1,3\\}\to\\{1,3,4\\}\leftarrow\\{1,4\\}\to\\{0,1,4\\}.$

Now consider $\mathcal{P}_{\text{Fin}}(\mathbb{N})$, the category of finite
ordered subsets of $(\mathbb{N},\leq)$ and order-preserving functions,
$\mathcal{P}_{\text{Fin}}(\mathbb{N})$. The diagram above can be extended to
the commutative diagram

${{[2]}}$${{[1]}}$${{[2]}}$${{[1]}}$${{[2]}}$${{\\{1,2,3\\}}}$${{\\{1,3\\}}}$${{\\{1,3,4\\}}}$${{\\{1,4\\}}}$${{\\{0,1,4\\}}}$${{[4]}}$${{=}}$${{[4]}}$${{=}}$${{[4]}}$${{=}}$${{[4]}}$${{=}}$${{[4]}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

in $\mathcal{P}_{\text{Fin}}(\mathbb{N})$ where the top vertical morphisms are
unique isomorphisms. Deleting the middle row we get the diagram

${{[2]}}$${{[1]}}$${{[2]}}$${{[1]}}$${{[2]}}$${{[4]}}$${{=}}$${{[4]}}$${{=}}$${{[4]}}$${{=}}$${{[4]}}$${{=}}$${{[4]}}$$\scriptstyle{\\{1,2,3\\}}$$\scriptstyle{\\{1,3\\}}$$\scriptstyle{\\{1,3,4\\}}$$\scriptstyle{\\{1,4\\}}$$\scriptstyle{\\{0,1,4\\}}$$\scriptstyle{\delta^{2}_{1}}$$\scriptstyle{\delta^{2}_{2}}$$\scriptstyle{\delta^{2}_{2}}$$\scriptstyle{\delta^{2}_{0}}$

in $\Delta$ where the vertical morphisms are labeled by their images. Now if
$X$ is a simplicial object in $\mathcal{C}$ we have a corresponding diagram

${{X_{2}}}$${{X_{1}}}$${{X_{2}}}$${{X_{1}}}$${{X_{2}}}$${{X_{4}}}$${{=}}$${{X_{4}}}$${{=}}$${{X_{4}}}$${{=}}$${{X_{4}}}$${{=}}$${{X_{4}.}}$$\scriptstyle{\\{1,2,3\\}}$$\scriptstyle{\\{1,3\\}}$$\scriptstyle{\\{1,3,4\\}}$$\scriptstyle{\\{1,4\\}}$$\scriptstyle{\\{0,1,4\\}}$$\scriptstyle{d^{2}_{1}}$$\scriptstyle{d^{2}_{2}}$$\scriptstyle{d^{2}_{1}}$$\scriptstyle{d^{2}_{0}}$

Thus there is an induced map $X_{4}\to
X_{2}\times_{X_{1}}X_{2}\times_{X_{1}}X_{2}$, which we write as $X_{4}\to
X_{\\{1,2,3\\}}\times_{X_{\\{1,3\\}}}X_{\\{1,3,4\\}}\times_{X_{\\{1,4\\}}}X_{\\{0,1,4\\}}.$
If $\mathcal{D}$ is a sufficiently nice model category, then the induced map
fits into the commutative square

${{\text{Map}^{h}(\Delta[4],X)}}$${{\text{Map}^{h}(\Delta[2]\sqcup_{\Delta[1]}\Delta[2]\sqcup_{\Delta[1]}\Delta[2],X)}}$${{X_{4}}}$${{X_{\\{1,2,3\\}}\times_{X_{\\{1,3\\}}}^{h}X_{\\{1,3,4\\}}\times_{X_{\\{1,4\\}}}^{h}X_{\\{0,1,4\\}}.}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

We have associated a map to a given polygonal subdivision. In general the
procedure is as follows. Suppose we are given a simplicial object in
$\mathcal{D}$, $X$ and a polygonal subdivision $\mathcal{P}$ of an $n$-gon.
First, we extend $X$ to a functor
$X\colon\mathcal{P}_{\text{Fin}}(\mathbb{N})\to\mathcal{D}$. After identifying
objects of $\mathcal{P}_{\text{Fin}}(\mathbb{N})$ with order-preserving
bijections of the form $f_{I}\colon[k]\to\\{i_{0},\ldots,i_{k}\\}=I$, $X$ is
given by $X(f_{I})=X_{k}$ and by sending $\alpha\colon
I=\\{i_{0}<i_{1}<\ldots<i_{k}\\}\to J=\\{j_{0}<j_{1}<\ldots<j_{m}\\}$ to
$X(f^{-1}_{J}\alpha f_{I})$. And now the $2$-Segal map associated to the
polygonal subdivision $\mathcal{P}$ and the functor $X$, $f_{\mathcal{P}}$, is
the map from $X_{n}$ to the homotopy limit over $X$ of the poset category of
vertex sets appearing in the subdivision.

######  Definition 4.15.

A simplicial object $X$ in a model category $\mathcal{C}$ is $2$-_Segal_ if
for every polygonal subdivision $\mathcal{P}$ of $P_{n}$, the map
$f_{\mathcal{P}}$ is a weak equivalence. The collection of maps
$f_{\mathcal{P}}$ for $\mathcal{P}$ a polygonal subdivision of $P_{n}$ for
some $n$ are called the $2$-_Segal maps_.

######  Proposition 4.16.

The simplicial topological space $|S_{\bullet}\mathcal{C}|$ is $2$-Segal for
any category with cofibrations $\mathcal{C}$.

###### Proof.

For all $n$, $S_{n}\mathcal{C}$ has a zero object given by the diagram with
all objects $0$. Thus $|S_{n}\mathcal{C}|$ is contractible. The homotopy
pullback respects weak homotopy equivalence so the $2$-Segal maps of
$|S_{\bullet}\mathcal{C}|$ are between contractible spaces. Such maps are
automatically weak homotopy equivalences. ∎

##  5\. Homotopy limits and projective 2-limits

The homotopy limit of a diagram of spaces is generally more difficult to
compute than the limit of a diagram of spaces. We replace homotopy limits with
something a little easier to work with, what Dyckerhoff and Kapranov call
projective $2$-limits [3, 1.3.6]. Finally, we give a technical result on
homotopy limits of diagrams of categories.

######  Definition 5.1.

Let $A$ be a small category and $(\mathcal{C}_{a})_{a\in A}$ be a diagram of
categories. The _projective 2-limit_ $2\lim_{a\in A}\mathcal{C}_{a}$ is the
category where an object is data consisting of:

(0) for all $a\in\text{ob}(A)$, an object $y_{a}$ of $\mathcal{C}_{a}$; and,

(1) for all $u\colon a\rightarrow b$ in $A$, an isomorphism $y_{u}\colon
u_{*}(y_{a})\rightarrow y_{b}$ in $\mathcal{C}_{b}$; such that

(2) if $a\xrightarrow{u}b\xrightarrow{b}c$ is a composable pair of morphisms
in $A$, then $y_{vu}=y_{v}\circ v_{*}(y_{u})$.

A morphism in $2\lim_{a\in A}\mathcal{C}_{a}$ from $(y_{a},y_{u})$ to
$(y^{\prime}_{a},y^{\prime}_{u})$ is a system of morphisms $y_{a}\rightarrow
y^{\prime}_{a}$ in $\mathcal{C}_{a}$ commuting with the $y_{u}$ and
$y^{\prime}_{u}$.

We note the similarity between the projective $2$-limit of a diagram into
$\mathcal{C}at$ and a certain characterization of the ordinary limit of a
diagram into $\mathcal{C}at$. Let $A$ be a small category and
$(\mathcal{C}_{a})$ a small diagram of categories. A limit object of the
diagram is given by the category where an object consists of:

(0) for all $a\in\operatorname{ob}(A)$, an object $y_{a}$ of
$\mathcal{C}_{a}$; such that

(1) for all $u\colon a\to b$ in $A$, $u_{*}(y_{a})=y_{b}$.

A morphism in $\lim_{A}\mathcal{C}_{a}$ from $(y_{a})$ to $(y_{a}^{\prime})$
is a system of morphisms $y_{a}\to y_{a}^{\prime}$ such that for all $u\colon
a\to b$, $u_{*}(y_{a}\to y_{a}^{\prime})=y_{b}\to y_{b}^{\prime}$.

So the difference between the ordinary limit and the projective $2$-limit is
that in the projective $2$-limit, instead of having $u_{*}(y_{a})=y_{b}$ for
all $u\colon a\to b$ we replace equalities with isomorphisms and require these
isomorphisms to be coherent.

The projective $2$-limit of a diagram of the form $A\rightarrow B\leftarrow C$
is denoted by $A\times^{(2)}_{B}C$ and called a _2-fiber product_.

######  Example 5.2.

Consider the category $[0]$ with one object and just the identity morphism.
Let $F\colon[0]\to\mathcal{C}at$ be a $[0]$-diagram of categories. Write
$\mathcal{C}$ for the unique category in the image of the diagram. Then
$2\lim_{a\in[0]}\mathcal{C}_{a}=\mathcal{C}$ because an object of
$2\lim_{a\in[0]}\mathcal{C}_{a}$ is an object $y_{a}$ of
$\mathcal{C}_{a}=\mathcal{C}$ together with an automorphism $y_{u}$ of $y_{a}$
such that $y_{u}=y_{u}^{2}$. Applying $y_{u}^{-1}$ to both sides shows that
$y_{u}$ must be the identity.

######  Example 5.3.

We describe the $2$-fiber product $A\times_{B}^{(2)}C$ of the diagram
$A\xrightarrow{F}B\xleftarrow{G}C$. By definition, an object is data
consisting of objects $a\in\operatorname{ob}(A)$, $b\in\operatorname{ob}(B)$,
and $c\in\operatorname{ob}(C)$, and isomorphisms $y_{id_{A}}\colon a\cong a$,
$y_{id_{B}}\colon b\cong b$, $y_{id_{C}}\colon c\cong c$, $y_{F}\colon
F(a)\cong b$, and $y_{G}\colon G(c)\cong b$. We write this object of
$A\times_{B}^{(2)}C$ as the tuple
$(a,b,c,y_{id_{A}},y_{id_{B}},y_{id_{C}},y_{F},y_{G})$. Some of this data is
actually superfluous. The same argument in the previous example shows that
$y_{id_{A}}$, $y_{id_{B}}$, and $y_{id_{C}}$ are all the identity. Thus we may
describe an object by only giving the data in the tuple $(a,b,c,y_{F}\colon
F(a)\cong b,y_{G}\colon G(c)\cong b)$. A morphism from $(a,b,c,y_{F}\colon
F(a)\cong b,y_{G}\colon G(c)\cong b)$ to
$(a^{\prime},b^{\prime},c^{\prime},y^{\prime}_{F}\colon F(a^{\prime})\cong
b^{\prime},y^{\prime}_{G}\colon G(c^{\prime})\cong b^{\prime})$ consists of
morphisms $f_{A}\colon a\to a^{\prime}$, $f_{B}\colon b\to b^{\prime}$, and
$f_{C}\colon c\to c^{\prime}$ such that the diagram

${{F(a)}}$${b}$${{G(c)}}$${{F(a^{\prime})}}$${{b^{\prime}}}$${{G(c^{\prime})}}$$\scriptstyle{F(f_{A})}$$\scriptstyle{G(f_{C})}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

commutes.

For the homotopy limits we want to replace, the model structure for
$\mathcal{C}at$ that we have in mind is the so-called "canonical model
structure". We begin by defining the fibrations for this model category

######  Definition 5.4.

[11, §2] A functor $F\colon\mathcal{C}\to\mathcal{D}$ is an _isofibration_ if
whenever $c$ is an object of $\mathcal{C}$ and $f\colon F(c)\to d$ is an
isomorphism, there is a $g$ such that $F(g)=f$.

######  Proposition 5.5.

[11, 3.1] There is a model category structure on $\mathcal{C}at$, called the
canonical model category structure, such that the fibrations are the
isofibrations, the cofibrations are the functors injective on objects, and the
weak equivalences are equivalences.

######  Proposition 5.6.

[3, 1.3.8] Let $|-|\colon\mathcal{C}at\to\mathcal{T}op$ denote geometric
realization. For any diagram of categories $A\to\mathcal{C}at$,
$a\mapsto\mathcal{C}_{a}$, we have a natural morphism of spaces
$f\colon|2\lim_{a\in\text{ob}(A)}\mathcal{C}_{a}|\to\text{holim}_{a\in\text{ob}(A)}|\mathcal{C}_{a}|$.
If $A\to\mathcal{C}at$ has image in the category of groupoids, then
$2\lim_{a\in\text{ob}(A)}\mathcal{C}_{a}$ is a groupoid and $f$ is a weak
equivalence.

Let $C\rightarrow D\leftarrow E$ be a diagram of groupoids. In light of the
previous proposition, if we had a weak equivalence between
$|C\times^{2}_{D}E|$ and $|C\times^{h}_{D}E|$ then we would have a weak
equivalence between $|C|\times_{|D|}^{h}|E|$ and $|C\times^{h}_{D}E|$. There
are some strong conditions under which there is a weak equivalence between
$|C\times^{2}_{D}E|$ and $|C\times^{h}_{D}E|$. We do not use these conditions
in what follows but believe they are of independent interest, so we give them
here. To arrive at these conditions we need to pick a particular model for the
homotopy pullback of categories, which we do now.

There is a functorial factorization of each morphism in $\mathcal{C}at$ into a
trivial cofibration followed by a fibration. In this factorization, the
functor $F\colon C\to D$ factors through $L(C)$, a category whose objects are
tuples $(c,d,\phi)$ where $c\in\text{ob}(C)$, $d\in\text{ob}(D)$, and
$\phi\colon F(c)\xrightarrow{\cong}d$ is an isomorphism in $D$. In the
canonical model structure, every category is fibrant, so by [6, 13.1.3],
$\mathcal{C}at$ is right proper. Thus the homotopy pullback of $C\rightarrow
D\leftarrow E$ may be defined by first factoring as $C\to L(C)\to D\leftarrow
L(E)\leftarrow E$ and then taking the pullback of $L(C)\to D\leftarrow L(E)$
[6, 13.3.2]. We conclude that a particular model for the homotopy pullback of
$C\xrightarrow{F}D\xleftarrow{K}E$ is presented by a category with objects
$(c,d,e,\phi\colon F(c)\xrightarrow{\cong}d,\psi\colon
F(c)=K(e)\xrightarrow{\cong}d)$, and a morphism from $(c,d,e,\phi\colon
F(c)\xrightarrow{\cong}d,\psi\colon F(c)=K(e)\xrightarrow{\cong}d)$ to
$(c^{\prime}d^{\prime},e^{\prime},\phi^{\prime}\colon F(c^{\prime})\to
d^{\prime},\psi^{\prime}\colon K(e^{\prime})\to d^{\prime})$ is given by a
pair $c\to c^{\prime}$ in $C$ and $e\to e^{\prime}$ in $E$ such that $F(c\to
c^{\prime})=K(e\to e^{\prime})$.

######  Proposition 5.7.

If $C\xrightarrow{F}D\xleftarrow{K}E$ is a a diagram of groupoids with either
$F$ or $K$ full and surjective on objects, then $|C\times^{h}_{D}E|$ is weakly
homotopy equivalent to $|C|\times^{h}_{|D|}|E|$.

###### Proof.

Without loss of generality assume $F$ is full and surjective on objects. We
define a functor $\mathcal{H}\colon C\times^{h}_{D}E\to C\times^{2}_{D}E$ by
sending $(c,d,e,\phi\colon F(c)\to d,\psi\colon K(e)\to d)$ to
$(c,d,e,\phi,\phi)$. Notice that $\mathcal{H}$ forgets about $\psi$. For
$L\colon A\to B$ we write $L/b$ for the slice category whose objects are
morphisms $L(a)\to b$. We claim that for any
$(c^{\prime},d^{\prime},e^{\prime},\phi^{\prime},\psi^{\prime})$ in
$C\times^{2}_{D}E$, the category
$\mathcal{H}/(c^{\prime},d^{\prime},e^{\prime},\phi^{\prime},\psi^{\prime})$
has an initial object. Then Quillen’s Theorem A [9, 1] applies to the
discussion after Proposition 5.6 to give the result. Let $a$ be some object
that $F$ sends to $K(e^{\prime})$, $h$ be some morphism $F$ sends to
$\phi^{\prime-1}\psi^{\prime}$. We claim that for
$\mathcal{H}/(c^{\prime},d^{\prime},e^{\prime},\phi^{\prime},\psi^{\prime})$,
an initial object is $(a,d^{\prime},e^{\prime},\psi^{\prime},\psi^{\prime})$
with the morphism part of the object given by
$h,id_{d^{\prime}},id_{e^{\prime}}$. To verify this data gives an initial
object we take any object of
$\mathcal{H}/(c^{\prime},d^{\prime},e^{\prime},\phi^{\prime},\psi^{\prime})$,
meaning a tuple
$(c^{\prime\prime},d^{\prime\prime},e^{\prime\prime},\phi^{\prime\prime},\psi^{\prime\prime})$
with $F(c^{\prime\prime})=K(e^{\prime\prime})$, $g_{C}\colon
c^{\prime\prime}\to c^{\prime}$, $g_{D}\colon d^{\prime\prime}\to d^{\prime}$,
and $g_{E}\colon e^{\prime\prime}\to e^{\prime}$ such that

${{F(c^{\prime\prime})}}$${d^{\prime\prime}}$${{K(e^{\prime\prime})}}$${{F(c^{\prime})}}$${{d^{\prime}}}$${{K(e^{\prime})}}$$\scriptstyle{F(g_{C})}$$\scriptstyle{g_{D}}$$\scriptstyle{K(g_{E})}$$\scriptstyle{\phi^{\prime\prime}}$$\scriptstyle{\phi^{\prime\prime}}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{\psi^{\prime}}$

commutes.

The unique morphism needed is given by the pair $(g_{C}^{-1}h,g_{E}^{-1})$.

∎

##  6\. 2-Segality of discrete variants of the $S_{\bullet}$-construction

Let $\mathcal{C}$ be a category with cofibrations. We determine which
$2$-Segal maps for $\text{iso}(s_{\bullet}\mathcal{C})$ are necessarily
bijections. These $2$-Segal maps turn out to be those from triangulations of
$P_{n}$ for which each triangle contains the vertex $0$.

First, we remark that the $2$-Segal maps for $s_{\bullet}\mathcal{C}$ are
almost never bijective. An example illustrates the general problem. Consider
the $2$-Segal map $s_{3}\mathcal{C}\to
s_{\\{0,1,2\\}}\mathcal{C}\times_{s_{\\{0,2\\}}\mathcal{C}}s_{\\{0,2,3\\}}\mathcal{C}$.
If this map were injective then any diagram of the form

${{A_{2,3}}}$${{A_{1,2}}}$${\bullet}$${{A_{0,1}}}$${{A_{0,2}}}$${{A_{0,3}}}$

could be uniquely completed to an object of $s_{3}\mathcal{C}$. Any completion
of the above diagram amounts to a choice of cokernel of $A_{01}\to A_{03}$ but
there is no reason a priori for there to be a unique cokernel for this
morphism in $\mathcal{C}$. The general failure of any $2$-Segal map for
$s_{\bullet}\mathcal{C}$ to be bijective is why we look at
$\text{iso}(s_{\bullet}\mathcal{C})$ instead.

######  Lemma 6.1.

Suppose that for every $n\geq 3$ and $0\leq j\leq n$, the map $X_{n}\to
X_{\\{0,j,j+1,\ldots,n\\}}\times_{X_{\\{0,j\\}}}X_{\\{0,1,\ldots,j\\}}$ is a
bijection. Then the $2$-Segal map, coming from the triangulation $\mathcal{T}$
where each triangle contains the vertex labeled $0$ under some cyclic
labeling, is bijective.

###### Proof.

We use the hypothesis repeatedly. The bijections

${{X_{n}}}$${{\xrightarrow{\cong}}}$${{X_{\\{0,n-1,n\\}}\times_{X_{\\{0,n-1\\}}}X_{\\{0,1,\ldots,n-1\\}}}}$${{X_{\\{0,1,\ldots,n-1\\}}}}$${{\xrightarrow{\cong}}}$${{X_{\\{0,n-2,n-1\\}}\times_{X_{\\{0,n-2\\}}}X_{\\{0,1,\ldots,n-2\\}}}}$${\vdots}$${{X_{\\{0,1,2,3\\}}}}$${{\xrightarrow{\cong}}}$${{X_{\\{0,2,3\\}}\times_{X_{\\{0,2\\}}}X_{\\{0,1,2\\}}}}$

combine to give that $f_{\mathcal{T}}\colon X_{n}\to
X_{\\{0,n-1,n\\}}\times_{X_{\\{0,n-1\\}}}X_{\\{0,n-2,n-1\\}}\times_{X_{\\{0,n-2\\}}}\cdots\times_{X_{\\{0,2\\}}}X_{\\{0,1,2\\}}$
is a bijection. ∎

Similarly, if the map $X_{n}\to
X_{\\{0,1,\ldots,j,n\\}}\times_{X_{\\{j,n\\}}}X_{\\{j,j+1,\ldots,n\\}}$ is a
bijection for every $n\geq 3$ and $0\leq j\leq n$, then the $2$-Segal map
coming from the triangulation $\mathcal{T}$ where each triangle contains the
vertex labeled $n$ under some cyclic labeling is bijective.

The previous lemma motivates us to define a weakened version of $2$-Segal
sets.

######  Definition 6.2.

Suppose $X$ is a simplicial object in $\mathcal{C}$.

  * •

We say $X$ is _left $2$-Segal_ if for every $n\geq 3$ and $0\leq j\leq n$, the
map $X_{n}\to
X_{\\{0,j,j+1,\ldots,n\\}}\times_{X_{\\{0,j\\}}}^{h}X_{\\{0,1,\ldots,j\\}}$ is
a weak equivalence.

  * •

We say $X$ is _right $2$-Segal_ if for every $n\geq 3$ and $0\leq j\leq n$,
the map $X_{n}\to
X_{\\{0,1,\ldots,j,n\\}}\times_{X_{\\{j,n\\}}}^{h}X_{\\{j,j+1,\ldots,n\\}}$ is
a weak equivalence.

The last fact we use to prove that certain $f_{\mathcal{T}}$ are always
bijective is the following.

######  Lemma 6.3.

Let $M_{n}$ be the set of isomorphism classes of chains of $(n-1)$
cofibrations. The function
$\mu_{n}\colon\text{iso}(s_{n}\mathcal{C})\longrightarrow M_{n}$ that sends
$[A]$ to $[A_{0,1}\rightarrowtail A_{0,2}\rightarrowtail\cdots\rightarrowtail
A_{0,n}]$ is a bijection.

###### Proof.

An inverse is defined as follows. Suppose [$A_{0,1}\rightarrowtail
A_{0,2}\rightarrowtail\cdots\rightarrowtail A_{0,n}]$ is an element of
$M_{n}$. Let $A_{k,k}=0$ for all $k$. Then for all other pairs $i,j$ with
$0<i\leq n$ and $0\leq j\leq n$, let $A_{i,j}$ be a pushout object of the
diagram $0\leftarrow A_{0,i}\rightarrow A_{0,j}$ noting that all pushouts of a
given diagram are isomorphic. Now, there are induced morphisms that make $[A]$
into an element of $\text{iso}(s_{n}\mathcal{C})$. ∎

Similarly, if $\mathcal{C}$ is a category with fibrations and $E_{n}$ is the
set of isomorphisms classes of chains of $(n-1)$ fibrations, then the function
$\text{iso}(s_{n}\mathcal{C})\to E_{n}$ which sends $[A]$ to
$[A_{0,n}\twoheadrightarrow A_{1,n}\twoheadrightarrow\cdots\twoheadrightarrow
A_{n-1,n}]$ is a bijection.

######  Proposition 6.4.

Let $\mathcal{T}$ be a triangulation of $P_{n}$. If each triangle of
$\mathcal{T}$ contains the vertex $0$, then the $2$-Segal map
$f_{\mathcal{T}}$ of $\text{iso}(s_{\bullet}\mathcal{C})$ associated to
$\mathcal{T}$ is a bijection.

###### Proof.

Our proof is an adaptation of Dyckerhoff and Kapranov’s proof of a similar
result [3, 2.4.8]. By Lemma 6.1, it suffices to show that the $2$-Segal map
$\text{iso}(s_{n}\mathcal{C})\rightarrow\text{iso}(s_{\\{0,1,\ldots,j\\}}\mathcal{C})\times_{\text{iso}(s_{\\{0,j\\}}\mathcal{C})}\text{iso}(s_{\\{0,j,j+1,\ldots,n\\}}\mathcal{C})$
is a bijection for any $j$. This $2$-Segal map fits into the commutative
diagram

${{\text{iso}(s_{n}\mathcal{C})}}$${{\text{iso}(s_{\\{0,1,\ldots,j\\}}\mathcal{C})\times_{\text{iso}(s_{\\{0,j\\}}\mathcal{C})}\text{iso}(s_{\\{0,j,j+1,\ldots,n\\}}\mathcal{C})}}$${{M_{n}}}$${{M_{j}\times_{\text{iso}(s_{\\{0,j\\}}\mathcal{C})}M_{n-j+1}}}$$\scriptstyle{\mu_{j}\times\mu_{n-j+1}}$$\scriptstyle{\mu_{n}}$$\scriptstyle{\phi_{j}}$

where $\phi_{j}$ is a bijection given by $[A_{0,1}\rightarrowtail
A_{0,2}\rightarrowtail\cdots\rightarrowtail
A_{0,n}]\mapsto([A_{0,1}\rightarrowtail
A_{0,2}\rightarrowtail\cdots\rightarrowtail A_{0,j}],[A_{0,j}\rightarrowtail
A_{0,j+2}\rightarrowtail\cdots\rightarrowtail A_{0,n}])$ so the result follows
by Lemma 6.3. ∎

In essence, the above argument worked because having all polygons in the
subdivision containing the vertex 0 means that objects in the right hand side
of the $2$-Segal map have complete top rows. A complete top row is enough to
specify an object in $S_{\bullet}\mathcal{C}$ up to isomorphism because all
the lower parts of an object of $S_{n}\mathcal{C}$ are pushouts of the
cofibrations in the top row, and taking pushouts of cofibrations is exactly
what we can do in a category with cofibrations.

Call the $2$-Segal maps corresponding to triangulations where each triangle
has the vertex $0$ the left $2$-Segal maps. Call the $2$-Segal maps
corresponding to triangulations where each triangle has the vertex $n$ the
right $2$-Segal maps. We prove that the left $2$-Segal maps are the _only_
$2$-Segal maps corresponding to triangulations for
$\text{iso}(s_{\bullet}\mathcal{C})$ that are always bijective. Specifically
we prove the following proposition.

######  Proposition 6.5.

For $\text{iso}(s_{\bullet}\mathcal{C})$ the $2$-Segal maps $f_{\mathcal{T}}$
which are not left $2$-Segal are not necessarily surjective.

First we provide a category with cofibrations such that some of its $2$-Segal
maps of the type in Proposition 6.5 are not surjective. We then use an
inductive argument relying on the structure of triangulations of polygons to
show that for this category with cofibrations none of the $2$-Segal maps of
Proposition 6.5 are surjective.

We want a category with cofibrations with as few cofibrations as possible. We
are thus led to a notion of generated categories with cofibrations.

######  Definition 6.6.

Let $\mathcal{D}$ be a subcategory of $\mathcal{C}$ that contains a $0$ object
of $\mathcal{C}$ and all morphisms between its objects and $0$ that are in
$\mathcal{C}$. Then we denote the intersection of all subcategories with
cofibrations of $\mathcal{C}$ containing $\mathcal{D}$ by
$\langle\mathcal{D}\rangle$. We call $\langle\mathcal{D}\rangle$ the _category
with cofibrations generated by $\mathcal{D}$ in $\mathcal{C}$_.

Now we justify the terminology.

######  Lemma 6.7.

The intersection of a collection of subcategories with cofibrations of
$\mathcal{C}$ is a category with cofibrations.

###### Proof.

Let $\mathcal{D}_{a}$ be a subcategory with cofibrations of $\mathcal{C}$ for
all $a\in I$. Then $\cap_{a\in I}\mathcal{D}_{a}$ contains $0$ and it is a
zero object for this category. Suppose $f$ is a cofibration in $\mathcal{C}$
which is in $\cap_{a\in I}\mathcal{D}_{a}$ and $g$ is a morphism in
$\cap_{a\in I}\mathcal{D}_{a}$. Then the pushout of $f$ along $g$ is in all
$\mathcal{D}_{a}$. ∎

######  Remark 6.8.

We can also describe $\langle\mathcal{D}\rangle$ as a colimit. Let
$\mathcal{D}=\mathcal{D}_{0}$ be a subcategory of $\mathcal{C}$ containing a
zero object of $\mathcal{C}$ and all morphisms between its objects and $0$
that are in $\mathcal{C}$. For all $i\geq 0$, let $\mathcal{D}_{i+1}$ be the
smallest subcategory of $\mathcal{C}$ containing $\mathcal{D}_{i}$, all
pushouts of a cofibration in $\mathcal{D}_{i}$ along a morphism in
$\mathcal{D}_{i}$ and all morphisms between $0$ and objects of $\mathcal{D}$
and these pushouts. Then we claim that
$\text{colim}_{i}\mathcal{D}_{i}\cong\langle\mathcal{D}\rangle$. Each
$\mathcal{D}_{i}$ is necessarily in $\langle\mathcal{D}\rangle$ so we just
need to show that $\text{colim}_{i}\mathcal{D}_{i}$ is a subcategory with
cofibrations of $\mathcal{C}$. The zero object condition follows from each
$\mathcal{D}_{i}$ satisfying the same condition. Suppose $f$ is a morphism of
$\mathcal{D}_{i}$ and $g$ is a cofibration of $\mathcal{D}_{j}$. Then $f$ and
$g$ are morphisms in $\mathcal{D}_{\max(i,j)}$ so the pushouts of $f$ along
$g$ and $g$ along $f$ are in $\mathcal{D}_{\max(i,j)+1}$.

The category $\mathcal{D}$ appearing in the following proposition is used to
prove Proposition 6.5.

######  Proposition 6.9.

There is a category with cofibrations $\mathcal{D}$ such that the $2$-Segal
maps
$\text{iso}(s_{n}\mathcal{D})\to\text{iso}(s_{\\{0,1,n\\}}\mathcal{D})\times_{\text{iso}(s_{\\{1,n\\}}\mathcal{D})}\text{iso}(s_{\\{1,2,\ldots,n\\}}\mathcal{D})$
are not surjective.

###### Proof.

We first consider the $n=3$ case. Showing that

$\text{iso}(s_{3}\mathcal{D})\to\text{iso}(s_{\\{0,1,3\\}}\mathcal{D})\times_{\text{iso}(s_{\\{1,3\\}}\mathcal{D})}\text{iso}(s_{\\{1,2,3\\}}\mathcal{D})$

is not surjective amounts to showing that there is a diagram

${{A_{0,1}}}$${\bullet}$${{A_{0,3}}}$${{A_{1,2}}}$${{A_{1,3}}}$${{A_{2,3}}}$

that cannot be completed to an object of $s_{3}\mathcal{D}$.

Consider the subcategory of $\mathcal{R}_{f}$ generated by the diagram

${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${{T^{2}}}$${{S^{1}}}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${{T^{2}/S^{1}}}$${{\hat{\Delta}}}$${{S^{2}}}$${\shortmid}$${\shortmid}$$\scriptstyle{||}$${\shortmid}$$\scriptstyle{||}$$\scriptstyle{||}$$\scriptstyle{||}$$\scriptstyle{||}$

where the same marking on different edges indicates that those edges should be
identified with each other or is used to indicate the image of an object. The
notation $\hat{\Delta}$ is used because the corresponding CW complex looks
like a triangle turned into a hat by pinching two corners together. Let
$\mathcal{D}_{0}$ be the smallest category containing the preceding diagram
and all morphisms between its objects and $0$. To complete the above diagram
to an object of $s_{3}\langle\mathcal{D}_{0}\rangle$, we must insert an
subcomplex of $T^{2}$, $X$, with $1$ $0$-cell, $3$ $1$-cells, and $1$ $2$-cell
in the first row in order for $\hat{\Delta}$ to be the cokernel object of the
inclusion of $S^{1}$ into $X$. Up to homeomorphism, the only subcomplex of
$T^{2}$ with this number of cells in each dimension is $P$, a triangle viewed
as a CW complex in the natural way with all of its vertices identified. Recall
how $\mathcal{D}_{i+1}$ (see Remark 6.8) is constructed from $\mathcal{D}_{i}$
and that $\langle\mathcal{D}_{0}\rangle=\text{colim}_{i}\mathcal{D}_{i}$. We
claim that $P$ is not in $\langle\mathcal{D}_{0}\rangle$. First, note that
every object in $\langle\mathcal{D}_{0}\rangle$ has one $0$-cell. This is
because all the objects in $\mathcal{D}_{0}$ have one $0$-cell as their
basepoint and all the morphisms in $\mathcal{D}_{0}$ are basepoint preserving.
Also, note that for all $j$, the objects that are in $\mathcal{D}_{j+1}$ but
not $\mathcal{D}_{j}$ are pushout objects of pushout diagrams like

${X}$${Y}$${Z}$${{X\sqcup Y/(x\sim f(x))}}$$\scriptstyle{f}$

whose left vertical morphisms and top horizontal morphisms are in
$\mathcal{D}_{j}$. This implies that if a $2$-cell in $Y$ is attached along
$k$ $1$-cells then the image of that $2$-cell in $X\sqcup Y/(x\sim f(x))$ is
attached along at most $k$ $1$-cells. Since the bottom map is a cellular
inclusion, a $2$-cell in the pushout that is also in $Z$ is attached along the
same number of $1$-cells that it is in $Z$. Since the most $1$-cells that a
$2$-cell is attached along among the objects of $\mathcal{D}_{0}$ is $3$, it
follows that all $2$-cells in any object of $\langle\mathcal{D}_{0}\rangle$
are attached along at most $3$ $1$-cells.

Now suppose to reach a contradiction that $j$ is the smallest number for which
an object of the form $W\vee P$ is in $\mathcal{D}_{j}$ (where $W$ might be a
point). Then there is a pushout diagram of the form

${X}$${Y}$${Z}$${{W\vee P}}$$\scriptstyle{f}$

where the left vertical morphism and top horizontal morphism are in
$\mathcal{D}_{j-1}$. The $2$-cell $C$ in $P$ in $W\vee P$ is attached along
$3$ $1$-cells. There is a $2$-cell in either $Z$ or $Y$ that gets mapped via
the bottom horizontal or right vertical map onto $C$. Suppose a $2$-cell in
$Z$ gets mapped onto $C$. Then since the bottom horizontal map is a cellular
inclusion, $P$ is a wedge summand of $Z$ which is a contradiction. Suppose a
$2$-cell $C^{\prime}$ in $Y$ gets mapped onto $C$ by the right vertical map.
Then $C^{\prime}$ must be attached in $Y$ along $3$ $1$-cells by earlier
remarks. If there is another $2$-cell $C^{\prime\prime}$ in $Y$ that is
attached along one of the $1$-cells that $C^{\prime}$ is attached along, then
that common $1$-cell gets contracted by the right vertical map since there is
not a $2$-cell attached to $C$ along a common $1$-cell in $W\vee P$. But then
there would be at most $2$ $1$-cells that $P$ is attached along in $W\vee P$.
This is a contradiction so we conclude that no $2$-cell shares a common
bounding $1$-cell with $C^{\prime}$. Thus $P$ is a wedge summand of $Y$, a
contradiction. We conclude that $P$ is not in $\langle\mathcal{D}_{0}\rangle$
so we have the $n=3$ case of the proposition.

Now in any subcategory with cofibrations of $\mathcal{R}_{f}$, the $n=3$ case
implies the other cases because then the diagram

${{A_{0,1}}}$${\bullet}$${\bullet}$${\cdots}$${\bullet}$${\bullet}$${{A_{0,3}}}$${0}$${0}$${\cdots}$${0}$${{A_{1,2}}}$${{A_{1,3}}}$${0}$${\cdots}$${0}$${{A_{1,2}}}$${{A_{1,3}}}$${\ddots}$${\vdots}$${\vdots}$${\vdots}$${0}$${{A_{1,2}}}$${{A_{1,3}}}$${{A_{1,2}}}$${{A_{1,3}}}$${{A_{2,3}}}$

cannot be completed to an object of $s_{n}\mathcal{D}$.

∎

######  Remark 6.10.

In a conversation, Maxine Calle pointed out that for any finite CW complex $X$
we can do the above argument in an analogous subcategory of
$\mathcal{R}_{f}(X)$ since the $X$ gets carried along in taking pushouts.

We now take a detour into the structure of triangulations of polygons needed
to set up an inductive argument for the proof of Proposition 6.5.

######  Lemma 6.11.

Every triangulation of a $P_{n}$ has a triangle whose vertices are labeled by
$j-1,j,j+1$ for some $j$.

###### Proof.

The vertices of $P_{2}$ are labeled by $0,1,2$ so we can take $j=1$. There are
two triangulations of $P_{3}$. The triangulation with a diagonal from $0$ to
$2$ contains the triangle labeled by $0$,$1$,$2$. The triangulation with a
diagonal from $1$ to $3$ contains the triangle labeled by $1$,$2$,$3$. We
proceed by induction. Suppose the result is true for $n=k$. Consider a
triangulation of the regular $(k+1)$-gon. Let $i<j$ be the labels for the
vertices of a diagonal of the triangulation. If $j=i+2$ then we are done. If
$j\neq i+2$ then consider the induced triangulation on the polygon with
vertices $i,i+1,\ldots j$. By inductive assumption, this triangulation
contains a triangle of the desired form which is then in the larger
triangulation. The result follows by induction. ∎

Note in the previous lemma that $j-1,j,j+1$ are strictly consecutive, not just
consecutive modulo $n$. So what is the difference in significance between
triangles with consecutive vertices and triangles with vertices consecutive
modulo $n$? Consider the following example. The triangulation

01234

can be built up as

012012301234

Notice that at each step the last triangle added had its vertices consecutive
modulo $n$ and that the last triangle added always has a vertex not connected
to any diagonals.

######  Lemma 6.12.

Viewing the edges and vertices of a triangulation of a $P_{n}$ as a graph,
every vertex has valency at least $2$. The vertices of valency $2$ are exactly
those $j$ for which $j-1,j,j+1$ mod $n$ is a triangle in the triangulation.

###### Proof.

Every triangulation can be constructed by attaching triangles to each other
along edges like the example above. Starting with a triangle with vertices
cyclically labeled, we add a triangle glued along one of the edges of the
original triangle. Then we relabel the vertices keeping $0$ at the same place.
Now we add another triangle and relabeling while keeping 0 at the same place
and repeat. The construction of the triangulation of the pentagon above gives
an example. At the end of each relabeling the last triangle added has become
one with vertices labeled $j-1,j,j+1$ mod $k$ for some $j$ and $k$. The
process of attaching a triangle as above increases the degree of two of the
vertices by $1$ and adds a vertex of degree $2$; namely, the middle vertex in
a triangle with consecutive vertices modulo $n$. ∎

######  Lemma 6.13.

When $\mathcal{T}$ is a triangulation containing the triangle $T_{0}$ which we
identify with the label set of its vertices $\\{j-1,j,j+1\\}$, the natural
morphisms of the form
$\lim_{\mathcal{T}}\text{iso}(s_{\bullet}\mathcal{C})\to\lim_{\mathcal{T},T_{0}}\text{iso}(s_{\bullet}\mathcal{C})$
are surjective. Here, $\lim_{\mathcal{T},T_{0}}\text{iso}$ is abbreviated
notation for $\lim_{\mathcal{T}-\\{T_{0},\\{j-1,j+1\\}\\}}$.

###### Proof.

Suppose the triangles of $\mathcal{T}$ are $T_{1},T_{2},\ldots,T_{n-2},$ and
$T_{0}:=\\{j-1,j,j+1\\}$ where we identify each triangle with the set of
lables on its vertices. Then
$\lim_{\mathcal{T}}\text{iso}(s_{\bullet}\mathcal{C})$ may be represented by
the set whose elements are tuples of isomorphism classes
$([A_{1}],[A_{2}],\ldots,[A_{n-2}],[A_{0}])$ where $A_{i}\in
s_{T_{i}}\mathcal{C}$ for all $i$, and if $T_{i}\cap T_{k}=\\{\ell,m\\}$, then
the images of $[A_{i}]$ and $[A_{k}]$ in
$\text{iso}(s_{\\{\ell,m\\}}\mathcal{C})$ are the same. Similarly,
$\lim_{\mathcal{T},T_{0}}\text{iso}(s_{\bullet}\mathcal{C})$ may be
represented by the set of tuples of isomorphism classes
$([A_{1}],[A_{2}],\ldots,[A_{n-2}])$ where $A_{i}\in s_{T_{i}}\mathcal{C}$ for
all $i$, and if $T_{i}\cap T_{k}=\\{\ell,m\\}$, then the images of $[A_{i}]$
and $[A_{k}]$ in $\text{iso}(s_{\\{\ell,m\\}})$ are the same. Consider a
generic element $([A_{1}],[A_{2}],\ldots,[A_{n-2}])$ of
$\lim_{\mathcal{T},T_{0}}\text{iso}(s_{\bullet}\mathcal{C})$. There is a
unique $k$ and $\alpha$ such that $\\{k,j-1,j+1\\}=T_{\alpha}$ is the only
other triangle of $\mathcal{T}$ that shares an edge with $\\{j-1,j,j+1\\}$.
Assume $k<j-1$. The other case ($k>j+1$) is similar. A preimage of this
generic element is given by
$([A_{1}],[A_{2}],\ldots,[A_{n-2}],[0\rightarrowtail(A_{\alpha})_{j-1,j+1}\twoheadrightarrow(A_{\alpha})_{j-1,j+1}])$.
∎

###### Proof of Proposition 6.5.

We proceed by induction on the number of sides of the polygon being
subdivided. For $n=3$ the result follows from Proposition 6.9. Let
$\mathcal{C}$ be the subcategory of $\mathcal{R}_{f}$ from the proof of
Proposition 6.9. Suppose the result is true for $n=k$. Let $\mathcal{T}$ be a
triangulation of the regular $(k+1)$-gon with cyclically labeled vertices
where some triangle in the triangulation does not contain the $0$ vertex. We
view the vertices and edges of the triangulation as a graph. There are two
cases: either the degree of the vertex $0$ is $2$ or it is greater than $2$.

Suppose the degree of the vertex $0$ is $2$. Then by Lemma 6.12, the triangle
given by $0,1,k+1$ is in $\mathcal{T}$. The induced triangulation on the
polygon with vertices labeled from $1$ to $k+1$ contains a triangle with
vertices of the form $j-1,j,j+1$ by Lemma 6.11. We have a commutative diagram

${{\text{iso}(s_{k+1}\mathcal{C})}}$${{\text{lim}_{\mathcal{T}}\text{iso}(s_{\bullet}\mathcal{C})}}$${{\text{iso}(s_{\\{0,1,\ldots,j-1,j+1,\ldots
k+1\\}}\mathcal{C})}}$${{\text{lim}_{\mathcal{T}-\\{j,j-1,j+1\\}-\\{j,j+1\\}}\text{iso}(s_{\bullet}\mathcal{C})}}$

where by $\\{j,j-1,j+1\\}$ we mean the triangle with those vertices. The top
and bottom maps are $2$-Segal maps, the left vertical map is the appropriate
face map of $\text{iso}(s_{\bullet}\mathcal{C})$ which is surjective, and the
right vertical map is projection. The right vertical map is surjective by
Lemma 6.13. The inductive hypothesis gives that the bottom horizontal map is
not surjective so the top horizontal map cannot be surjective.

Now suppose the degree of the vertex labeled $0$ is greater than $2$. Then
there is a $j\neq 1,k+1$ such that the edge from $0$ to $j$ is a diagonal of
the triangulation. This diagonal splits $P_{n}$ into two cyclically labeled
triangulated polygons. One of these two triangulated polygons must now contain
a triangle that does not contain the vertex labeled $0$. Proceeding like above
we can reduce our attention to the $2$-Segal map corresponding to the
triangulated sub-polygon with a triangle not containing $0$. Look at the
degree of the vertex $0$; if it is not $2$ repeat this procedure until the
degree of the vertex $0$ is $2$. Then proceed as in the first case. ∎

We have shown that for all categories with cofibrations $\mathcal{C}$,
$\text{iso}(s_{\bullet}\mathcal{C})$ is left $2$-Segal but need not be fully
$2$-Segal.

##  7\. 2-Segality of topological versions of the $S_{\bullet}$-construction

Let $\mathcal{C}$ be a category with cofibrations. The goals of this chapter
are threefold. First, we show that the left $2$-Segal maps for
$|iS_{\bullet}\mathcal{C}|$ are weak equivalences. Second, we show these are
the only $2$-Segal maps, corresponding to triangulations, which are
necessarily weak equivalences. The proofs given are more categorical versions
of the proofs given for $\text{iso}(s_{\bullet}\mathcal{C})$. Finally, we give
a partial result for $|wS_{\bullet}\mathcal{C}|$.

######  Proposition 7.1.

Let $\mathcal{C}$ be a category with cofibrations and $\mathcal{T}$ be the
triangulation of $P_{n}$ where each triangle of $\mathcal{T}$ contains the
vertex $0$. Then $f_{\mathcal{T}}$ is a weak equivalence.

######  Lemma 7.2.

Let $M_{n}\mathcal{C}$ be the groupoid whose objects are chains of $(n-1)$
cofibrations in $\mathcal{C}$ and whose morphisms are isomorphisms of such
chains. The functor $\mu_{n}\colon iS_{n}\mathcal{C}\rightarrow
M_{n}\mathcal{C}$ which sends $A$ to $A_{0,1}\rightarrowtail
A_{0,2}\rightarrowtail\cdots\rightarrowtail A_{0,n}$ is an equivalence.

The proof is analogous to that of Lemma 6.3.

###### Proof of Proposition 7.1.

By Proposition 5.6 it suffices to show that the functor $\Phi_{j}\colon
iS_{n}\mathcal{C}\rightarrow
iS_{\\{0,1,\ldots,j\\}}\mathcal{C}\times^{(2)}_{S_{\\{0,j\\}}\mathcal{C}}iS_{\\{j,j+1,\ldots,n\\}}\mathcal{C}$
is an equivalence. There is a commutative diagram

${{iS_{n}\mathcal{C}}}$${{iS_{\\{0,1,\ldots,j\\}}\mathcal{C}\times^{(2)}_{iS_{\\{0,j\\}}\mathcal{C}}iS_{\\{j,j+1,\ldots,n\\}}\mathcal{C}}}$${M_{n}}$${M_{j}\times^{(2)}_{iS_{\\{0,j\\}}\mathcal{C}}M_{n-j+1}}$$\scriptstyle{\mu_{n}}$$\scriptstyle{\Phi_{j}}$$\scriptstyle{\mu_{j}\times\mu_{n-j+1}}$$\scriptstyle{\phi_{j}}$

where $\phi_{j}$ is an equivalence so $\Phi_{j}$ is an equivalence. ∎

Now we turn to the "only" part of the result.

######  Lemma 7.3.

A functor between groupoids that is not essentially surjective cannot induce a
weak homotopy equivalence between realizations.

###### Proof.

Suppose $F\colon\mathcal{C}\rightarrow\mathcal{D}$ is a functor between
groupoids that induces a weak homotopy equivalence
$|\mathcal{C}|\rightarrow|\mathcal{D}|$. Then $|F|$ gives a surjection (in
fact a bijection) between path components of the $1$-skeleta. Since all edges
in the $1$-skeleta come from isomorphisms this cannot be the case unless $F$
is essentially surjective. ∎

Because of Lemma 7.3, it suffices to show that the maps from
$iS_{n}\mathcal{C}$ to $2$-fiber products of categories corresponding to
$2$-Segal maps are not necessarily essentially surjective.

######  Lemma 7.4.

When $\mathcal{T}$ is a triangulation containing the triangle
$\\{j-1,j,j+1\\}$, the natural morphisms of the form
$2\lim_{\mathcal{T}}(iS_{\bullet}\mathcal{C})\to
2\lim_{\mathcal{T}-\\{j-1,j,j+1\\}-\\{j-1,j+1\\}}(iS_{\bullet}\mathcal{C})$
are surjective on objects.

The proof is the same as for Lemma 6.13 with "bijection" replaced by
"essentially surjective" except that one must keep track of the isomorphisms
that are part of the data of an object in a $2$ limit. We use the fact that a
composite of essentially surjective functors is essentially surjective and if
$G$ and $F$ are functors and $GF$ is essentially surjective, then so is $G$.

######  Proposition 7.5.

There is a category with cofibrations $\mathcal{D}$ such that the natural map
$iS_{n}\mathcal{D}\rightarrow
iS_{\\{0,1,n\\}}\mathcal{D}\times^{(2)}_{iS_{\\{1,n\\}}\mathcal{D}}iS_{\\{1,2,\ldots,n\\}}\mathcal{D}$
is not essentially surjective.

The proof is essentially the same as the proof of Proposition 6.9.

######  Proposition 7.6.

The $2$-Segal maps of $|iS_{\bullet}\mathcal{C}|$ which are not left $2$-Segal
are not necessarily weak equivalences.

###### Proof.

By Lemma 7.3 and Proposition 5.6, it suffices to show that the corresponding
maps into the projective $2$-limits of categories are not necessarily
essentially surjective. The proof is the same as the proof of Proposition 6.5,
but with limits replaced by projective $2$-limits.

∎

Now having the weak equivalences all be isomorphisms is an unusually special
situation. Another unusually special situation is having all morphisms be weak
equivalences. We have seen that $|S_{\bullet}\mathcal{C}|$ is $2$-Segal and
$|iS_{\bullet}\mathcal{C}|$ is left $2$-Segal and may be $2$-Segal. What can
we say in the middle cases of arbitrary Waldhausen categories? So far we have
not been able to say anything, but if $\mathcal{C}$ is a nice Waldhausen
category we can say the following.

######  Proposition 7.7.

If $\mathcal{C}$ is a saturated Waldhausen category satisfying the mapping
cylinder axiom, then a $2$-Segal map of $|wS_{\bullet}\mathcal{C}|$ is a weak
equivalence if and only if the corresponding $2$-Segal map for
$|wcofM_{\bullet}\mathcal{C}|$ is a weak equivalence.

###### Proof.

By Proposition 3.15, the maps $wcofS_{n}\mathcal{C}\to S_{n}\mathcal{C}$ are
homotopy equivalences. Analogously to Lemma 7.2, the maps
$wcofM_{n}\mathcal{C}\to wcofS_{n}\mathcal{C}$ are homotopy equivalences. All
of these maps commute with face and degeneracy maps. Since in $\mathcal{T}op$
all spaces are fibrant, the result follows. ∎

##  8\. 2-Segality of categorical versions of the $S_{\bullet}$-construction

We have seen that $|S_{\bullet}\mathcal{C}|$ is always $2$-Segal (Proposition
4.16). Here we show that $S_{\bullet}\mathcal{C}$ is always left $2$-Segal for
$\mathcal{C}$ a category with cofibrations but is not always $2$-Segal. In
particular, a simplicial object in $\mathcal{C}at$ can be left but not fully
$2$-Segal while its geometric realization is fully $2$-Segal. We will also see
that $wS_{\bullet}\mathcal{C}$ behaves similarly to $S_{\bullet}\mathcal{C}$.
Some of our arguments are parallel to the ones given in the previous two
sections.

######  Lemma 8.1.

Let $M_{n}$ be the category whose objects are chains of $(n-1)$ cofibrations
in $\mathcal{C}$ and whose morphisms are morphisms of such chains. The functor
$\mu_{n}\colon S_{n}\mathcal{C}\to M_{n}$ which sends $A$ to
$A_{0,1}\rightarrowtail A_{0,2}\rightarrowtail\cdots\rightarrowtail A_{0,n}$
is an equivalence.

######  Lemma 8.2.

If $i\colon\\{0,j\\}\to\alpha=\\{\alpha_{1},\ldots,\alpha_{l}\\}$ is injective
then $S_{i}\mathcal{C}\colon S_{\alpha}\mathcal{C}\to
S_{\\{0,j\\}}\mathcal{C}$ is an isofibration.

###### Proof.

Let $A$ be an object of $S_{\alpha}\mathcal{C}$ and
$\tilde{f}\colon(S_{i}\mathcal{C})(A)=A_{0,j}\to B_{0,j}$ be an isomorphism in
$S_{\\{0,j\\}}\mathcal{C}$. We want to produce an object $B$ of
$S_{\alpha}\mathcal{C}$ and a morphism $f\colon A\to B$ such that
$(S_{i}\mathcal{C})(f)=\tilde{f}$. We gain both $B$ and $f$ by extending the
diagram

${{A_{0,\alpha_{1}}}}$${{A_{0,\alpha_{2}}}}$${\cdots}$${{A_{0,\alpha_{k}}}}$${{B_{0,j}}}$${{A_{0,\alpha_{k+1}}}}$${\cdots}$${{A_{0,\alpha_{l}}}}$${{A_{0,\alpha_{1}}}}$${{A_{0,\alpha_{2}}}}$${\cdots}$${{A_{0,\alpha_{k}}}}$${{A_{0,j}}}$${{A_{0,\alpha_{k+1}}}}$${\cdots}$${{A_{0,\alpha_{l}}}}$$\scriptstyle{a_{2}}$$\scriptstyle{a_{3}}$$\scriptstyle{a_{k}}$$\scriptstyle{a}$$\scriptstyle{a_{k+1}}$$\scriptstyle{a_{k+2}}$$\scriptstyle{a_{l}}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{a_{2}}$$\scriptstyle{a_{3}}$$\scriptstyle{a_{k}}$$\scriptstyle{\tilde{f}}$$\scriptstyle{\tilde{f}a}$$\scriptstyle{a_{k+1}\tilde{f}^{-1}}$$\scriptstyle{=}$$\scriptstyle{a_{k+2}}$$\scriptstyle{a_{l}}$$\scriptstyle{=}$

using the equivalence of the previous lemma.

∎

We conclude that for any $j$,
$S_{\\{0,1,\ldots,j\\}}\mathcal{C}\times_{S_{\\{0,j\\}}\mathcal{C}}^{h}S_{\\{0,j,j+1,\ldots,n\\}}\mathcal{C}$
is equivalent to the ordinary pullback. Now in the same way as Lemma 6.3 and
Proposition 7.2, there is a commutative diagram

${{S_{n}\mathcal{C}}}$${{S_{\\{0,1,\ldots,j\\}}\mathcal{C}\times_{S_{\\{0,j\\}}\mathcal{C}}S_{\\{0,j,j+1,\ldots,n\\}}\mathcal{C}}}$${{M_{n}}}$${{M_{j}\times_{S_{\\{0,j\\}}\mathcal{C}}M_{n-j+1}.}}$

The bottom and vertical maps are equivalences, so the top map is an
equivalence.

Now suppose $\mathcal{C}$ is a Waldhausen category. Consider the map
$wS_{n}\mathcal{C}\to wS_{\\{0,1,\ldots,j\\}}\mathcal{C}$
$\times_{wS_{\\{0,j\\}}\mathcal{C}}^{h}wS_{\\{0,j,j+1,\ldots,n\\}}\mathcal{C}$.
The isomorphisms in $wS_{m}\mathcal{C}$ are the same as the isomorphisms in
$S_{m}\mathcal{C}$. Thus $wS_{\\{0,1,\ldots,j\\}}\mathcal{C}\to
wS_{\\{0,j\\}}\mathcal{C}$ is an isofibration and the homotopy pullback can be
replaced by an ordinary pullback. We can for each $n$, consider the category
$wM_{n}$ whose objects are chains of $n-1$ cofibrations in $\mathcal{C}$ and
whose morphisms are componentwise weak equivalences. We get, like above for
$S_{\bullet}\mathcal{C}$, a commutative diagram

${{wS_{n}\mathcal{C}}}$${{wS_{\\{0,1,\ldots,j\\}}\mathcal{C}\times_{wS_{\\{0,j\\}}\mathcal{C}}wS_{\\{0,j,j+1,\ldots,n\\}}\mathcal{C}}}$${{wM_{n}}}$${{wM_{j}\times_{wS_{\\{0,j\\}}\mathcal{C}}wM_{n-j+1}}}$

where the bottom and vertical maps are equivalences..

From the above considerations in $S_{\bullet}\mathcal{C}$ and
$wS_{\bullet}\mathcal{C}$ we get the following proposition.

######  Proposition 8.3.

The left $2$-Segal maps of $S_{\bullet}\mathcal{C}$ and
$wS_{\bullet}\mathcal{C}$, are equivalences of categories.

######  Remark 8.4.

The nerve functor applied to a morphism which is the image of an injection
under $S_{\bullet}\mathcal{C}$ is not a fibration of simplicial sets since
this nerve is not cofibered over groupoids [7, 2.1.1.2]. Since
$N(S_{i}\mathcal{C})$ is not a fibration we do not have a reason to be able to
replace a homotopy pullback with a pullback when studying the $2$-Segal maps
of $|S_{\bullet}\mathcal{C}|$. Just like the $S_{\bullet}\mathcal{C}$ case,
$wS_{\\{0,1,\ldots,j\\}}\mathcal{C}\to wS_{\\{0,j\\}}\mathcal{C}$ is never
fibered in groupoids.

######  Proposition 8.5.

There is a category with cofibrations $\mathcal{D}$ such that the natural map
$S_{n}\mathcal{D}\to
S_{\\{0,1,n\\}}\mathcal{D}\times_{S_{\\{1,n\\}}\mathcal{D}}S_{\\{1,2,\ldots,n\\}}\mathcal{D}$
is not essentially surjective.

The same category $\mathcal{D}$ used for Proposition 6.9 can be used here, and
the proof is the same as the one there. Since equivalences of categories must
be essentially surjective, we get that the $2$-Segal map $S_{n}\mathcal{D}\to
S_{\\{0,1,n\\}}\mathcal{D}\times_{S_{\\{1,n\\}}\mathcal{D}}^{h}S_{\\{1,2,\ldots,n\\}}\mathcal{D}$
is not an equivalence. Proposition 6.5 has an analogue with the same proof
here as well.

######  Proposition 8.6.

The $2$-Segal maps $f_{\mathcal{T}}$ corresponding to triangulations where
some triangle does not contain 0 are not necessarily weak equivalences.

Because $iS_{\bullet}\mathcal{C}$ is a special case of
$wS_{\bullet}\mathcal{C}$, and we have replaced the homotopy pullback with an
ordinary pullback, we have a $wS_{\bullet}\mathcal{C}$ version of Proposition
8.6.

######  Proposition 8.7.

The $2$-Segal maps $f_{\mathcal{T}}$ with source $wS_{\bullet}\mathcal{C}$
corresponding to triangulations where some triangle does not contain $0$ are
not necessarily weak equivalences.

##  9\. A sufficient condition for $S_{\bullet}\mathcal{C}$ to be 2-Segal

We now have some examples of possible strict left $2$-Segal spaces :
$\text{iso}(s_{\bullet}\mathcal{C})$, $S_{\bullet}\mathcal{C}$,
$wS_{\bullet}\mathcal{C}$, and $|iS_{\bullet}\mathcal{C}|$. Therefore, we now
look for sufficient conditions under which some of these simplicial spaces are
necessarily $2$-Segal. Analyzing our previous work we find the following
criterion.

######  Proposition 9.1.

Let $\mathcal{C}$ be a category with cofibrations. Then
$S_{\bullet}\mathcal{C}$ is $2$-Segal if any diagram in $\mathcal{C}$ of the
form

${A}$${\bullet}$${X}$${0}$${B}$${Y}$

can be extended to

${A}$${C}$${X}$${0}$${B}$${Y}$

where the squares are pushouts and the right square is a pullback.

###### Proof.

By [3, 2.3.2(4)] and since we argued in Section 8 that we can replace homotopy
pullbacks in the $2$-Segal maps of $S_{\bullet}\mathcal{C}$ with ordinary
pullbacks, it suffices to show that the $2$-Segal maps of the forms
$S_{n}\mathcal{C}\to
S_{\\{0,j,j+1,\ldots,n\\}}\mathcal{C}\times_{S_{\\{0,j\\}}\mathcal{C}}S_{\\{0,1,\ldots
j\\}}\mathcal{C}$ and $S_{n}\mathcal{C}\to S_{\\{0,1,\ldots
j,n\\}}\mathcal{C}\times_{S_{\\{j,n\\}}\mathcal{C}}S_{\\{j,j+1,\ldots,n\\}}\mathcal{C}$
are equivalences. The $2$-Segal maps of the first of these two forms are
equivalences by the work in Section 8. We focus on $2$-Segal maps of the
second form. To show essential surjectivity of these it suffices to show that
the diagram

${{A_{0,1}}}$${{A_{0,2}}}$${\cdots}$${{A_{0,j}}}$${\bullet}$${\bullet}$${\cdots}$${\bullet}$${{A_{0,n}}}$${{A_{1,2}}}$${\cdots}$${{A_{1,j}}}$${\bullet}$${\bullet}$${\cdots}$${\bullet}$${{A_{1,n}}}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$${{A_{j-1,j}}}$${\bullet}$${\bullet}$${\cdots}$${\bullet}$${{A_{j-1,n}}}$${{A_{j,j+1}}}$${{A_{j,j+2}}}$${\cdots}$${{A_{j,n-1}}}$${{A_{j,n}}}$${{A_{j+1,j+2}}}$${\cdots}$${{A_{j+1,n-1}}}$${{A_{j+1,n}}}$${\vdots}$${\vdots}$${{A_{n-2,n-1}}}$${{A_{n-2,n}}}$${{A_{n-1,n}}}$

can be extended to an object of $S_{n}\mathcal{C}$ compatible with the given
morphisms. We make this extension one object at a time. First, we use our
hypothesis to extend

${{A_{0,j}}}$${\bullet}$${{A_{0,n}}}$${0}$${{A_{j,j+1}}}$${{A_{j,n}.}}$

The newly added object we denote by $A_{0,j+1}$. Next, we use our hypothesis
to extend

${{A_{0,j+1}}}$${\bullet}$${{A_{0,n}}}$${0}$${{A_{j+1,j+2}}}$${{A_{j+1,n}.}}$

Proceeding in this way fill out the first row. Now we add in the rest of the
required objects by taking cokernels of composites of morphisms in the first
row. Compatibility with preexisting morphisms in the first $j$ rows of the
diagram comes from the uniqueness part of the universal property for pushouts.
That the last $n-j$ rows may come from choices of cokernels in the first row
comes from the fact that if $A\rightarrowtail B\twoheadrightarrow D$,
$(A\rightarrowtail B\rightarrowtail C)\twoheadrightarrow E$, and
$D\rightarrowtail E\twoheadrightarrow F$ are cofibration sequences, then so is
$B\rightarrowtail C\twoheadrightarrow F$. To show fullness, it suffices to
show that

${{B_{0,j}}}$${{B_{0,n}}}$${{A_{0j}}}$${{A_{0n}}}$${0}$${{B_{j,j+1}}}$${{B_{j,n}}}$${0}$${{A_{j,j+1}}}$${{A_{j,n}}}$

can be extended to

${{B_{0,j}}}$${{B_{0,j+1}}}$${{B_{0,n}}}$${{A_{0j}}}$${{A_{0,j+1}}}$${{A_{0n}}}$${0}$${{B_{j,j+1}}}$${{B_{j,n}}}$${0}$${{A_{j,j+1}}}$${{A_{j,n}}}$

where the extension on the front and back faces comes from applying the
hypothesis. This extension is obtained by applying the hypothesis to the front
and back faces and then using that the back right square is a pullback square.
We must check commutativity of the three squares containing the morphism
$A_{0,j+1}\to B_{0,j+1}$. The top right square and vertical middle square
commute because the morphism $A_{0,j+1}\to B_{0,j+1}$ was induced by the
commutative square

$\leavevmode\hbox to102.59pt{\vbox
to69.36pt{\pgfpicture\makeatletter\hbox{\hskip
51.29697pt\lower-34.73221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-51.29697pt}{-34.63237pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
15.12444pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.8189pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{A_{0,j+1}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
15.12444pt\hfil&\hfil\hskip 23.91397pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.60844pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{A_{0,n}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
11.91399pt\hfil&\hfil\hskip 24.20737pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.90184pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{B_{0,n}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
12.20738pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
14.88222pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.57668pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{A_{j,j+1}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
14.88222pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
15.17561pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.87007pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{B_{j,j+1}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
15.17561pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 23.96515pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.65962pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{B_{j,n}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
11.96516pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-18.71704pt}{20.04408pt}\pgfsys@lineto{-9.54572pt}{20.04408pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-18.51706pt}{20.04408pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.34573pt}{20.04408pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{15.08221pt}{20.04408pt}\pgfsys@lineto{26.28224pt}{20.04408pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.48222pt}{20.04408pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-18.66586pt}{-32.13237pt}\pgfsys@lineto{26.52446pt}{-32.13237pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{-18.46588pt}{-32.13237pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.72444pt}{-32.13237pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{39.08958pt}{10.79106pt}\pgfsys@lineto{39.08958pt}{-23.53938pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{39.08958pt}{-22.29945pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{39.08958pt}{-23.73936pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-36.12135pt}{10.74883pt}\pgfsys@lineto{-36.12135pt}{2.54884pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-36.12135pt}{3.78877pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-36.12135pt}{2.34886pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-36.12135pt}{-15.3394pt}\pgfsys@lineto{-36.12135pt}{-23.53938pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-36.12135pt}{-23.73936pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
Commutativity of the top left square comes from the uniqueness part of the
universal property of the pullback: $A_{0,j}\to B_{0,j}\rightarrowtail
B_{0,j+1}$ and $A_{0,j}\rightarrowtail A_{0,j+1}\to B_{0,j+1}$ both can both
be added to give a commutative diagram

$\leavevmode\hbox to138.34pt{\vbox
to96.36pt{\pgfpicture\makeatletter\hbox{\hskip
69.17183pt\lower-48.23212pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-69.17183pt}{-48.13228pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
11.54666pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.24112pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{A_{0,j}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
11.54666pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 23.84004pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.53452pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{B_{0,j}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
11.84006pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
6.80554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${0}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 27.41782pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.11229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{B_{0,j+1}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
15.41783pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 24.20737pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.90184pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{B_{0,n}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
12.20738pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil\cr\vskip 8.99994pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 27.1756pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.87007pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{B_{j,j+1}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
15.17561pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 11.99998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 23.96515pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.65962pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{B_{j,n}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
11.96516pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}{{}}{{}}{{}}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{10.79478pt}{-45.63228pt}\pgfsys@lineto{44.39934pt}{-45.63228pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{10.99477pt}{-45.63228pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{44.59932pt}{-45.63228pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{56.96446pt}{-10.7972pt}\pgfsys@lineto{56.96446pt}{-37.03929pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{56.96446pt}{-35.79936pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{56.96446pt}{-37.23927pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{11.037pt}{-1.54417pt}\pgfsys@lineto{44.15712pt}{-1.54417pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{11.23698pt}{-1.54417pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{44.3571pt}{-1.54417pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-6.6607pt}{-10.83943pt}\pgfsys@lineto{-6.6607pt}{-37.03929pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-6.6607pt}{-35.79936pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-6.6607pt}{-37.23927pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}\pgfsys@setdash{2.79985pt,1.59991pt}{0.0pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-45.87851pt}{25.45975pt}\pgfsys@lineto{-18.88538pt}{6.87564pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.82367}{-0.56706}{0.56706}{0.82367}{-18.72066pt}{6.76225pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-45.87851pt}{33.54399pt}\pgfsys@lineto{-19.10072pt}{33.54399pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.90074pt}{33.54399pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{7.20056pt}{25.90253pt}\pgfsys@lineto{44.20686pt}{5.49106pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.87564}{0.48297}{-0.48297}{-0.87564}{7.37569pt}{25.80594pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.87564}{-0.48297}{0.48297}{0.87564}{44.38197pt}{5.39447pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-57.62517pt}{24.24873pt}\pgfsys@lineto{-57.62517pt}{6.65993pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-57.62517pt}{7.89986pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-57.62517pt}{6.45995pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{-49.04665pt}{-8.96353pt}\pgfsys@lineto{-16.43098pt}{-37.1776pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.75629}{0.65424}{-0.65424}{-0.75629}{-48.8954pt}{-9.09438pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.75629}{-0.65424}{0.65424}{0.75629}{-16.27974pt}{-37.30843pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
The uniqueness part of the universal property of pullbacks and pushouts gives
faithfulness. ∎

######  Example 9.2.

The category $\mathcal{R}_{f}$ satisfies the hypothesis of Proposition 9.1.
The missing object in

${A}$${\bullet}$${X}$${0}$${B}$${X/A}$$\scriptstyle{g}$$\scriptstyle{q}$$\scriptstyle{f}$

can be filled in with $q^{-1}(g(B))$ which inherits its CW structure from $X$.
The object $A$ is a subcomplex of $q^{-1}(g(B))$ since it is a subcomplex of
$X$ that $q$ maps to the basepoint of $X/A$ which is contained in $g(B)$. The
needed map from $q^{-1}(g(B))$ to $B$ is $g^{-1}q$ where $g^{-1}$ is defined
on $g(B)$ and $q$ is restricted to $q^{-1}(g(B))$. We have a commutative
square of the right shape and it remains to check that the left square is a
pushout and the right square is biCartesian. The middle vertical map is the
restriction of the map which quotients by $A$ so the left square is a pushout.
A model for the pushout of $B\leftarrow q^{-1}(g(B))\rightarrowtail X$ is
$B\sqcup_{x\sim g^{-1}q(x)}X$. The process of quotienting by this relation can
be described in two steps. First $A\subset X$ gets identified with the point
$A\in X/A$. Next $q^{-1}(g(B))$ is bijectively identified with $B-\\{A\\}$.
The resulting space is $X/A$. The right square is pullback since given an
extension problem of the form

${Z}$${{q^{-1}(g(B))}}$${X}$${B}$${{X/A}}$$\scriptstyle{h}$$\scriptstyle{k}$$\scriptstyle{q}$$\scriptstyle{g}$

the unique solution is given by $k$ with restricted codomain. The morphism $k$
has image in $q^{-1}(g(B))$ since $q(k(Z))\subset g(B)$ by commutativity.

As a special case, we get that the category of based finite sets with
injective maps as cofibrations satisfies the hypothesis of Proposition 9.1. In
fact, the arguments of this example go through for the category of based sets
with cofibrations the injective maps in this category.

######  Remark 9.3.

Proto-exact categories with admissible monomorphisms as cofibrations as
defined in [3, 2.4.2] are close to satisfying the hypothesis of Proposition
9.1. The part about the left square being a pushout is where this hypothesis
conceivably might fail for a proto-exact category. Conversely, a category with
cofibrations need not be exact if we say the admissible monomorphisms are
cofibrations and their cokernels are admissible epimorphisms. This is because
the cokernels of cofibrations do not need to be closed under composition.

##  Appendix A Upper and lower 2-Segal

Here we prove that what we have been calling left and right $2$-Segal are the
same as lower and upper $2$-Segal respectively.

######  Definition A.1.

Let $\mathcal{D}$ be a model category, $X\colon\Delta^{op}\to\mathcal{D}$ a
simplicial object, and $0<i<n$. Consider the squares

${{X_{n+1}}}$${{X_{n}}}$${{X_{n+1}}}$${{X_{n}}}$and${{X_{n}}}$${{X_{n-1}}}$${{X_{n}}}$${{X_{n-1}}}$$\scriptstyle{d_{n+1}}$$\scriptstyle{d_{i}}$$\scriptstyle{d_{n}}$$\scriptstyle{d_{i}.}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{i}}$$\scriptstyle{d_{i+1}}$

We say $X$ is _upper_ $2$-Segal if for any choices of $i$ and $n$, the left
square is homotopy cartesian. We say $X$ is _lower_ $2$-Segal if for any
choices of $i$ and $n$, the right square is homotopy cartesian.

######  Remark A.2.

We use the formulation that appears in [4, 1] but the first place the
definition appears in the literature is [8, 2.2].

######  Proposition A.3.

A simplicial object in a model category is upper $2$-Segal if and only if it
is right $2$-Segal, and it is lower $2$-Segal if and only if it is left
$2$-Segal.

###### Proof.

We prove the first statement; the second statement is dual. It is known that a
simplicial object $Y$ is lower $2$-Segal if and only if for each $n$ the map
$Y_{n}\to Y_{\\{0,1,n\\}}\times^{h}_{Y_{\\{1,n\\}}}Y_{\\{1,2,\ldots,n\\}}$ is
a weak equivalence. So if $X$ is right $2$-Segal then by duality and
definition we can conclude that $X$ is upper $2$-Segal. Now we turn to the
converse.

We can restate our definition of right $2$-Segal in terms of homotopy pullback
squares as follows. A simplicial object $X$ in a model category is right
$2$-Segal exactly when the diagram

${{X_{k}}}$${{X_{j+1}}}$${{X_{k-j}}}$${{X_{1}}}$$\scriptstyle{(d_{j+1})^{k-j-1}}$$\scriptstyle{(d_{0})^{j}}$$\scriptstyle{(d_{1})^{k-j-1}}$$\scriptstyle{(d_{0})^{j}}$

is homotopy cartesian for all $0<j<k$.

Assume that $X$ is upper $2$-Segal. Then

${{X_{3}}}$${{X_{2}}}$${{X_{2}}}$${{X_{1}}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$

is homotopy cartesian by assumption. In other words $X_{3}\simeq
X_{\\{0,1,3\\}}\times^{h}_{X_{\\{1,3\\}}}X_{\\{1,2,3\\}}$. By assumption, we
have homotopy cartesian squares

${{X_{4}}}$${{X_{3}}}$${{X_{3}}}$${{X_{2}}}$and${{X_{3}}}$${{X_{2}}}$${{X_{2}}}$${{X_{1}}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$

so that

${{X_{4}}}$${{X_{2}}}$${{X_{3}}}$${{X_{1}}}$$\scriptstyle{d_{2}d_{2}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}d_{1}}$

is homotopy cartesian. Similarly, we have the homotopy cartesian square

${{X_{4}}}$${{X_{2}}}$${{X_{3}}}$${{X_{1}.}}$$\scriptstyle{d_{0}d_{0}}$$\scriptstyle{d_{0}d_{0}}$$\scriptstyle{d_{3}}$$\scriptstyle{d_{1}}$

For $n\leq 5$ we have verified that for $0<j<k<n$, the square

${{X_{k}}}$${{X_{j+1}}}$${{X_{k-j}}}$${{X_{1}}}$$\scriptstyle{(d_{j+1})^{k-j-1}}$$\scriptstyle{(d_{0})^{j}}$$\scriptstyle{(d_{1})^{k-j-1}}$$\scriptstyle{(d_{0})^{j}}$

is homotopy cartesian. To use induction, assume that for some $n\geq 5$ the
above square is homotopy cartesian for all $0<j<k<n$. Because $X$ is upper
$2$-Segal,

${{X_{n}}}$${{X_{n-1}}}$${{X_{n-1}}}$${{X_{n-2}}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{3}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{2}}$

is homotopy cartesian. By inductive assumption, the square

${{X_{n-1}}}$${{X_{2}}}$${{X_{n-2}}}$${{X_{1}}}$$\scriptstyle{(d_{2})^{n-3}}$$\scriptstyle{d_{0}}$$\scriptstyle{(d_{1})^{n-3}}$$\scriptstyle{d_{0}}$

is homotopy cartesian so we get the homotopy cartesian square

${{X_{n}}}$${{X_{n-1}}}$${{X_{2}}}$${{X_{1}.}}$$\scriptstyle{d_{0}}$$\scriptstyle{(d_{2})^{n-2}}$$\scriptstyle{(d_{1})^{n-2}}$$\scriptstyle{d_{0}}$

If $j\neq 1$, then by inductive assumption the square

${{X_{n-1}}}$${{X_{j}}}$${{X_{n-j}}}$${{X_{1}}}$$\scriptstyle{(d_{j})^{n-j-1}}$$\scriptstyle{(d_{0})^{j-1}}$$\scriptstyle{(d_{1})^{n-j-1}}$$\scriptstyle{(d_{0})^{j-1}}$

is homotopy cartesian. By inductive assumption the square

${{X_{j+1}}}$${{X_{2}}}$${{X_{j}}}$${{X_{1}}}$$\scriptstyle{d_{0}}$$\scriptstyle{(d_{2})^{j-1}}$$\scriptstyle{d_{0}}$$\scriptstyle{(d_{1})^{j-1}}$

is homotopy cartesian. Putting the previous three squares together we have the
commutative diagram

${{X_{n}}}$${{X_{j+1}}}$${{X_{n-1}}}$${{X_{2}}}$${{X_{j}}}$${{X_{n-j}.}}$${{X_{1}}}$${{X_{1}}}$$\scriptstyle{(d_{j+1})^{n-j-1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{0}}$$\scriptstyle{(d_{j})^{n-j-1}}$$\scriptstyle{(d_{2})^{j-1}}$$\scriptstyle{(d_{1})^{j-1}}$$\scriptstyle{d_{0}}$$\scriptstyle{(d_{0})^{j-1}}$$\scriptstyle{(d_{0})^{j-1}}$$\scriptstyle{(d_{1})^{n-j-1}}$

The left and center squares together form a homotopy cartesian rectangle, and
the left and right squares are homotopy cartesian, so the rectangle formed by
the right and center squares is homotopy cartesian. By induction we conclude
that $X$ is right $2$-Segal.

∎

## References

  * [1] J. Bergner, A. Osorno, V. Ozornova, M. Rovelli, C. Scheimbauer, 2-Segal objects and the Waldlhausen construction, Algebraic and Geometric Topology, Volume 21, 2021.
  * [2] J. Bergner, A. Osorno, V. Ozornova, M. Rovelli, C.Scheimbauer, 2-Segal sets and the Waldhausen construction, Topology and its Applications, Volume 235, 2018, Pages 445-484.
  * [3] T. Dyckerhoff and M. Kapranov, Higher Segal Spaces, Lecture Notes in Mathematics. 2244, Springer, 2012.
  * [4] M. Feller, R. Garner, J. Kock, M. Underhill-Proulx, and M. Weber. Every 2-Segal space is unital. Commun. Contemp. Math., Volume 23, 2021.
  * [5] Imma Gálvez-Carrillo, Joachim Kock, Andrew Tonks, Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory, Advances in Mathematics, Volume 331, 2018, Pages 952-1015,
  * [6] P. Hirschhorn, Model Categories and Their Localization, Mathematical Surveys and Monographs, Volume 99, 2003.
  * [7] J. Lurie, Higher Topos Theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009
  * [8] T. Poguntke, Higher Segal structures in algebraic K-theory, arXiv, 2017
  * [9] D. Quillen, Higher algebraic K-theory: I. In: Bass, H. (eds) Higher K-Theories, Lecture Notes in Mathematics, vol 341. Springer, Berlin, Heidelberg, 1973.
  * [10] D. Quillen, Higher K-theory for categories with exact sequences, New Developments in Topology, London Math. Soc. Lecture Notes no. 11, Cambridge Univ. Press, 1974, pp. 95-110
  * [11] C. Rezk, A model category for categories (2000), Unpublished notes, https://faculty.math.illinois.edu/~rezk/papers.html
  * [12] F. Waldhausen, Algebraic K-theory of spaces, Lecture Notes in Math. 1126, Springer-Verlag, 1985.
  * [13] G. Warner, Topics in Topology and Homotopy Theory, arXiv, 2022
  * [14] C. Weibel, The K-Book: An Introduction to Algebraic K-Theory, Graduate Studies in Math, Volume 145, 2013.