Claude Code transcript - page 1/1

1	This folder has several files:
2	
3	decomp.tex / decomp.bib (don't need to read bib probably) : an overview paper I wrote about decomposition spaces / 2-Segal spaces. Includes décalage / edgewise subdivision criteria for 2-Segality, which is what you should be interested in.
4	
5	from_hsd.tex : about d-Segal spaces, from a paper on higher Segal spaces and partial groups
6	
7	from_coskel.tex / higher.bib : excerpt from a recent preprint I wrote, with slightly different conventions to the above
8	
9	higher_segal_spheres.tex / spheres.tex : a calculation of a couple examples, written around the same time as the coskeletality paper
10	
11	Look over these and familiarize yourself with them. Feel free to ask me any questions.
12	
13	The thing I'd like you to investigate: 
14	Is there a version of the edgewise subdivision criterion that is true for d-Segal spaces for d > 2? 
15	
16	My instinct is that if anything like this were to be true it would be for 2k-Segal spaces, since edgewise subdivision contains in a sense both upper and lower décalage, and the path space criterion already relates upper or lower 2k-Segality to lower (2k-1)-Segality of these décalages.
17	
18	(But it also may just be false for d > 2, who knows!)

total 352
drwxr-xr-x  11 user  staff    352 Jul 31 16:07 .
drwx------@ 95 user  staff   3040 Jul 31 16:07 ..
drwxr-xr-x@ 14 user  staff    448 Jul 31 16:07 .git
-rw-r--r--@  1 user  staff  27810 Mar 29  2025 decomp.bib
-rw-r--r--@  1 user  staff  67208 Jun  6  2025 decomp.tex
-rw-r--r--   1 user  staff  21351 Jul 31 16:00 from_coskel.tex
-rw-r--r--   1 user  staff  20106 Jul 31 15:54 from_hsd.tex
-rw-r--r--@  1 user  staff   2760 Jul 27 05:10 higher_segal_spheres.tex
-rw-r--r--@  1 user  staff   9019 Jul 27 09:35 higher.bib
-rw-r--r--   1 user  staff   1194 Jul 31 16:06 instr.md
-rw-r--r--   1 user  staff  13031 Jul 27 05:16 spheres.tex
    1126 decomp.tex
     337 from_coskel.tex
     278 from_hsd.tex
      67 higher_segal_spheres.tex
     183 spheres.tex
     834 decomp.bib
     268 higher.bib
    3093 total

1	\documentclass{amsart}
2	\usepackage{amssymb}
3	\usepackage{mathtools}
4	\usepackage[only,llbracket,rrbracket]{stmaryrd}
5	\usepackage[svgnames]{xcolor}
6	\usepackage[unicode,
7	  colorlinks=true,
8	  linktocpage=true,
9	  citecolor=OliveDrab,
10	  linkcolor=DarkMagenta,
11	  urlcolor=DarkMagenta,
12	  pdftitle={Coskeletality and the higher Segal conditions},
13	  pdfauthor={Philip Hackney}
14	  ]{hyperref}
15	\usepackage[capitalise,noabbrev]{cleveref}
16	\usepackage{tikz-cd}
17	\usepackage{microtype}
18	\frenchspacing
19	\usepackage[T1]{fontenc}
20	\usepackage{enumitem}
21	
22	\setlist[enumerate,1]{left=0pt, label={\arabic*.}, ref=\arabic*}
23	
24	\usepackage{comment}
25	
26	\newcommand{\cube}[1]{\llbracket #1 \rrbracket}
27	
28	\newcommand{\lowev}{\mathsf{le}}
29	\newcommand{\lowod}{\mathsf{lo}}
30	\newcommand{\uppev}{\mathsf{ue}}
31	\newcommand{\uppod}{\mathsf{uo}}
32	
33	\newcommand{\ps}{\mathcal{P}}
34	\newcommand{\op}{\textup{op}}
35	
36	\newcommand{\sset}{\mathsf{sSet}}
37	
38	\newcommand{\bim}{{\bar \imath}}
39	\newcommand{\bjm}{{\bar \jmath}}
40	
41	\newcommand{\decbot}{\operatorname{dec}_\bot}
42	\newcommand{\dectop}{\operatorname{dec}_\top}
43	
44	% Only for a small example
45	\DeclareMathOperator{\cosk}{cosk}
46	\DeclareMathOperator{\sk}{sk}
47	\DeclareMathOperator{\map}{map}
48	
49	\newtheorem{theorem}{Theorem}
50	\newtheorem{proposition}[theorem]{Proposition}
51	\newtheorem{corollary}[theorem]{Corollary}
52	\newtheorem{lemma}[theorem]{Lemma}
53	
54	\theoremstyle{definition}
55	\newtheorem{definition}[theorem]{Definition}
56	
57	\theoremstyle{remark}
58	\newtheorem{remark}[theorem]{Remark}
59	
60	\begin{document}
61	
62	\title{Coskeletality and the higher Segal conditions}
63	
64	\author{Philip Hackney}
65	\address{Department of Mathematics, University of Louisiana at Lafayette}
66	\email{philip@phck.net} 
67	\urladdr{http://phck.net}
68	
69	\thanks{
70	This work was supported by a grant from the Simons Foundation (\#850849, PH). 
71	The author was partially supported by the Louisiana Board of Regents through the Board of Regents Support fund LEQSF(2024-27)-RD-A-31.
72	}
73	
74	\date{\today}
75	
76	\begin{abstract}
77	A simplicial set is $d$-Segal if and only if it is $(d{+}1)$-coskeletal and satisfies the $d$-Segal condition in the two lowest relevant simplicial dimensions.
78	\end{abstract}
79	
80	\maketitle
81	
82	A simplicial set $X$ is $n$-coskeletal if it is determined by its simplices of dimension at most $n$, together with unique fillers for higher-dimensional simplex boundaries.\footnote{Precisely, $X$ is right Kan extended along $\Delta_{\leq n} \hookrightarrow \Delta$ from its restriction to $\Delta_{\leq n}$.} 
83	It is well-known that a simplicial set is the nerve of a category if and only if it is 2-coskeletal\footnote{In the reverse direction, one may actually assume that the simplicial set is 3-coskeletal rather than 2-coskeletal.} and has unique fillers for inner horns in simplicial dimension 2 and 3.\footnote{\cite{nlab:simplicial_skeleton}, \cite[Lemma 5.2]{Riehl:SSCAQC}}
84	There is a similar result for 2-Segal sets \cite{DyckerhoffKapranov:HSS} (also known as discrete decomposition spaces \cite{GKT1}):
85	a simplicial set is 2-Segal if and only if it is 3-coskeletal and satisfies the 2-Segal condition for the square and the pentagon (that is, it satisfies the 2-Segal conditions in simplicial dimensions 3 and 4).\footnote{\cite[Corollary 1.7]{BOORS:2SSetsWC}, \cite[\S4.1]{Stern:P2SC} -- the latter source observes that one may replace 3-coskeletal by 4-coskeletal in the reverse direction.}
86	In this paper, we prove an analogous result for $d$-Segal sets, resolving a question of Walker Stern: a simplicial set is upper (resp.\ lower) $d$-Segal if and only if it is $(d{+}1)$-coskeletal and satisfies the upper (resp.\ lower) $d$-Segal conditions in simplicial dimensions $d+1$ and $d+2$ (the lowest two non-vacuous dimensions).
87	See \cref{thm main theorem} for a slightly stronger statement.
88	The corresponding statement for simplicial spaces is false (\cref{rmk simplicial anima}).
89	
90	The higher Segal conditions were introduced by Dyckerhoff and Kapranov \cite{Dyckerhoff:CPOC,DyckerhoffKapranov:HSS} as exactness conditions arising from triangulations of cyclic polytopes, generalizing the classical Segal conditions underlying models for $(\infty,1)$-categories.
91	The case $d=2$ has proven especially fruitful, with applications to Hall algebras, incidence coalgebras, and other constructions in representation theory, algebraic combinatorics, and algebraic K-theory \cite{Dyckerhoff:HCAHA,GKT:DSC}. 
92	The situation for general $d$ is considerably less developed: besides \cite{Dyckerhoff:CPOC,Poguntke:HSSAKT}, see \cite{DyckerhoffJassoWalde:SSHART,HackneyLynd:HSSPG,Walde:HSSHE} for some of the first steps in this direction.
93	
94	Our criterion allows for an effective, finite check of $d$-Segality for simplicial sets with finitely many nondegenerate simplices (for $d$ suitably large).
95	This rests on \cite[Theorem 3.19]{KRRZ:LTSS}, which states that $n$-skeletal simplicial sets (with $n > 1$) are automatically $(2n{-}1)$-coskeletal.
96	This was used to compute that $\Delta^n/\partial \Delta^n$ is $2n$-Segal but not lower $(2n{-}1)$-Segal for $2 \leq n \leq 7$; we proceeded to prove this for general $n \geq 1$ \cite{Hackney:SdSS}. 
97	Thus the strictness of the hierarchy of higher Segal conditions is visible in rather simple and familiar examples.
98	
99	\section{Simplicial conventions}
100	In this paper $\Delta$ will refer to the category whose objects are nonempty finite sets of integers and whose morphisms are order-preserving maps.
101	The usual simplicial indexing category is the skeletal subcategory of $\Delta$ whose objects are the intervals $[n] = [0,n] = \{0, 1, \dots, n\}$ for $n\geq 0$.
102	We'll write $X(S)$ for $X$ evaluated at $S\in \Delta$; we occasionally abbreviate $X([n])$ to $X_n$.
103	
104	The opposite $X^\op$ of a simplicial object $X$ is given by $X^\op(S) = X(-S)$, and sends $f\colon S \to T$ to
105	\[
106		X(f^-) \colon X(-T) \to X(-S),
107	\]
108	where $f^- \colon (-S) \to (-T)$ is given by $x\mapsto -f(-x)$ for $x\in -S$.
109	
110	If $S\in \Delta$ is a finite ordered set with more than one element and $i\in S$, we'll write $S_i = S\setminus \{i\}$. 
111	If $X$ is a simplicial set, we'll write $e_i \colon X(S) \to X(S_i)$ for the restriction.
112	This is essentially the face map $d_i \colon X_n \to X_{n-1}$ (where $n+1 = |S|$), but has the more convenient interchange property $e_i e_j = e_j e_i$ for $i\neq j$ in $S$.
113	
114	\subsection{D\'ecalage} 
115	The \emph{upper d\'ecalage} of $X$, denoted $\dectop X$, is obtained by shifting simplicial dimension by one (i.e.\ $(\dectop X)_n = X_{n+1}$ for $n\geq 0$) and forgetting about the top face and degeneracy maps \cite[VI.1]{Illusie:CCD2}.
116	More concretely in our setting: $(\dectop X)(S)$ is defined to be $X(S^\triangleright)$ where $S^\triangleright = S \cup \{ \max(S) + 1 \}$, and, if $f\colon S \to T$ is in $\Delta$, then $(\dectop X) (f) \colon (\dectop X)(T) \to (\dectop X)(S)$ is defined to be $X(f^\triangleright)$ where $f^\triangleright \colon S^\triangleright \to T^\triangleright$ is the extension of $f$ which preserves maximal elements.
117	Similarly the \emph{lower d\'ecalage} $\decbot X$ is given by $(\decbot X)(S) = X(S^\triangleleft)$ and $(\decbot X)(f) = X(f^\triangleleft)$, where $S^\triangleleft = \{\min(S) - 1\} \cup S$, and $f^\triangleleft$ is the extension which  preserves minimal elements.
118	Since $(f^\triangleright)^- = (f^-)^\triangleleft$, 
119	the simplicial objects $\dectop (X^\op)$ and $(\decbot X)^\op$ are equal. 
120	
121	\subsection{Coskeletality}
122	
123	If $S \in \Delta$ and a simplicial set $X$ are fixed in a discussion, a symbol like $f_i$ will generally be used for an element of $X(S_i)$.
124	
125	\begin{definition}
126	Let $X$ be a simplicial set, $U \subset S$, and $f_i \in X(S_i)$ a collection of elements (for $i\in U$).
127	\begin{itemize}
128	\item If $e_i f_j = e_j f_i \in X(S_{i,j})$ for all $i\neq j \in U$, we say $\{f_i\}_{i\in U}$ is \emph{compatible}.
129	\item Suppose additionally that $t \in S \setminus U$ and $f_t \in X(S_t)$. If $e_i f_t = e_t f_i$ for all $i\in U$, then we say $f_t$ is \emph{compatible with} the collection $\{f_i\}_{i\in U}$.
130	\end{itemize}
131	\end{definition}
132	
133	We'll use the following simple compatibility lemma repeatedly.
134	
135	\begin{lemma}\label{triv lemma}
136	Let $f_i \in X(S_i)$ for each $i$ in a three element subset $\{p,q,t\} \subset S$. 
137	If $e_qf_t = e_tf_q$ and $e_qf_p = e_p f_q$, then $e_q e_p f_t = e_q e_t f_p$.
138	\end{lemma}
139	\begin{proof} $e_q e_p f_t = e_p e_q f_t = e_p e_t f_q = e_t e_p f_q = e_t e_q f_p = e_q e_t f_p$.
140	\end{proof}
141	
142	A compatible collection where $U=S=[m]$ is called an \emph{$m$-sphere in $X$} in \cite[Definition 3.2]{KRRZ:LTSS}.
143	Coskeletality is about unique filling of $m$-spheres: 
144	
145	\begin{definition}\label{def n-coskel}
146	A simplicial set $X\in \sset$ is \emph{$n$-coskeletal} if, whenever $S = [m]$ for $m > n$ and $\{ f_i \}_{i\in S}$ is a compatible collection, there exists a unique $F \in X_m$ such that $e_i F = f_i$ for all $0\leq i \leq m$.
147	\end{definition}
148	
149	There are multiple equivalent formulations of $n$-coskeletality. 
150	For instance we could ask that $X$ is orthogonal to the boundary inclusions $\partial \Delta^m \to \Delta^m$ for all $m > n$, or we could ask that it is in the image of the right Kan extension from $n$-truncated simplicial sets (presheaves on $\Delta_{\leq n}$).
151	See also \cite[\href{https://kerodon.net/tag/051Z}{Tag 051Z}]{kerodon}.
152	
153	
154	\section{Background: Higher Segal Conditions}
155	
156	We closely follow the presentation of the higher Segal conditions from \cite{HackneyLynd:HSSPG}, which is based on Walde's theorem \cite{Walde:HSSHE}.
157	See \cite{Dyckerhoff:CPOC} for a recent survey on higher Segal spaces.
158	
159	A \emph{gapped subset} $I$ of a finite totally ordered set $S$ is a proper subset which does not contain a pair of adjacent elements of $S$.
160	We will often abbreviate `gapped set of cardinality $k+1$' by `gapped set' when the number $k$ is fixed in the discussion.
161	
162	Given a gapped set $I\subset S$ of cardinality $k+1$, there is a cube
163	\[
164		\cube{I} = \cube{I\subset S} \colon \ps(I) \to \Delta^\op
165	\]
166	sending $U \subset I$ to $S\setminus U$.
167	If $X \colon \Delta^\op \to \mathcal{C}$ is a simplicial object in a category or $\infty$-category $\mathcal{C}$, we write $X\cube{I} \colon \ps(I) \to \mathcal{C}$ for the corresponding composite cube.
168	
169	\begin{definition}[Higher Segal conditions]\label{def higher Segal}
170	Let $k$ be a nonnegative integer and $X$ a simplicial object in $\mathcal{C}$.
171	Consider the collection of gapped subsets $I \subset [n]$ of cardinality $k+1$ as $n$ varies and the associated collection of cubes $X\cube{I} \colon \ps(I) \to \mathcal{C}$.
172	We say that $X$ is
173	\begin{enumerate}
174	\item \emph{lower $(2k{-}1)$-Segal} if $X\cube{I}$ is cartesian for all such $I$,
175	\item \emph{lower $2k$-Segal} if $X\cube{I}$ is cartesian whenever $0 \notin I$,
176	\item \emph{upper $2k$-Segal} if $X\cube{I}$ is cartesian whenever $n \notin I$, and
177	\item \emph{upper $(2k{+}1)$-Segal} if $X\cube{I}$ is cartesian whenever $0 \notin I$ and $n \notin I$.
178	\end{enumerate}
179	\end{definition}
180	
181	We say $X$ is $d$-Segal if it is both upper and lower $d$-Segal.
182	The lower 1-Segal condition is equivalent to the usual Segal condition.
183	When $k=0$, all four cases coincide, and amount to $X$ being a constant simplicial object (\cite[Example~3.9]{Dyckerhoff:CPOC}, \cite[Remark~3.11]{HackneyLynd:HSSPG}).
184	Poguntke showed that if $X$ is upper or lower $d$-Segal, then it is also both upper and lower $(d{+}1)$-Segal (\cite[Theorem~5.1]{Dyckerhoff:CPOC}, \cite[Proposition~3.14]{HackneyLynd:HSSPG}, \cite[Proposition~2.10]{Poguntke:HSSAKT}).
185	
186	\begin{remark}
187	As mentioned in \cite[Remark 3.8]{HackneyLynd:HSSPG}, if $X$ is a simplicial set, then $X$ is lower $(2k{-}1)$-Segal if and only if for each gapped set $I\subset S$ of cardinality $k+1$ and each compatible collection $\{f_i\}_{i\in I}$, there exists a unique $F\in X(S)$ with $e_i F = f_i$ for all $i\in I$.
188	A similar statement holds for the other higher Segal conditions above, and for definitions \ref{def segal properties} and \ref{def d critical} below.
189	This is the form we most often use these properties later.
190	\end{remark}
191	
192	We separate the conditions from \cref{def higher Segal} by simplicial dimension.
193	
194	\begin{definition}\label{def segal properties}
195	Let $k,n$ be nonnegative integers and $X$ a simplicial object.
196	Consider the collection of gapped subsets $I \subset [n]$ of cardinality $k+1$ and the associated collection of cubes $X\cube{I} \colon \ps(I) \to \mathcal{C}$.
197	We say that $X$ has property
198	\begin{enumerate}
199	\item $\lowod^k_n$ if $X\cube{I}$ is cartesian for all $I \subset [n]$,
200	\item $\lowev^k_n$ if $X\cube{I}$ is cartesian whenever $0 \notin I \subset [n]$,
201	\item $\uppev^k_n$ if $X\cube{I}$ is cartesian whenever $n \notin I \subset [n]$, and
202	\item $\uppod^k_n$ if $X\cube{I}$ is cartesian whenever $0,n \notin I \subset [n]$.
203	\end{enumerate}
204	\end{definition}
205	
206	Notice that $X$ satisfies $\lowod^k_n$ or $\uppod^k_n$ if and only if $X^\op$ does so, and $X$ satisfies $\lowev^k_n$ if and only if $X^\op$ satisfies $\uppev^k_n$.
207	This allows us to infer some theorems by \emph{duality}.
208	If $k$ is positive,\footnote{Condition $\lowod^0_0$ is vacuous since gapped sets are proper, while $\lowev_1^0$, $\uppev_1^0$, and $\uppod_2^0$ are non-vacuous.} the first non-vacuous conditions are $\lowod^k_{2k}$, $\lowev^k_{2k+1}$, $\uppev^k_{2k+1}$, and $\uppod^k_{2k+2}$.
209	It will be important to isolate the first \emph{two} simplicial dimensions where there is something to check; see also \cref{rmk d crit}.
210	
211	\begin{definition}\label{def d critical}
212	We say that a simplicial set is \emph{upper (resp.\ lower) $d$-critical} if it satisfies the upper (resp.\ lower) $d$-Segal conditions in the lowest two non-vacuous simplicial dimensions ($d+1$ and $d+2$).
213	This means that our simplicial set is
214	\begin{enumerate}
215	\item lower $(2k{-}1)$-critical if it satisfies $\lowod^k_{2k}$ and $\lowod^k_{2k+1}$,
216	\item lower $2k$-critical if it satisfies $\lowev^k_{2k+1}$ and $\lowev^k_{2k+2}$,
217	\item upper $2k$-critical if it satisfies $\uppev^k_{2k+1}$ and $\uppev^k_{2k+2}$, and 
218	\item upper $(2k{+}1)$-critical if it satisfies $\uppod^k_{2k+2}$ and $\uppod^k_{2k+3}$.
219	\end{enumerate}
220	\end{definition}
221	
222	We next have Poguntke's \emph{path space criterion} \cite[Proposition 2.7]{Poguntke:HSSAKT}.\footnote{See also \cite[Theorem 6.3.2]{DyckerhoffKapranov:HSS} and \cite[Theorem 4.10]{GKT1}}
223	The similar statement replacing `$d$-Segal' with `$d$-critical' holds by \cref{easy psc} below.
224	
225	\begin{theorem}[Path Space Criterion]\label{path space criterion}
226	Let $X$ be a simplicial object and $k\geq 1$.
227	The simplicial object 
228	$X$ is lower (resp.\ upper) $2k$-Segal if and only if $\decbot X$ (resp.\ $\dectop X$) is lower $(2k{-}1)$-Segal.
229	The simplicial object $X$ is upper $(2k{+}1)$-Segal if and only if $\decbot X$ is upper $2k$-Segal if and only if $\dectop X$ is lower $2k$-Segal if and only if $\dectop \decbot X = \decbot \dectop X$ is lower $(2k{-}1)$-Segal. \qed
230	\end{theorem}
231	
232	\begin{proposition}\label{easy psc}
233	Let $X$ be a simplicial object and $k\geq 1$.
234	\begin{enumerate}
235			\item $X$ satisfies $\lowev^k_n$ if and only if $\decbot X$ satisfies $\lowod^k_{n-1}$.\label{PSC ldec}
236			\item $X$ satisfies $\uppev^k_n$ if and only if $\dectop X$ satisfies $\lowod^k_{n-1}$.\label{PSC udec}
237			\item The following are equivalent:
238			\begin{enumerate}[label={\alph*.}, ref=\alph*]
239			\item $X$ satisfies $\uppod^k_n$.
240			\item $\decbot X$ satisfies $\uppev^k_{n-1}$.
241			\item $\dectop X$ satisfies $\lowev^k_{n-1}$.
242			\item $\decbot \dectop X = \dectop \decbot X$ satisfies $\lowod^k_{n-2}$.
243			\end{enumerate}
244	\end{enumerate}
245	\end{proposition}
246	\begin{proof}
247	As in the proof of Proposition 3.13 of \cite{HackneyLynd:HSSPG}.
248	\end{proof}
249	
250	
251	
252	\begin{remark}\label{rmk d crit}
253	In formulating upper or lower $d$-critical in \cref{def d critical}, one could instead use (for simplicial dimension $n=d+1,d+2$) the geometric conditions in \cite[Definition 3.7]{Dyckerhoff:CPOC} (or \cite[Definition 2.2]{Poguntke:HSSAKT}) expressed in terms of triangulations of cyclic polytopes.
254	This yields an equivalent definition -- it's not so arduous to prove this directly, but one can also see it by a careful reading of Walde's proof of the equivalence of the $(2k{-}1)$-Segal conditions (especially the proof of Theorem 7.2.1 of \cite{Walde:HSSHE}; see \cite[Remark 3.6]{HackneyLynd:HSSPG}) and Poguntke's account of the path space criterion \cite[Proposition 2.7]{Poguntke:HSSAKT}.
255	Thus, for example, $2$-critical simplicial objects can be described as those local with respect to triangulations of squares and pentagons.
256	\end{remark}
257	
258	We end this section with the Wiggle Lemma, whose strategy of proof informs the argument in \cref{prop lower odd implies coskel}.
259	Immediate consequences of the Wiggle Lemma are \cite[Lemma 3.9]{HackneyLynd:HSSPG} (on which its proof is based) and \cite[Lemma 3.6]{GKT1}.
260	
261	\begin{lemma}[Wiggling]\label{wiggling}
262	If $X$ has property $\lowod^k_{n-1}$ and there is a gapped set $I \subset [n]$ of size $k+1$ such that $X\cube{I}$ is cartesian, then $X$ has property $\lowod^k_n$.
263	Similarly, if $X$ has property $\lowev_{n-1}^k$ (resp.\ $\uppev_{n-1}^k$, resp.\ $\uppod_{n-1}^k$) and there is a gapped set %(of size $k+1$) 
264	$I\subset [n]$ avoiding $0$ (resp.\ avoiding $n$, resp.\ avoiding $0$ and $n$) such that $X\cube{I}$ is cartesian, then $X$ has property $\lowev_n^k$ (resp.\ $\uppev_n^k$, resp.\ $\uppod_n^k$).
265	\end{lemma}
266	\begin{proof}
267	If $n < 2k$ then $\lowod^k_n$ is a vacuous condition, while if $n=2k$ then there is exactly one gapped set in $[n]$, so the condition is satisfied by hypothesis. Assume $n > 2k$.
268	
269	Suppose we have two gapped sets $I, J \subset [n] = S$ which differ in only one position.
270	In other words, $I\setminus J = \{ \bim \}$ and $J\setminus I = \{ \bjm \}$ such that the successor (resp.\ predecessor) of $\bim$ in $I \cup \{-1, n+1\}$ coincides with the successor (resp.\ predecessor) of $\bjm$ in $J \cup \{-1, n+1\}$.
271	Notice that $J\subset S_\bim$ and $I \subset S_\bjm$ are also gapped sets.
272	The following square of maps of cubes commutes.
273	\[ \begin{tikzcd}[row sep=0.25cm]
274	& 
275	X\cube{I_\bim \subset S_\bim} \rar[equals] &[-0.6cm] 
276	X\cube{J_\bjm \subset S_\bim}
277		\drar[bend left=15,"X\cube{J\subset S_\bim}"]
278	\\
279	X\cube{I_\bim \subset S} \dar[equals] 
280		\urar[bend left=15,"X\cube{I\subset S}"]
281	& 
282	&
283	&
284	X\cube{J_\bjm \subset S_{\bim,\bjm}} \dar[equals]
285	\\%[-0.4cm]
286	X\cube{J_\bjm \subset S}
287		\drar[bend right=15,"X\cube{J\subset S}"']
288	& 
289	&
290	&
291	X\cube{I_\bim \subset S_{\bim,\bjm}}
292	\\
293	& 
294	X\cube{J_\bjm \subset S_\bjm} \rar[equals] &[-0.6cm] 
295	X\cube{I_\bim \subset S_\bjm}
296		\urar[bend right=15,"X\cube{I\subset S_\bjm}"']
297	\end{tikzcd} \]
298	The cubes $X\cube{J\subset S_\bim}$ and $X\cube{I\subset S_\bjm}$ are cartesian by $\lowod^k_{n-1}$, so $X\cube{I\subset S}$ is cartesian if and only if $X\cube{J\subset S}$ is cartesian by \cite[Lemma 3.3]{HackneyLynd:HSSPG}.
299	Iterating this process, we see that $X$ is cartesian for a gapped set $I \subset [n]$ if and only if it is cartesian for $\{0 < 2 < \dots < 2k \}$, and the result follows.
300	The statements in the second sentence follow from the first via \cref{easy psc}.
301	\end{proof}
302	
303	
304	
305	==============
306	
307	
308	
309	
310	
311	\begin{theorem}\label{thm main theorem}
312	Let $X$ be a simplicial set and $d \geq 0$.
313	If $X$ is $(d{+}2)$-coskeletal and is upper (resp.\ lower) $d$-critical, then it is upper (resp.\ lower) $d$-Segal.
314	Conversely, if $X$ is upper or lower $d$-Segal, then it is $(d{+}1)$-coskeletal.
315	\end{theorem}
316	\begin{proof}
317	The second statement is \cref{hsc implies coskel}, while the first combines \cref{lem k=0 case}, \cref{thm lowod case coskel to segal}, and \cref{thm other cases coskel to segal}.
318	\end{proof}
319	
320	\begin{remark}\label{rmk simplicial anima}
321	The corresponding statement for simplicial spaces is false.
322	Let $M$ be a nonempty space, and $X$ the associated constant simplicial space, which is automatically $0$-Segal (hence $d$-Segal for all $d$).
323	The $p$-coskeleton of $X$ (see \cite[\href{https://kerodon.net/tag/05GU}{Tag 05GU}]{kerodon}) is given in dimension $n$ by 
324	\[
325		(\cosk_p X)_n \simeq \lim_{(\Delta_{/[n]} \times_\Delta \Delta_{\leq p})^\op} M \simeq \map(|\Delta_{/[n]} \times_\Delta \Delta_{\leq p}|, M),
326	\]
327	the second equivalence because $X$ is constant, 
328	where $\Delta_{\leq p} \subset \Delta$ is the full subcategory on $[q]$ for $q \leq p$.
329	Since $|\Delta_{/[n]} \times_\Delta \Delta_{\leq p}| \simeq |\sk_p \Delta^n|$, taking $n=p+1$ gives $|\partial \Delta^{p+1}| \simeq S^p$, and the unit map $X_{p+1} \to (\cosk_p X)_{p+1}$ is identified with restriction along $\sk_p \Delta^{p+1} \subset \Delta^{p+1}$, i.e.\ with the inclusion of constant maps $M \to M^{S^p}$.
330	This is a section of evaluation at a basepoint, whose fibers are $\Omega^p M$, hence is an equivalence if and only if $\Omega^p M$ is contractible.
331	
332	Now suppose $M$ has infinitely many nontrivial homotopy groups (e.g.\ $M=S^2$).
333	Then $X$ immediately gives a counterexample to the second statement, as $\Omega^p M$ is not contractible for any $p$, but $X$ is $d$-Segal for all $d\geq 0$.
334	
335	For the first statement, set $Y \coloneq \cosk_{d+2} X$, which is $(d{+}2)$-coskeletal. 
336	As the unit $X \to Y$ is an equivalence in simplicial dimensions up to $d+2$, we have that $Y$ is $d$-critical since $X$ is so.
337	But naturality implies that $Y$ cannot satisfy the $d$-Segal condition in simplicial dimension $d+3$, since $X$ is known to, and $M \simeq X_{d+3} \to Y_{d+3} \simeq M^{S^{d+2}}$ is not an equivalence.
338	\end{remark}

55:\DeclareMathOperator{\sd}{sd}
100:Key results include the path space criterion, which characterizes decomposition spaces in terms of their upper and lower décalages, and the edgewise subdivision criterion. 
126:The next three sections (\S\ref{sec segal spaces}, \S\ref{sec path space}, \S\ref{sec edgewise}) are all about relationships between decomposition spaces and Segal spaces.
130:Finally, in \cref{sec edgewise} we prove the closely related edgewise subdivision criterion, which says that $X$ is a decomposition space if and only if $\sd X$ is a Segal space.
783:\section{The edgewise subdivision criterion}\label{sec edgewise}
787:This proof relies on the path space criterion (\cref{cor path space}), which is appropriate as there is a close connection between the edgewise subdivision and the upper/lower d\'ecalage pair.\phnote{Additionally, this theorem can be interpreted in terms of the double-categorical perspective from \cite{BOORS:2SSWC}.}
788:\begin{theorem}\label{thm edgewise}
789:A simplicial space $X$ is a decomposition space if and only if its edgewise subdivision $\sd X$ is Segal.
792:As with the d\'ecalage functors, the edgewise subdivision is the result of precomposition with an endofunctor of $\Delta$, namely the one that sends $[n]$ to $[n]^\op \star [n] \cong [2n+1]$.
808:If $X$ is a simplicial space, then the simplicial space $Z = \sd X$ has $Z_n = X_{2n+1}$ and
813:One can check that $\sd X = \sd (X^\op)$. %May 30 note.
814:If $X$ is the nerve of a category $C$, then $\sd X$ is (isomorphic to) the nerve of the \emph{twisted arrow category} of $C$, whose objects are the morphisms of $C$, and where a morphism $f \to g$ is given by a factorization of $g$ as below:
822:One manifestation of the connection between d\'eclage and edgewise subdivision is that iterated bottom and top degeneracies yield simplicial maps
824:\ldec X \rar & \sd X & (\udec X)^\op. \lar
829:\begin{equation}\label{eq ldec to sd}
830:  \ldec X_n = X_{n+1} \xrightarrow{s_\bot^n} X_{2n+1} = \sd X_n.
834:  \sd X_n = X_{2n+1} \xleftarrow{s_\top^n} X_{n+1} = (\udec X)^\op_n,
838:\begin{lemma}\label{lem decomp implies sd Segal}
839:If $X$ is a decomposition space, then $\sd X$ is Segal.
842:Let $Z = \sd X$.
858:Let $X$ be a simplicial space and write $Y = \ldec X$, $Z = \sd X$. For each $n \geq 1$, the square below left is a retract of the square below right:
867:Consequently, if $\sd X$ is Segal, so is $\ldec X$.
871:The map from left to right is given by the simplicial map $Y \to Z$ from \eqref{eq ldec to sd}.
917:If $\sd X$ is Segal, so is the upper d\'ecalage $\udec X$.
920:We have $\sd X = \sd (X^\op)$ for any simplicial space $X$. 
924:\begin{proof}[Proof of \cref{thm edgewise}]
925:The forward direction is \cref{lem decomp implies sd Segal}.
926:If $\sd X$ is Segal, then the preceding two lemmas show that $\ldec X$ and $\udec X$ are Segal as well, so the path space criterion implies that $X$ is a decomposition space.
929:% We have already observed that the edgewise subdivision and d\'ecalage constructions are closely related, which is corroborated by the following, already prefigured in \cite{BOORS:2SSWC}.
931:\begin{remark}\label{rmk sd vs dec}
932:If $X$ is a discrete decomposition space, then the three simplicial sets $\sd X$, $\ldec X$, and $\udec X$ are (nerves of) categories.
933:We have inclusions of simplicial sets $\ldec X \hookrightarrow \sd X$ and $(\udec X)^\op \hookrightarrow \sd X$.
934:Identifying each of these subcategories with its image, the category $\sd X$ admits two (strict) factorization systems $(\ldec X, (\udec X)^\op)$ and $((\udec X)^\op, \ldec X)$; see \cite{Hackney:OC2SS} for one explanation.% (including for simplicial spaces).
937:There is also an interesting story about culf maps (\cref{def culf}) and edgewise subdivision, which gives the following theorem from \cite{KockSpivak:DSST} (see \cite{HackneyKock:CMES} for a generalization). 
940:If $X$ is a discrete decomposition space, then there is an equivalence of categories between the category of culf maps over $X$ and the category of presheaves on $\sd X$.

640	\end{proof}
641	
642	\begin{exercise}\label{exc 2-Segal defs}
643	Recall that one characterization of 2-Segal spaces given in \cite{Stern:BIRS} is that $X$ sends the pushout diagram of finite ordered sets
644	\begin{equation}
645	\label{diag ij}
646	\begin{tikzcd}
647	\{ i,j\} \rar[-act] \dar[tail] & \{i,i+1, \dots, j \} \dar[tail] \\
648	\{0,1,\dots, i,j,j+1, \dots, n\} \rar[-act] & {[n]}
649	\end{tikzcd} \end{equation}
650	to a pullback for each $0 \leq i < j \leq n$.
651	Prove this condition is equivalent to the definition of 2-Segal space from \cref{def 2-segal}.
652	\end{exercise}
653	
654	The squares \eqref{diag ij} correspond to those squares appearing in \eqref{diag ai pushout} of \cref{lem ai pushout} where $n=1$ and $\alpha$ is injective.
655	As in \cite[Proposition 2.3.2]{DyckerhoffKapranov:HSS}, one only needs to require these squares be sent to pullbacks when $i=0$ or $j=n$.
656	
657	\section{Segal spaces are decomposition spaces}\label{sec segal spaces}
658	
659	In \cite{Stern:BIRS}, we saw that the nerve of a category is a 2-Segal set, under the characterization from \cref{exc 2-Segal defs}, a fact that is true more generally: Segal spaces \cite{Rezk:MHTHT,Segal:CCT} are always decomposition spaces.
660	In this brief section we use the pasting law to establish this fact, and to characterize the Segal spaces among decomposition spaces.
661	
662	\begin{definition}\label{def Segal}
663	A simplicial space $X$ is a \emph{Segal space} if the following equivalent conditions hold.
664	\begin{enumerate}
665	\item \label{item pullback Segal} For each $n\geq 1$, the following square is a pullback. 
666	\[ \begin{tikzcd}
667	X_{n+1} \ar[dr, phantom, "\lrcorner" very near start] \rar{d_\bot} \dar[swap]{d_\top} & X_n \dar{d_\top} \\
668	X_n \rar[swap]{d_\bot} & X_{n-1}
669	\end{tikzcd} \]
670	\item For each $n\geq 2$, the map
671	\[
672	\begin{tikzcd}[column sep=2.2cm]
673	X_n \rar["(d_\top^{n-i} d_\bot^{i-1})_{i=1}^n" swap, "\simeq"] & X_1 \times_{X_0} X_1 \times_{X_0} \dots  \times_{X_0} X_1
674	\end{tikzcd} 
675	\]
676	is an equivalence. \label{item usual Segal}
677	\end{enumerate}
678	\end{definition}
679	
680	\begin{exercise}
681	Prove that \eqref{item pullback Segal} and \eqref{item usual Segal} in \cref{def Segal} are equivalent. (For a hint or solution, see \cite[Lemma 2.10]{GKT1}.)
682	\end{exercise}
683	
684	\begin{proposition}\label{segal implies decomposition}
685	Every Segal space is a decomposition space.
686	\end{proposition}
687	\begin{proof}
688	We show that every Segal space is upper 2-Segal.
689	Since the opposite of a Segal space is a Segal space, and the opposite of an upper 2-Segal space is a lower 2-Segal space, this argument suffices.
690	By \cref{prop fewer squares}, we need only show that the left square in diagram 
691	\[ \begin{tikzcd}
692	X_{n+1} \rar{d_n} \dar[swap]{d_\bot} &
693	X_n \rar{d_\top} \dar[swap]{d_\bot} \ar[dr, phantom, "\lrcorner" very near start] & X_{n-1} \dar{d_\bot}
694	\\
695	X_n \rar[swap]{d_{n-1}} & 
696	X_{n-1} \rar[swap]{d_\top} & X_{n-2}
697	\end{tikzcd} \]
698	is a pullback.
699	The right square is a pullback since $X$ is Segal.
700	The outer rectangle is equal to the outer rectangle of the diagram
701	\[ \begin{tikzcd}
702	X_{n+1} \rar{d_\top} \dar[swap]{d_\bot} \ar[dr, phantom, "\lrcorner" very near start] &
703	X_n \rar{d_\top} \dar[swap]{d_\bot} \ar[dr, phantom, "\lrcorner" very near start] & X_{n-1} \dar{d_\bot}
704	\\
705	X_n \rar[swap]{d_\top} & 
706	X_{n-1} \rar[swap]{d_\top} & X_{n-2}.
707	\end{tikzcd} \]
708	Since both squares are pullbacks, so is the outer rectangle, which implies that the left square in first diagram is a pullback, as desired.
709	\end{proof}
710	
711	\begin{proposition}
712	A simplicial space $X$ is a Segal space if and only if it is a decomposition space and the following square is a pullback:
713	\[ \begin{tikzcd}
714	X_2 \rar{d_0} \dar[swap]{d_2} & X_1 \dar{d_1} \\
715	X_1 \rar[swap]{d_0} & X_0.
716	\end{tikzcd} \]
717	\end{proposition}
718	\begin{proof}
719	The forward direction is immediate from \cref{segal implies decomposition}.
720	For the reverse direction, one shows via induction that the squares in \cref{def Segal} \eqref{item pullback Segal} are pullbacks; the case $n=1$ is an assumption.
721	For the inductive step, use the two rectangles from the proof of \cref{segal implies decomposition} in a different way, assuming that both squares in the first rectangle and the right square in the second rectangle are pullbacks, in order to deduce that the left square in the second rectangle is a pullback.
722	\end{proof}
723	
724	\section{The path space criterion}\label{sec path space}
725	
726	We now turn to the path space criterion, which is originally due to Dyckerhoff and Kapranov \cite[Theorem 6.3.2]{DyckerhoffKapranov:HSS}.
727	Our presentation separates the lower and upper 2-Segal conditions, following Poguntke \cite{Poguntke:HSSAKT}, who also gave higher-dimensional versions.
728	A version for decomposition spaces appeared as \cite[Theorem 4.10]{GKT1}.
729	
730	\begin{theorem}\label{thm upper path}
731	A simplicial space $X$ is upper 2-Segal if and only if its upper d\'ecalage $\udec X$ is Segal.
732	\end{theorem}
733	
734	The upper d\'ecalage $\udec X$ is given by precomposition of $X \colon \Delta^\op \to \spaces$ with the endofunctor $\Delta \to \Delta$ that sends $[n]$ to $[n]\star [0] \cong [n+1]$.\footnote{
735	Here $\star$ denotes the \emph{join} of categories \cite[\href{https://kerodon.net/tag/0160}{Tag 0160}]{kerodon}, where the set of objects of $C \star D$ is the disjoint union of the sets of objects of $C$ and $D$, the categories $C$ and $D$ are full subcategories of $C\star D$, and, for $c\in C$ and $d\in D$, there are no morphisms $d\to c$ and exactly one morphism $c\to d$.}
736	This endofunctor takes $f\colon [n] \to [m]$ to $g\colon [n+1] \to [m+1]$ satisfying $g(n+1) = m+1$ and $g(k) = f(k)$ for $0\leq k \leq n$.
737	
738	More explicitly, we have $(\udec X)_n = X_{n+1}$ and the face and degeneracy operators are given for $0 \leq k \leq n$ by
739	\begin{align*} 
740	\big( d_k \colon (\udec X)_n \to (\udec X)_{n-1} \big) &= \big( d_k \colon X_{n+1} \to X_n \big)
741	\\
742	\big( s_k \colon (\udec X)_n \to (\udec X)_{n+1} \big) &= \big( s_k \colon X_{n+1} \to X_{n+2} \big).
743	\end{align*}
744	Notably, we have forgotten about the top face $d_\top = d_{n+1} \colon X_{n+1} \to X_n$ and top degeneracy $s_\top = s_n \colon X_n \to X_{n+1}$.
745	The upper d\'ecalage is also called the \emph{final path space} and denoted by $P^\triangleright X$.
746	
747	\begin{proof}[Proof of \cref{thm upper path}]
748	For $n\geq 2$, the following two squares are the same:
749	\[ \begin{tikzcd}
750	(\udec X)_n \rar{d_\top} \dar[swap]{d_\bot}  & (\udec X)_{n-1} \dar{d_\bot} 
751	&  
752	X_{n+1} \rar{d_n} \dar[swap]{d_\bot}  & X_n \dar{d_\bot}
753	\\
754	(\udec X)_{n-1} \rar[swap]{d_\top} & (\udec X)_{n-2}
755	&  
756	X_n \rar[swap]{d_{n-1}} & X_{n-1}.
757	\end{tikzcd} \]
758	\cref{prop fewer squares} says that the right square is a pullback for all $n\geq 2$ if and only if $X$ is upper 2-Segal.
759	\end{proof}
760	
761	Similarly, there is a lower d\'ecalage $\ldec X$ (or initial path space $P^\triangleleft X$) with $(\ldec X)_n = X_{n+1}$ obtained by forgetting the bottom face and degeneracy operators (and shifting indices by one).
762	It can be formally obtained using precomposition with the endofunctor $[n] \mapsto [0] \star [n]$ of $\Delta$, sending $f \colon [n] \to [m]$ to $g \colon [n+1] \to [m+1]$ satisfying $g(0) = 0$ and $g(k) = f(k-1) + 1$ for $1 \leq k \leq n+1$.
763	In other words, the face and degeneracy operators are as follows:
764	\begin{align*} 
765	\big( d_k \colon (\ldec X)_n \to (\ldec X)_{n-1} \big) &= \big( d_{k+1} \colon X_{n+1} \to X_n \big)
766	\\
767	\big( s_k \colon (\ldec X)_n \to (\ldec X)_{n+1} \big) &= \big( s_{k+1} \colon X_{n+1} \to X_{n+2} \big).
768	\end{align*}
769	One checks that $(\udec X)^\op = \ldec(X^\op)$, so that applying \cref{thm upper path} to $X^\op$ we obtain the following.
770	
771	\begin{corollary}
772	A simplicial space $X$ is lower 2-Segal if and only if its lower d\'ecalage $\ldec X$ is Segal. \qed
773	\end{corollary}
774	
775	Using \cref{thm unitality}, we conclude the original form of the path space criterion.
776	
777	\begin{corollary}\label{cor path space}
778	A simplicial space $X$ is a decomposition space if and only if both $\udec X$ and $\ldec X$ are Segal. \qed
779	\end{corollary}
780	
781	When $X$ is a decomposition space, it turns out that the maps $d_\top \colon \udec X \to X$ and $d_\bot \colon \ldec X \to X$ are culf (in the sense of \cref{def culf}); see \cite[Theorem~4.10]{GKT1}.
782	
783	\section{The edgewise subdivision criterion}\label{sec edgewise}
784	
785	In this section we explain the following theorem from \cite{BOORS:ESC}.
786	The proof presented here utilizes the exact same techniques that we have been using since \cref{sec 2-Segl decomp}.
787	This proof relies on the path space criterion (\cref{cor path space}), which is appropriate as there is a close connection between the edgewise subdivision and the upper/lower d\'ecalage pair.\phnote{Additionally, this theorem can be interpreted in terms of the double-categorical perspective from \cite{BOORS:2SSWC}.}
788	\begin{theorem}\label{thm edgewise}
789	A simplicial space $X$ is a decomposition space if and only if its edgewise subdivision $\sd X$ is Segal.
790	\end{theorem}
791	
792	As with the d\'ecalage functors, the edgewise subdivision is the result of precomposition with an endofunctor of $\Delta$, namely the one that sends $[n]$ to $[n]^\op \star [n] \cong [2n+1]$.
793	A map $f \colon [n] \to [m]$ is sent to $g \colon [2n+1] \to [2m+1]$ with
794	\[
795	g(t) = \begin{cases}
796	m - f(n-t) & t \leq n \\
797	m+1 + f(t-n-1) & n+1 \leq t.
798	\end{cases}
799	\]
800	If we write $[2n+1]$ in the following way
801	\[
802	\begin{tikzcd}[sep=small]
803	  0 \rar & 1 \rar & \cdots \rar & n \\
804	  0' \uar & 1' \lar & \cdots \lar & n' \lar,
805	\end{tikzcd}
806	\]
807	then $g$ can be interpreted as $t' \mapsto f(t)'$ and $t \mapsto f(t)$.
808	If $X$ is a simplicial space, then the simplicial space $Z = \sd X$ has $Z_n = X_{2n+1}$ and
809	\begin{align*}
810	  \big(d_i \colon Z_n \to Z_{n-1}\big) &= \big(d_{n-i} d_{n+i+1} \colon X_{2n+1} \to X_{2n-1}\big) \\
811	  \big(s_i \colon Z_n \to Z_{n+1}\big) &= \big(s_{n-i} s_{n+i+1} \colon X_{2n+1} \to X_{2n+3}\big).
812	\end{align*}
813	One can check that $\sd X = \sd (X^\op)$. %May 30 note.
814	If $X$ is the nerve of a category $C$, then $\sd X$ is (isomorphic to) the nerve of the \emph{twisted arrow category} of $C$, whose objects are the morphisms of $C$, and where a morphism $f \to g$ is given by a factorization of $g$ as below:
815	\[
816	\begin{tikzcd}  
817	  c' \rar{k} & d' \\
818	  c \uar{f} & d \lar{h} \uar[swap]{g}.
819	\end{tikzcd}
820	\]
821	
822	One manifestation of the connection between d\'eclage and edgewise subdivision is that iterated bottom and top degeneracies yield simplicial maps
823	\[ \begin{tikzcd}[sep=small]
824	\ldec X \rar & \sd X & (\udec X)^\op. \lar
825	\end{tikzcd} \]
826	For the map on the left, recall that $\ldec X$ is given by precomposition with the endofunctor $[n] \mapsto [0] \star [n]$. 
827	The maps $[n]^\op \star [n] \to [0] \star [n]$ collapsing the first component to a point gives a natural transformation of functors, and hence the indicated map.
828	It is specified by
829	\begin{equation}\label{eq ldec to sd}
830	  \ldec X_n = X_{n+1} \xrightarrow{s_\bot^n} X_{2n+1} = \sd X_n.
831	\end{equation}
832	Likewise, the leftward arrow is given by
833	\[
834	  \sd X_n = X_{2n+1} \xleftarrow{s_\top^n} X_{n+1} = (\udec X)^\op_n,
835	\]
836	which arises from the natural transformation $[n]^\op \star [n] \to [n]^\op \star [0]$.
837	
838	\begin{lemma}\label{lem decomp implies sd Segal}
839	If $X$ is a decomposition space, then $\sd X$ is Segal.
840	\end{lemma}
841	\begin{proof}
842	Let $Z = \sd X$.
843	For $n\geq 1$, the following two squares are the same:
844	\[ \begin{tikzcd}
845	Z_{n+1} \rar{d_{n+1}} \dar[swap]{d_0}  & Z_n \dar{d_0} 
846	&  
847	X_{2n+3} \rar{d_0d_{2n+3}} \dar[swap]{d_{n+1}d_{n+2}}  & X_{2n+1} \dar{d_n d_{n+1}}
848	\\
849	Z_n \rar[swap]{d_n} & Z_{n-1}
850	&  
851	X_{2n+1} \rar[swap]{d_0d_{2n+1}} & X_{2n-1}.
852	\end{tikzcd} \]
853	The result follows since the right-hand square is an active-inert pullback square.
854	\end{proof}
855	
856	
857	\begin{lemma}\label{lem ldec square as retract}
858	Let $X$ be a simplicial space and write $Y = \ldec X$, $Z = \sd X$. For each $n \geq 1$, the square below left is a retract of the square below right:
859	\[ \begin{tikzcd}
860	Y_{n+1} \rar{d_\bot} \dar[swap]{d_\top} & Y_n \dar{d_\top} 
861	& Z_{n+1} \rar{d_\bot} \dar[swap]{d_\top} & Z_n \dar{d_\top} 
862	\\
863	Y_n \rar[swap]{d_\bot} & Y_{n-1}
864	&
865	Z_n \rar[swap]{d_\bot} & Z_{n-1}.
866	\end{tikzcd} \]
867	Consequently, if $\sd X$ is Segal, so is $\ldec X$.
868	\end{lemma}
869	\begin{proof}
870	The conclusion follows since pullbacks are stable under retracts.
871	The map from left to right is given by the simplicial map $Y \to Z$ from \eqref{eq ldec to sd}.
872	We explicitly give the retraction from right to left (which does \emph{not} come from a simplicial map $Z \to Y$).
873	Utilizing the simplicial identities, all faces in the following cube commute:
874	\[ \begin{tikzcd}
875	X_{2n+3} \ar[rr,"d_0d_{2n+3}"] \ar[dd,"d_{n+1}d_{n+2}"'] \ar[dr,"d_0d_2^n" description] & & 
876	X_{2n+1} \ar[dd, "d_nd_{n+1}"' very near start] \ar[dr,"d_1^n"]  \\
877	& 
878	X_{n+2} \ar[rr,"d_{n+2}" near end, crossing over] & & 
879	X_{n+1} \ar[dd,"d_1"] \\
880	X_{2n+1} \ar[rr,"d_0d_{2n+1}" very near start] \ar[dr,"d_0d_2^{n-1}"'] & & X_{2n-1} \ar[dr,"d_1^{n-1}" description] \\
881	& X_{n+1} \ar[rr,"d_{n+1}"] \ar[from=uu, crossing over, "d_1" very near start] & & X_n.
882	\end{tikzcd} \]
883	In terms of the face maps of $Y$ and $Z$, the front and back faces of the cube are as follows:
884	\[ \begin{tikzcd}
885	Z_{n+1} \ar[rr,"d_{n+1}^Z"] \ar[dd,"d_0^Z"'] \ar[dr,"d_0d_2^n" description] & & 
886	Z_n \ar[dd, "d_0^Z"' very near start] \ar[dr,"d_1^n"]  \\
887	& 
888	Y_{n+1} \ar[rr,"d_{n+1}^Y" near end, crossing over] & & 
889	Y_n \ar[dd,"d_0^Y"] \\
890	Z_n \ar[rr,"d_n^Z" very near start] \ar[dr,"d_0d_2^{n-1}"'] & & Z_{n-1} \ar[dr,"d_1^{n-1}" description] \\
891	& Y_n \ar[rr,"d_n^Y"] \ar[from=uu, crossing over, "d_0^Y" very near start] & & Y_{n-1}.
892	\end{tikzcd} \]
893	% That is, we are using the maps of squares
894	% \[
895	% \setlength\arraycolsep{1pt}
896	% S = \,
897	% \begin{matrix}
898	% s_\bot^{n+1} & \vline & s_\bot^n \\
899	% \hline
900	% s_\bot^n & \vline & s_\bot^{n-1}
901	% \end{matrix}
902	% \qquad
903	% \text{and}
904	% \qquad
905	% D = 
906	% \, \begin{matrix}
907	% d_0d_2^n & \vline & d_1^n \\
908	% \hline
909	% d_0d_2^{n-1} & \vline & d_1^{n-1}
910	% \end{matrix}
911	% \]
912	% with $DS = \id$.
913	We thus see that the $Y$ square is indeed a retract of the $Z$ square since $d_1^n s_0^n = \id_{X_{n+1}} = d_0 d_2^{n-1} s_0^n$ for all $n\geq 1$.
914	\end{proof}
915	
916	\begin{lemma}
917	If $\sd X$ is Segal, so is the upper d\'ecalage $\udec X$.
918	\end{lemma}
919	\begin{proof}
920	We have $\sd X = \sd (X^\op)$ for any simplicial space $X$. 
921	By \cref{lem ldec square as retract} we conclude that $\ldec(X^\op) = (\udec X)^\op$ is Segal.
922	\end{proof}
923	
924	\begin{proof}[Proof of \cref{thm edgewise}]
925	The forward direction is \cref{lem decomp implies sd Segal}.
926	If $\sd X$ is Segal, then the preceding two lemmas show that $\ldec X$ and $\udec X$ are Segal as well, so the path space criterion implies that $X$ is a decomposition space.
927	\end{proof}
928	
929	% We have already observed that the edgewise subdivision and d\'ecalage constructions are closely related, which is corroborated by the following, already prefigured in \cite{BOORS:2SSWC}.
930	
931	\begin{remark}\label{rmk sd vs dec}
932	If $X$ is a discrete decomposition space, then the three simplicial sets $\sd X$, $\ldec X$, and $\udec X$ are (nerves of) categories.
933	We have inclusions of simplicial sets $\ldec X \hookrightarrow \sd X$ and $(\udec X)^\op \hookrightarrow \sd X$.
934	Identifying each of these subcategories with its image, the category $\sd X$ admits two (strict) factorization systems $(\ldec X, (\udec X)^\op)$ and $((\udec X)^\op, \ldec X)$; see \cite{Hackney:OC2SS} for one explanation.% (including for simplicial spaces).
935	\end{remark}
936	
937	There is also an interesting story about culf maps (\cref{def culf}) and edgewise subdivision, which gives the following theorem from \cite{KockSpivak:DSST} (see \cite{HackneyKock:CMES} for a generalization). 
938	
939	\begin{theorem}\label{thm kock spivak}

1	\section{Higher Segal spaces}\label{sec higher Segal}
2	
3	At the beginning of this section, we provide background material on higher Segal spaces. 
4	Our approach highlights that the types of arguments used in the decomposition space literature can also be used for higher Segal spaces, by replacing pullback squares with cartesian cubes of larger dimension.
5	This relies on work of Walde, who recast the higher Segal conditions in terms of cartesian cubes. 
6	We'll begin with preliminaries on cartesian cubes before turning to the higher Segal conditions for simplicial objects in \cref{ss HSC Walde}.
7	Arguments in this section are in the spirit of those in \cite{Hackney:DSP}.
8	In \cref{ss SSO Segal} we discuss the case of symmetric simplicial objects, where a number of subtleties vanish.
9	
10	In this section, $\mathcal{C}$ will denote a fixed category (or $\infty$-category) with finite limits.
11	All subsequent sections of the paper will take $\mathcal{C}$ to be the category of sets, and the reader is welcome to make this replacement immediately.
12	
13	\subsection{Cubical diagrams}
14	We recall basics about cube-shaped diagrams in a category or $\infty$-category; references include \cite[\S3.3]{Walde:HSSHE} and \cite[\S6.1.1]{LurieHA}.
15	The \emph{generic cube} of dimension $n$ is the $n$-fold product $[1]^n \in \cat$ of the generic arrow $ \{ 0 \to 1 \} = [1] \in \ord \subset \cat$.
16	An \emph{$n$-dimensional cubical diagram} in $\mathcal{C}$, or briefly an \emph{$n$-cube}, is a functor $[1]^n \to \mathcal{C}$.
17	The functor category $\fun([1]^n, \mathcal{C})$ is the associated category of cubes.
18	If $S$ is a set of cardinality $n$, then we may also think of a functor $\ps(S) \to \mathcal{C}$ from the powerset of $S$ as an $n$-cube in $\mathcal{C}$ by choosing an isomorphism $\ps(S) \cong [1]^n$ (and similarly for $\ps(S)^\op$).
19	
20	A map between $n$-cubes may be regarded as an $(n{+}1)$-cube.
21	Namely, we have the following description of the arrow category of the category of cubes, for each choice of isomorphism $[1] \times [1]^n \cong [1]^{n+1}$:
22	\[
23		\fun([1], \fun([1]^n, \mathcal{C})) \cong \fun([1] \times [1]^n, \mathcal{C}) \cong \fun([1]^{n+1}, \mathcal{C}).
24	\]
25	
26	An $n$-cube $Q \colon [1]^n \cong \ps(S) \to \mathcal{C}$ is \emph{cartesian} if it is a limit diagram.
27	Another way to say this is that $Q$ is cartesian if and only if it is right Kan extended from its restriction to the punctured cube $[1]^n \setmin 0 \cong \ps(S) \setmin \{ \varnothing \}$ (i.e.\ $Q \simeq i_*i^*Q$ where $i$ is in the inclusion of the punctured cube, $i_*$ is right Kan extension, and $i^*$ restriction).
28	We now recount several basic lemmas about cartesian cubes that we will need below.
29	
30	\begin{lemma}\label{lem retract}
31	Retracts of cartesian $n$-cubes are cartesian.
32	\end{lemma}
33	\begin{proof}
34	This is an instance of a general fact about closure of limit diagrams under retracts; see e.g.\ \cite[\href{https://kerodon.net/tag/05E6}{Tag 05E6}]{kerodon}.
35	\end{proof}
36	
37	The following two well-known lemmas likely first appear in the Goodwillie calculus literature (with $\mathcal{C}$ the $\infty$-category of spaces). 
38	See Proposition~1.6 and Proposition~1.8 of \cite{Goodwillie:CalcII}.
39	In this generality, the next lemma is \cite[Lemma 3.3.8]{Walde:HSSHE}.
40	
41	
42	\begin{lemma}\label{cube lemma 2}
43	Let $P$ and $Q$ be $n$-cubes, and $R \colon P \to Q$ an $(n{+}1)$-cube. 
44	If $Q$ is cartesian, then $P$ is cartesian if and only if $R$ is cartesian.
45	\end{lemma}
46	
47	
48	\begin{lemma}\label{generalized pasting law}
49	Suppose $P$, $Q$, and $R$ are $(n{+}1)$-cubes, which satisfy $R = Q \circ P$ when regarded as maps of $n$-cubes.
50	If $Q$ is cartesian, then $P$ is cartesian if and only if $R$ is cartesian.
51	\end{lemma}
52	This lemma generalizes the usual pasting law for pullbacks when $n=1$.
53	For completeness, we provide a proof in \cref{cube appendix} in the generality of $\mathcal{C}$ an arbitrary $\infty$-category with finite limits.
54	
55	\begin{remark}
56	When $\mathcal{C}$ is a complete and cocomplete $\infty$-category, these lemmas also follow from corresponding results for derivators (Theorem~8.7 and Proposition~8.11 of \cite{GrothStovicek:TTSHS}) applied to the homotopy derivator of $\mathcal{C}$. 
57	When $\mathcal{C}$ is a stable $\infty$-category, stronger statements hold -- see Corollary~A.16 and Corollary~A.18 of \cite{DyckerhoffJassoWalde}.
58	\end{remark}
59	
60	\subsection{Higher Segal conditions after Walde}\label{ss HSC Walde}
61	
62	In Walde's perspective on the higher Segal conditions \cite{Walde:HSSHE}, a simplicial object is lower $(2k{-}1)$-Segal if and only if it maps each strongly bicartesian $(k{+}1)$-dimensional cube in $\ord$ to a cartesian cube.
63	Strongly bicartesian means that each $2$-dimensional face is bicartesian.
64	On the other hand, not all strongly bicartesian cubes need be checked, only the ones with injective edges.
65	This collection of cubes corresponds precisely to intersection cubes associated with gapped subsets, as we now explain. 
66	
67	Given an object $S \in \ordalt$ and proper subset $I \subset S$, the associated \emph{intersection cube} in $\ordalt$ is the functor 
68	\[
69	\intcu{I} = \intcube{I}{S} \colon \ps(I)^\op \to \ordalt
70	\]
71	that sends a subset $J \subseteq I$ to its complement $S\setmin J$ in $S$. 
72	We use the abbreviation $\intcu{I}$ when $S$ is understood. 
73	To illustrate the terminology, note that if we let $S_i = S\setmin i$ for $i\in I$,
74	then the vertices of the cube are the intersections $\intcu{I}_J = \bigcap_{j\in J} S_j$ of the $S_i$, and the edges are the inclusions.
75	The initial vertex of the cube is $\bigcap_{i\in I} S_i = S \setmin I$, and the terminal vertex is $S$.
76	For example, if $I = \{i_0,\dots,i_k\} \subset [n] = S$, then the intersection cubes for $k = 0$ and $1$ are 
77	$[n]$ and $[n] \leftarrow [n]\setmin \{i_0\}$, while the ones for $k = 2$ and $3$ look like
78	\[
79	\begin{tikzcd}
80	{[n]} & {[n]\setmin i_1} \lar\\
81	{[n]\setmin i_0} \uar & {[n]\setmin \{i_0,i_1\}} \uar \lar
82	\end{tikzcd}
83	\,\, \text{and} \quad
84	\begin{tikzcd}[column sep=0.8ex, row sep=0.8ex]
85	{[n]} \ar[from=rr] \ar[from=dd] \ar[from=dr] & & 
86	{[n]\setmin i_2} \ar[from=dd] \ar[from=dr]  \\
87	& {[n]\setmin i_1} \ar[from=rr, crossing over] & & {[n]\setmin\{i_1,i_2\}} \ar[from=dd] \\
88	{[n]\setmin i_0} \ar[from=rr] \ar[from=dr] & & {[n]\setmin\{i_0,i_2\}} \ar[from=dr] \\
89	& {[n]\setmin\{i_0,i_1\}} \ar[from=rr] \ar[uu, crossing over] & & {[n]\setmin\{i_0, i_1,i_2\}}.
90	\end{tikzcd}
91	\]
92	
93	\smallskip
94	If $X \colon \ordalt^\op \to \mathcal{C}$ is a simplicial object, then composing $\intcu{I}$ with $X$ yields a cube
95	\[
96	X\intcu{I} = X\intcube{I}{S} \colon \ps(I) \to \mathcal{C}
97	\]
98	in $\mathcal{C}$ whose initial vertex is $X_S$ and whose terminal vertex is $X_{S \setmin I}$. 
99	In the special case where $S = [n]$ and $I$ has cardinality $k+1$, these amount to $X_S = X_n$ and $X_{S \setmin I} = X_{n-k-1}$. 
100	
101	\smallskip
102	A subset $I \subseteq S$ is \emph{gapped} if for each pair of elements $i < i'$ in $I$, there is $j \in S$ such that $i < j < i'$. 
103	If $S = [n]$, this just means that each pair of distinct elements of $I$ are at distance at least two from one another. 
104	We use the notation $i \ll i'$ if there exists such a gap $j$ between $i$ and $i'$, so that a gapped subset can be written as a sequence
105	\[
106	i_0 \ll i_1 \ll i_2 \ll \cdots \ll i_k
107	\]
108	where $k+1$ is the cardinality of $I$.
109	As an example, if $I$ is the gapped subset $0 \ll i \ll n$ of $[n]$, then 
110	\[
111	  \begin{tikzcd}[row sep=small]
112	    X_n \ar[rr,"d_n"] \ar[dd,"d_0"'] \ar[dr,"d_i"'] & & 
113	    X_{n-1} \ar[dd, "d_0"' very near start] \ar[dr,"d_i"]  \\
114	    & X_{n-1} \ar[rr,"d_{n-1}" near end, crossing over] & & X_{n-2} \ar[dd,"d_0"] \\
115	    X_{n-1} \ar[rr,"d_{n-1}" near start] \ar[dr,"d_{i-1}"'] & & X_{n-2} \ar[dr,"d_{i-1}" near start] \\
116	    & X_{n-2} \ar[rr,"d_{n-2}"] \ar[from=uu, crossing over, "d_0" very near start] & & X_{n-3}.
117	  \end{tikzcd}
118	\]
119	is the corresponding 3-dimensional cube $X\cube{I}$.
120	\begin{definition}\label{def lower odd Segal}
121	Let $k$ be a positive integer.
122	A simplicial object $X$ is \emph{lower $(2k{-}1)$-Segal} if for every $n \in \mathbb{N}$ and every gapped set $I \subset [n]$ of cardinality $k+1$, the associated cube $X\cube{I}\colon \ps(I) \to \mathcal{C}$ is cartesian.
123	\end{definition}
124	
125	These conditions generalize the usual Segal condition. 
126	Indeed, the lower 1-Segal condition coincides with the Segal condition (see e.g.\ \cite[\S1]{Walde:HSSHE} or \cite[Ex.~3.9]{Dyckerhoff:CPOC}), and if $X$ is lower $(2k{-}1)$-Segal, it is also lower $(2k{+}1)$-Segal (\cref{prop hierarchy}).
127	
128	\begin{remark}\label{rmk walde lower segal}
129	\Cref{def lower odd Segal} is a distillation of the main theorem of \cite{Walde:HSSHE}. 
130	Specifically, it is a combination of Corollary 4.1.5, Theorem 6.1.1, and Theorem 7.2.2 of \cite{Walde:HSSHE}, along with an unraveling of a compatible (Definition 4.1.1) and primitive (Definition 4.3.1) claw whose constituent maps are injective.
131	\end{remark}
132	
133	Each lower $(2k{-}1)$-Segal condition is a one-parameter family of conditions on $X$ involving gapped subsets of $[n]$ for each $n \geq 0$.
134	Observe however, that there are no gapped subsets $I$ of $[n]$ of cardinality $k+1$ if $n < 2k$;
135	the first nonvacuous condition involves the gapped sequence $0 \ll 2 \ll \cdots \ll 2k$ in $[2k]$. 
136	
137	\begin{example}
138	If $X$ is the nerve of a category, then the first lower $1$-Segal condition ($k = 1$, $n = 2$) says that 
139	a $2$-simplex amounts to a pair $(f,g)$ of morphisms agreeing target to source, which is of course the case. 
140	The first of the lower $3$-Segal conditions ($k = 2$, $n = 4$) says a $4$-simplex amounts to a 
141	triple of $3$-simplices $[f|g|h]$, $[g|h|k]$, and $[f|h \circ g|k]$, again the case.
142	\end{example}
143	
144	\begin{remark}\label{exercise sset}
145	Using a cofinality argument, the lower $(2k{-}1)$-Segal condition for a simplicial set $X$ can be reformulated as follows:
146	for every gapped set $I \subset [n]$ of cardinality $k+1$ and every list of $(n{-}1)$ simplices $(x_i) \in \prod_I X_{n-1}$ satisfying $d_i x_j = d_{j-1} x_i$ for $i < j$ in $I$, there exists a unique $x\in X_n$ such that $d_i x = x_i$ for all $i\in I$.
147	\end{remark}
148	
149	The following is a variant on similar results for pullback squares, e.g. \cite[Lemma 3.10]{GKT1}. 
150	It reduces further the number of cubes one needs to check for lower $(2k{-}1)$-Segality. 
151	
152	\begin{lemma}\label{segality top bottom}
153	Let $k$ be a positive integer and $X$ a simplicial object.
154	Assume that for each $n \in \mathbb{N}$ and each gapped subset $I \subset [n]$ of cardinality $k+1$ containing both $0$ and $n$, the cube $X\cube{I}$ is cartesian.
155	Then $X$ is lower $(2k{-}1)$-Segal.
156	\end{lemma}
157	\begin{proof}
158	By the cowidth of a subset $I \subset S \in \ordalt$ we mean the cardinality of the set $\{s \in S \mid s < \min(I) \text{ or } s > \max(I)\}$, 
159	so for $S = [n]$ we are in the hypotheses of the lemma just when the cowidth is $0$. 
160	Assume that for each $S' \in \ordalt$ and each gapped subset $I' \subset S'$ of cardinality $k+1$ and cowidth $0$, the associated cube $X\cube{I'}$ is cartesian. 
161	Fix $S \in \ordalt$ and a gapped subset $I \subset S$ of cardinality $k+1$. 
162	We wish to prove that $X\cube{I}$ is cartesian, and this is by induction first on $|S|$, and then on the cowidth of $I$ in $S$.
163	In the case $|S| = 2k+1$, the unique gapped subset has cowidth $0$ and so the result holds by hypothesis. 
164	
165	Assume now $|S| > 2k+1$. 
166	We may assume that either $\min(S) \notin I$ or $\max(S) \notin I$, say the latter, so that $\max(I) < \max(S)$.
167	Write $m = \max(I)$ and $n = \max(S)$ for short. 
168	A subscript on $I$ or $S$ indicates that the corresponding elements have been removed. 
169	For example, $I_m = I \setmin m$ and $S_{m,n} = S \setmin \{m,n\}$. 
170	We also set $J = I \setmin m \cup n$, which is gapped in both $S_m$ and in $S$. 
171	Observe that since $m < n$ by assumption, $J$ has strictly smaller cowidth in $S$ than $I$ does.
172	
173	Regard $\intcube{I}{S}$ as a map of cubes from $\intcube{I_m}{S_m}$ to $\intcube{I_m}{S}$. 
174	This is the top arrow in the commutative diagram
175	\[
176	\begin{tikzcd}[column sep=large]
177	\intcube{I_m}{S_m} \rar{\intcube{I}{S}} & \intcube{I_m}{S}\\
178	\intcube{I_m}{S_{m,n}} \uar{\intcube{J}{S_m}} \rar{\intcube{I}{S_n}} & \intcube{I_m}{S_n} \uar[swap]{\intcube{J}{S}}.
179	\end{tikzcd}
180	\]
181	By induction, $X\intcube{I}{S_n}$ and $X\intcube{J}{S}$ are cartesian, hence so is their composite by \cref{generalized pasting law}. 
182	By induction, the cube $X\intcube{J}{S_m}$ is cartesian as well. 
183	So by \cref{generalized pasting law} again, $X\intcube{I}{S}$ is cartesian.
184	\end{proof} 
185	
186	We are now ready to introduce the other higher Segal conditions.
187	
188	\begin{definition}[Other higher Segal conditions]\label{def other higher Segal}
189	Let $k$ be a positive integer and $X$ a simplicial object.
190	Consider the collection of gapped subsets $I \subset [n]$ of cardinality $k+1$ and the associated collection of cubes $X\cube{I} \colon \ps(I) \to \mathcal{C}$.
191	We say that $X$ is
192	\begin{enumerate}
193	\item \emph{lower $2k$-Segal} if $X\cube{I}$ is cartesian whenever $0 \notin I$,
194	\item \emph{upper $2k$-Segal} if $X\cube{I}$ is cartesian whenever $n \notin I$, and
195	\item \emph{upper $(2k{+}1)$-Segal} if $X\cube{I}$ is cartesian whenever $0 \notin I$ and $n \notin I$.
196	\end{enumerate}
197	\end{definition}
198	
199	If the definition seems a bit ad hoc, the reason is that all of our definitions are in terms of cartesian cubes, rather than the original geometric definitions (see \cite{poguntke} and \cite[p.\ xv]{DyckerhoffKapranov}) in terms of upper and lower triangulations of cyclic polytopes.
200	Walde proved in \cite{Walde:HSSHE} (see \cref{rmk walde lower segal}) that \cref{def lower odd Segal} is equivalent to the geometric definition of lower $(2k{-}1)$-Segal.
201	Independently, Poguntke proved the characterization in \cref{prop path space criterion} below for the original geometric definitions \cite[Proposition 2.7]{poguntke}.
202	As \cref{prop path space criterion} holds for the conditions defined in \cref{def other higher Segal}, this means that they coincide with the geometric ones.
203	
204	\begin{remark}
205	One could also take $k=0$ in \cref{def lower odd Segal} and \cref{def other higher Segal} to arrive at notions of lower $(-1)$-Segal, lower and upper $0$-Segal, and upper $1$-Segal.
206	The latter three appear in \cite{poguntke}.
207	An adaptation of the proof of \cref{prop symmetric higher Segal} below shows that all four of these conditions coincide for a simplicial object $X \colon \ord^\op \to \mathcal{C}$, and just mean that $X$ is constant \cite[Ex.\ 3.9]{Dyckerhoff:CPOC}.
208	We will not consider this `degree zero' (\cref{def degree}) case any further in this paper, always taking $k > 0$ and not distinguishing between discrete and nondiscrete groupoids.
209	\end{remark}
210	
211	The following is immediate from \cref{def lower odd Segal} and \cref{def other higher Segal}.
212	
213	\begin{lemma}[Opposites]\label{lem opposites}
214	Let $X$ be a simplicial object and $d$ a positive integer.
215	If $d$ is odd, then $X$ is lower or upper $d$-Segal if and only if $X^\op$ is so.
216	If $d$ is even, then $X$ is lower $d$-Segal if and only if $X^\op$ is upper $d$-Segal. \qed
217	\end{lemma}
218	
219	To state the next proposition, we need the \emph{d\'ecalage} functors of Illusie \cite[VI.1]{Illusie:CCD2}
220	\[
221	\ldec, \udec \colon \fun(\ord^\op, \mathcal{C}) \to \fun(\ord^\op, \mathcal{C}),
222	\]
223	which we now define (see also \cite[\S6]{Hackney:DSP}). 
224	There is a functor $\ord \to \ord$ which sends $[n]$ to the ordinal sum $[0] \star [n] = [n+1]$. 
225	Restriction along this functor induces the lower d\'ecalage functor $\ldec \colon \fun(\ord^\op, \mathcal{C}) \to \fun(\ord^\op, \mathcal{C})$.
226	If $X$ is a simplicial object, then $\ldec X$ is obtained from $X$ by deleting $X_0$, setting $\ldec X_n = X_{n+1}$, and deleting the bottom face and degeneracy maps (and renumbering the remaining ones by 1):
227	\[ \begin{tikzcd}
228	X: &  X_0 \rar["s_0" description] & X_1 \lar[shift left=2, "d_0"] \lar[shift right=2, "d_1"']  \rar[shift left=1.5] \rar[shift right=1.5] & X_2 \lar[shift left=3,"d_0"] \lar[shift right=3,"d_2"'] \lar
229	\rar[shift left=3] \rar[shift right=3] \rar &
230	X_3 
231	\lar[shift left=1.5] \lar[shift left=4.5] \lar[shift right=1.5] \lar[shift right=4.5] \cdots \\[+0.25cm]
232	\ldec X: &   & X_1  \rar[shift left=1.5] \rar[shift right=1.5, dotted] & X_2 \lar[shift left=3, dotted, "\color{gray} d_0"] \lar[shift right=3,"d_2"'] \lar
233	\rar[shift left=3] \rar[shift right=3, dotted] \rar &
234	X_3 
235	\lar[shift left=1.5] \lar[shift left=4.5, dotted] \lar[shift right=1.5] \lar[shift right=4.5] \cdots 
236	\end{tikzcd} \]
237	That is, $d_k \colon \ldec X_n \to \ldec X_{n-1}$ is equal to $d_{k+1} \colon X_{n+1} \to X_n$ (and similarly for degeneracies).
238	(In \cite{DyckerhoffKapranov}, $\ldec X$ is called the initial path space $P^\triangleleft X$.)
239	Likewise, there is a functor $\ord \to \ord$ sending $[n]$ to $[n] \star [0] = [n+1]$ and restriction along it induces the upper d\'ecalage functor $\udec \colon \fun(\ord^\op, \mathcal{C}) \to \fun(\ord^\op, \mathcal{C})$.
240	We again have $\udec X_n = X_{n+1}$ for $n\geq 0$, and this time we delete the top face and degeneracy maps (no renumbering of the remaining faces/degeneracies is necessary).
241	
242	\begin{proposition}[Path space criterion \cite{poguntke}]\label{prop path space criterion}
243	Let $X$ be a simplicial object and $k$ a positive integer.
244	\begin{enumerate}
245			\item $X$ is lower $2k$-Segal if and only if $\ldec X$ is lower $(2k{-}1)$-Segal.\label{PSC ldec}
246			\item $X$ is upper $2k$-Segal if and only if $\udec X$ is lower $(2k{-}1)$-Segal.\label{PSC udec}
247	    \item $X$ is upper $(2k{+}1)$-Segal if and only if $\ldec \udec X = \udec \ldec X$ is lower $(2k{-}1)$-Segal.\label{PSC double}
248	\end{enumerate}
249	\end{proposition}
250	Combining the criteria, $X$ is upper $(2k{+}1)$-Segal if and only if $\ldec X$ is upper $2k$-Segal if and only if $\udec X$ is lower $2k$-Segal.
251	These separate conditions are how \eqref{PSC double} is presented in \cite{poguntke,Dyckerhoff:CPOC}.
252	\begin{proof}
253	We prove \eqref{PSC udec}.
254	The natural inclusion $\delta^{n+1} \colon [n] \to [n+1]$ gives a bijection between gapped sets $I\subset [n]$ of cardinality $k+1$ and gapped sets $I' \subset [n+1]$ of cardinality $k+1$ such that $n+1 \notin I'$.
255	Under this correspondence, the cube $(\udec X)\cube{I}$ is equal to the cube $X\cube{\delta^{n+1}I}$, as $(\udec X)_{[n] \setmin J} = X_{[n+1] \setmin J}$ for $J\subseteq I \subset [n]$.
256	This establishes \eqref{PSC udec}.
257	The other statements are proved similarly, replacing $\delta^{n+1}$ by $\delta^0 \colon [n] \to [n+1]$ and $\delta^0 \delta^{n+1} \colon [n] \to [n+2]$.
258	\end{proof}
259	
260	In a sense, the path space criterion tells us that \cref{def lower odd Segal} is the most essential of the higher Segal conditions. 
261	In \cref{prop symmetric higher Segal} we will see that this is even more pronounced for symmetric sets.
262	
263	These higher Segal conditions fit into a hierarchy, due to the following proposition which appears as \cite[Proposition 2.10]{poguntke}; the cases that are not immediate from the definitions are that upper $(2k{-}1)$-Segal implies $2k$-Segal, and that lower or upper $2k$-Segal implies lower $(2k{+}1)$-Segal.
264	
265	\begin{proposition}[Poguntke]\label{prop hierarchy}
266	If $X$ is lower or upper $d$-Segal, then $X$ is both lower $(d{+}1)$-Segal and upper $(d{+}1)$-Segal.
267	\end{proposition}
268	\begin{proof}
269	Throughout $k$ is a positive integer.
270	We first show that if $X$ is lower $2k$-Segal, then $X$ is lower $(2k{+}1)$-Segal.
271	Let $I \subseteq S = [n]$ be a gapped subset of cardinality $k+2$  with $0,n \in I$.
272	By \cref{segality top bottom} it is enough to show that $X\cube{I}$ is cartesian.
273	The set $I_0 = I \setmin 0$ is gapped in both $S_0 = S\setmin 0$ and $S$ and contains the minimal element of neither.
274	By lower $2k$-Segality, the cubes $X\intcube{I_0}{S_0}$ and $X\intcube{I_0}{S}$ are cartesian. 
275	Since $X\cube{I}$ is the map $d_0$ between them, $X\cube{I}$ is cartesian by \cref{cube lemma 2}. 
276	
277	If $X$ is upper $2k$-Segal, then it is lower $(2k{+}1)$-Segal by \cref{lem opposites} and the previous paragraph. 
278	If $X$ is upper $(2k{+}1)$-Segal, then $X$ is both lower and upper $(2k{+}2)$-Segal by the path space criterion and the previous paragraph. 
279	\end{proof}

1	%!TEX root = higher_segal_spheres.tex
2	
3	\section{Simplex modulo boundary}
4	In this section we consider, for $n\geq 1$, the simplicial model for the sphere $\Delta^n / \partial \Delta^n$.
5	For which $d$ is this upper and/or lower $d$-Segal?
6	When $n=1$, this circle is the nerve of the partial monoid $\{1,x\}$ (with $xx$ undefined), hence is 2-Segal \cite{BOORS:2SSetsWC}.  
7	It is evidently not lower 1-Segal, and is not constant so is not upper 1-Segal.
8	In general: 
9	
10	\begin{theorem}
11	Let $n > 1$ be an integer.
12	Then $\Delta^n / \partial \Delta^n$ is $2n$-Segal, but not lower $(2n{-}1)$-Segal. It is upper $(2n{-}1)$-Segal if and only if $n$ is odd.
13	\end{theorem}
14	
15	This follows from \cref{ex nice counterexample}, \cref{ex ugly counterexample}, \cref{thm sphere odd case}, and \cref{thm sphere even case}.
16	
17	When talking about simplices in $X=\Delta^n / \partial \Delta^n$ (at $S\in \Delta$), we just regard them as functions $S \to [n]$, though strictly speaking we're identifying all of the nonsurjective, or \emph{trivial}, functions $S \to [n]$ to a single basepoint $\ast \in X(S)$.
18	We'll write $f\simeq g \colon S \to [n]$ to indicate that either $f=g$ or that $f$ and $g$ are both nonsurjective.
19	We'll also call surjective functions \emph{nontrivial}.
20	Notice that if $e_i f_j = e_j f_i$ is surjective, then there exists a unique $F$ such that $e_i F = f_i$ and $e_j F = f_j$ since $\Delta^n$ is lower 1-Segal.
21	
22	\begin{example}\label{ex nice counterexample}
23	Let $F, G \colon [2n] \to [n]$ be given by the formulas $F(t) = \max(0,t-n)$ and $G(t) = \min(t,n)$.
24	The function $F$ restricts to a bijection $[n+1, 2n] \cong [1,n]$ (with other elements sent to 0), and $G$ restricts to the identity on $[0,n-1]$ (with other elements sent to $n$).
25	When $n$ is even let $I$ be the set of odd elements in $[2n]$, and when $n$ is odd let $I$ be the set of even elements in $[2n]$. 
26	% If $n$ is even, set $I = \{1,3,\dots, 2n-1\}$, and if $n$ is odd set $I= \{0,2,\dots, 2n\}$
27	In either case define $f_i \coloneq e_i F$ for $i<n$ and $f_i = e_i G$ for $i > n$ (noting that $n\notin I$).
28	Each $f_i$ is surjective, and this is a compatible collection of nontrivial elements in $X$.
29	Compatibility is clear for $I\cap [0,n]$ and for $I \cap [n,2n]$, and cross terms $e_i f_j$ are trivial when $i$ and $j$ are on opposite sides of $n$ (since $e_j F = \ast$ when $j > n$ and $e_i G = \ast$ when $i < n$).
30	Now if $H\in X_{2n}$ is any element with $e_i H = f_i$ for all $i$, then for each $i < n$ we have $H(n) = f_i(n) = F(n) = 0$, while for each $i > n$ we have $H(n) = f_i(n) = G(n) = n$, an impossibility.
31	This establishes that $\Delta^n / \partial \Delta^n$ is not upper $(2n{-}1)$-Segal when $n$ is even, and $\Delta^n / \partial \Delta^n$ is not lower $(2n{-}1)$-Segal when $n$ is odd.
32	\end{example}
33	
34	\begin{example}\label{ex ugly counterexample}
35	Suppose $n$ is even, and let $F, G \colon [2n] \to [n]$ be given by the formulas
36	\begin{align*}
37		F(t) &= \min(\max(0,t-n+1), n) \\
38		G(t) &= \max(t-n, \min(t,n-1)).
39	\end{align*}
40	Expanding, these functions act as follows:
41	% \[
42	% \begin{matrix}
43	% 0 & 1 & \cdots & n-1 & n  & n+1 & \cdots & 2n-2& 2n-1& 2n \\
44	% 0 & 0 & \cdots & 0   & 1  & 2   & \cdots & n-1 & n   & n  \\
45	% 0 & 1 & \cdots & n-1 & n-1 & n-1& \cdots & n-1 & n-1 & n
46	% \end{matrix}
47	% \]
48	\[
49	\begin{array}{l|cccccccccc}
50	  t & 0 & 1 & \cdots & n-1 & n  & n+1 & \cdots & 2n-2& 2n-1& 2n \\
51	\hline
52	F(t) & 0 & 0 & \cdots & 0   & 1  & 2   & \cdots & n-1 & n   & n  \\
53	G(t) & 0 & 1 & \cdots & n-1 & n-1 & n-1& \cdots & n-1 & n-1 & n
54	\end{array}
55	\]
56	Let $I \subset [2n]$ be the set of even elements, $I_F = I \cap [0,n-2] \cup \{2n\}$, and $I_G = I \cap [n,2n-2]$.
57	Set $f_i = e_i F$ for $i\in I_F$ and $f_i = e_i G$ for $i\in I_G$.
58	This is a compatible collection of elements -- we only need to check the cross terms, and these vanish since $e_i G = \ast$ for $i\in I_F$ and $e_i F = \ast$ for $i\in I_G$.
59	If $H \in X_{2n}$ is an element with $e_i H = f_i$ for all $i\in I$, then for $i\in I_F$ we have $H(1) = f_i(1) = F(1) = 0$ and for $i\in I_G$ we have $H(1) = f_i(1) = G(1) = 1$, an impossibility.
60	We conclude that $\Delta^n / \partial \Delta^n$ is not lower $(2n{-}1)$-Segal when $n$ is even.
61	\end{example}
62	
63	% Could consider making this a lemma if we expand.
64	
65	If $I \subset S$ is gapped with more than $\frac{n+2}{2}$ elements and $\{f_i\}_{i\in I}$ consists of trivial elements, then $\ast \in X(S)$ is the unique filler.
66	Indeed, suppose $F \colon S \to [n]$ is surjective. 
67	If $\{i\} = F^{-1}F(i)$ for all $i \in I$, we also need somewhere outside of $\{F(i) \mid i\in I \} \subseteq [n]$ to send elements \emph{between} the elements of $I$, implying $|I| + (|I|-1) \leq n+1$.
68	
69	\begin{theorem}\label{thm sphere odd case}
70	If $n > 1$ is odd, then $X = \Delta^n / \partial \Delta^n$ is upper $(2n{-}1)$-Segal.
71	\end{theorem}
72	\begin{proof}
73	Let $I$ be the set of odd elements in $[2n]$ and for a fixed $p\geq 2n$ declare $S = [p]$. 
74	Let $\{f_i\}_{i\in I}$ be a compatible collection of $(p-1)$-simplices in $X = \Delta^n / \partial \Delta^n$ (i.e.\ $f_i \in X(S_i)$ with $e_i f_j \simeq e_j f_i$ for $i\neq j$ in $I$).
75	
76	Define the following relation on $I$: we say $i\sim j$ if $e_i f_j$ is nontrivial, which implies $e_i f_j = e_j f_i$.
77	As $\Delta^n$ is a category (and hence lower 1-Segal), $i\sim j$ implies there exists a unique $F$ such that $e_i F = f_i$ and $e_j F = f_j$.
78	Let's check that $\sim$ is transitive, and to that end suppose also $j\sim k$ and let $G$ be the element with $e_j G = f_j$ and $e_k G = f_k$.
79	By compatibility we either have $e_i f_k = e_k f_i$ or we have $e_i f_k$ is trivial.
80	If $e_i f_k$ is trivial, then $f_k(i)$ is only hit once by $f_k$.
81	This means $f_k(i\pm 1) = f_k(i)\pm 1$ by surjectivity, and the fact that $i\pm 1 \notin I$, so in particular is different from $k$.
82	But $f_k(i\pm 1) = G(i\pm 1) = f_j(i\pm 1) = F(i\pm 1) = f_i(i\pm 1)$.
83	This implies that $f_k(i)$ is not in the image of $f_i$, contrary to our assumption that $f_i$ is nontrivial.
84	Now each equivalence class $E$ of $I$ having more than one element will have a unique $F$ such that $e_i F = f_i$ for all $i\in E$ (since $\Delta^n$ is lower 1-Segal, and hence lower $(|E|-1)$-Segal).
85	
86	
87	
88	Suppose $i\in I$ is such that $f_i$ is nontrivial. We'd like to know how large its equivalence class $E$ is.
89	Define the following sets:
90	\begin{align*}
91	J &= \{ j \in I \mid e_j f_i = \ast\} \\
92	K &= \{ k \in I \mid e_k f_i \text{ surjective}\},
93	\end{align*}
94	which satisfy $I = \{i\} \amalg J \amalg K$ and $E = K \amalg \{i\}$.
95	
96	Notice for $j\in J$ that $f_i(j-1) < f_i(j) < f_i(j+1)$, and hence $f_i(j+1) - f_i(j-1) = 2$. 
97	Now
98	\[
99		2|J| = \sum_{j\in J} f_i(j+1) - f_i(j-1) \leq \sum_{t \in I} f_i(t+1) - f_i(t-1) = f_i(2n) - f_i(0) \leq n.
100	\]
101	Thus $|J| \leq \frac{n}{2}$.
102	Since $n$ is odd and $E\amalg J = I$ has $n$ elements, $E$ has at least $(n+1)/2$ elements.
103	Since $E$ contains more than half of the elements of $I$, there cannot be any other equivalence class containing an $i$ with $f_i$ nontrivial.
104	By the preceding paragraph we know there is a unique $F$ such that $e_i F = f_i$ for all $i\in E$.
105	
106	It just remains to show that $e_j F$ is trivial for all $j\in J$ (since we know that $f_j$ is trivial), but this is about $e_j F(j\pm 1) = f_i(j\pm 1) = (e_j f_i)(j\pm 1)$ being two apart.
107	Thus we have established $\uppod_p^{n-1}$.
108	\end{proof}
109	
110	In particular, when $n$ is odd then $\Delta^n / \partial \Delta^n$ is $2n$-Segal.
111	
112	\begin{theorem}\label{thm sphere even case}
113	If $n > 1$ is even, then $X = \Delta^n / \partial \Delta^n$ is $2n$-Segal.
114	\end{theorem}
115	\begin{proof}
116	Since $X \cong X^\op$, it suffices to show that $X$ is lower $2n$-Segal.
117	Let $I \subset [2n+1]$ be the set of odd elements (an $n+1$ element set), considered as a gapped subset of $S = [p]$ for a fixed $p \geq 2n+1$.
118	Let $\{f_i \in X(S_i)\}_{i\in I}$ be a compatible collection of $(p-1)$-simplices.
119	
120	The argument proceeds as in the proof of \cref{thm sphere odd case}.
121	We again define a relation $\sim$ on $I$, which is again transitive.
122	The proof of this works even when $p = 2n+1$ if we assume $i < k$, which is fine since $e_if_k \simeq e_k f_i$.
123	
124	We define $J$ and $K$ in an identical way, and we wish to show that $|E| \geq \frac{n}{2} + 1$, i.e. that $E$ contains more than half of the elements of the $(n+1)$ element set $I$.
125	This implies that there can be at most one nontrivial equivalence class.
126	If $p > 2n+1$ we obtain the inequality
127	\[
128		2|J| = \sum_{j\in J} f_i(j+1) - f_i(j-1) \leq \sum_{t \in I} f_i(t+1) - f_i(t-1) = f_i(2n+2) - f_i(0) \leq n
129	\]
130	exactly as before, so $|J| \leq \frac{n}{2}$ and hence $|E| \geq \frac{n}{2} + 1$.
131	If $p=2n+1$ is not a member of $J$, then we have (instead summing over $I_p$ the set of odds less than $p$) 
132	\[
133		2|J| = \sum_{j\in J} f_i(j+1) - f_i(j-1) \leq \sum_{t \in I_p} f_i(t+1) - f_i(t-1) = f_i(2n) - f_i(0) \leq n,
134	\]
135	and may again conclude $|E| \geq \frac{n}{2} +1$.
136	Finally, if $p=2n+1$ is a member of $J$, then we compute
137	\[
138		2|J_p| = \sum_{j\in J_p} f_i(j+1) - f_i(j-1) \leq \sum_{t \in I_p} f_i(t+1) - f_i(t-1) = f_i(2n) - f_i(0) = n -1,
139	\]
140	implying $|J_p| \leq \frac{n-1}{2}$, and, since $n$ is even, $|J| \leq \frac{n}{2}$.
141	Thus we again have $|E| \geq \frac{n}{2} + 1$.
142	
143	Since $|E|-1 \geq \frac{n}{2} \geq 1$ and $\Delta^n$ is lower 1-Segal, there exists a unique $F$ such that $e_i F = f_i$ for all $i\in E$.
144	As in the proof of \cref{thm sphere odd case} we see that $e_j F$ is trivial for all $j\in J$, unless $p=2n+1$ and $j=p$.
145	In this last case, since we know $e_p f_i$ is trivial, we have $f_i(2n) = n-1$, but $F(2n) = f_i(2n)$, hence $e_pF$ does not have $n$ in its image, hence is trivial.
146	We conclude that $X$ is lower $2n$-Segal.
147	\end{proof}
148	
149	
150	
151	\section{Simplex boundaries}\label{sec simplex boundaries}
152	
153	\begin{proposition} If $n\geq 0$, then 
154	$\partial \Delta^{n+1}$ is $(n{+}1)$-coskeletal but not $n$-coskeletal.
155	\end{proposition}
156	\begin{proof}
157	It's clear that it's not $n$-coskeletal, as the identity $\partial \Delta^{n+1} \to \partial \Delta^{n+1}$ does not have a filler $\Delta^{n+1} \to \partial \Delta^{n+1}$.
158	% If $n=-1$ then $\partial \Delta^0 = \varnothing$ is trivially $0$-coskeletal. 
159	If $n=0$ then $\partial \Delta^{n+1} = \partial \Delta^1$ is discrete, hence 1-coskeletal. We now assume $n \geq 1$.
160	Suppose $m > n+1$, $S= [m]$, and $f_i \in X(S_i)$ is a compatible collection (ranging over all $i\in S$).
161	There is a unique $F \colon [m] \to [n+1]$ with $e_i F = f_i$ for all $i$, this is using that $\Delta^{n+1}$ is 2-coskeletal since it's a category, and $m > n+1 \geq 2$. If $F$ is surjective, then there exists a $k$ such that $F^{-1}(k)$ has more than one element; if $i \in F^{-1}(k)$, then $e_i F = f_i$ is surjective, contrary to assumption. Thus $F \in (\partial \Delta^{n+1})_m$.
162	\end{proof}
163	
164	% We immediately see for $n\geq 1$ that $\partial \Delta^{n+1}$ is not upper or lower $(n{-}1)$-Segal, as this would imply that it's $n$-coskeletal.
165	
166	\begin{proposition}
167	For $n \geq 1$, $\partial \Delta^{n+1}$ is $(n{+}1)$-Segal but not upper or lower $n$-Segal.
168	\end{proposition}
169	\begin{proof}
170	Suppose $X = \partial \Delta^{n+1}$ is not $(n{+}1)$-critical. 
171	Then for $S = [n+2]$ or $[n+3]$ we can find a compatible collection whose unique filler in $\Delta^{n+1}$ is surjective. 
172	We may assume that $I \subset [n+2]$ is either the set of even elements\footnote{If $n$ is even this set has size $(n/2 + 1) + 1$ and may contain both endpoints, while if $n$ is odd this set has size $(n+1)/2+1$ and misses $n+2$.} or the set of odd elements.\footnote{
173	If $n$ is even this set has size $n/2 +1$ and misses $0,n+2$, while if $n$ is odd this set has size $(n+1)/2+1$ elements and misses $0$ but not $n+2$.}
174	Consider $I$ as a gapped subset of $S$, and let $\{f_i\}_{i\in I}$ be a compatible collection whose unique filler in $\Delta^{n+1}(S)$ is a surjective function $F\colon S \to [n+1]$.
175	If $S = [n+2]$ then there is exactly one $k$ such that $F^{-1}(k)$ contains two elements; by parity, one must be in $I$, which then gives $e_i F = f_i$ surjective, contrary to assumption.
176	For $S = [n+3]$, there are two possibilities for the surjective function $F\colon [n+3] \to [n+1]$.
177	If there is a $k < n+1$ having $F^{-1}(k)$ having at least two elements, then $I \cap F^{-1}(k)$ will be nonempty, so we can argue by parity as before. 
178	If $F^{-1}(n+1)$ has size three, then it will contain the largest element of $I$, and we again do the same thing.
179	We've thus established that $\partial \Delta^{n+1}$ is $(n{+}1)$-critical -- since it's $(n{+}1)$-coskeletal, it is $(n{+}1)$-Segal by the main theorem of \cite{Hackney:CHSC}.
180	
181	To see that $\partial \Delta^{n+1}$ is not upper or lower $n$-Segal, let $F \colon [n+1] \to [n+1]$ be the identity and $I \subset [n+1]$ to be either the set of even elements or the set of odd elements. 
182	Then $\{e_i F\}_{i\in I}$ is an appropriate compatible collection of $n$-simplices in $\partial \Delta^{n+1}$, but its only filler is surjective.
183	(If $n=1$, the argument against upper 1-Segal is simply that $\partial \Delta^2$ is not constant.)
184	\end{proof}

% (Without this assumption, one still obtains a semi-simplicial 2-Segal set.) 
\end{exercise}

\begin{remark}
Each partial monoid may be regarded as a monoid with a zero element, by adjoining a new absorbing element $0$ to $M$ and declaring that all undefined multiplications take $0$ as their value.
Likewise, for any monoid with a zero element different from $1$, we obtain a partial monoid by discarding $0$.
In a similar manner, the $\catpar$-enriched categories from the preceding exercise correspond to categories with (nonidentity) zero morphisms \cite[\S1.7]{Pareigis:CF}.
% (Also note that $\catpar$-enriched categories are a special case of the multivalued categories of \cite[\S3.3]{DyckerhoffKapranov:HSS}.)
\end{remark}

The examples of partial monoids and $\catpar$-enriched categories suggest a useful perspective: simplicial sets that are 2-Segal (or equivalently, discrete decomposition spaces) can be understood as encoding multivalued ``associative'' composition laws.
See \cite[\S3.3]{DyckerhoffKapranov:HSS} and \cite{Stern:BIRS} for precise statements.
The preceding examples cover the case of composition having at most one value, and the single-valued composition laws correspond to the Segal objects (\cref{sec segal spaces}).

We mention, in passing, a class of maps that plays an important role in the theory of decomposition spaces.
These maps do not make a significant appearance in this paper, but reappear elsewhere in this volume \cite{CooperYoung:BIRS,GKT:DSC}.
In particular, \cite{CooperYoung:BIRS} explains why a culf map between decomposition spaces induces a coalgebra homomorphism between the associated incidence coalgebras (originally from \cite[Lemma 8.2]{GKT1}).

\begin{definition}[Culf maps]\label{def culf}
A map $f \colon X \to Y$ between simplicial spaces is \emph{culf} if it is cartesian on active maps.
Concretely, we require that the following squares are pullbacks, for $0 < i < n$ and $0\leq j \leq n$:
\[ \begin{tikzcd}
X_n \rar{d_i} \dar[swap]{f} \ar[dr, phantom, "\lrcorner" very near start]  & X_{n-1} \dar{f} 
& &
X_n \rar{s_j} \dar[swap]{f} \ar[dr, phantom, "\lrcorner" very near start]  & X_{n+1} \dar{f} 
\\
Y_n  \rar[swap]{d_i} & Y_{n-1} 
& &
Y_n  \rar[swap]{s_j} & Y_{n+1}.
\end{tikzcd}
\]
\end{definition}

\section{2-Segal spaces are decomposition spaces}\label{sec 2-Segl decomp}

In this section we discuss the following fundamental result; as mentioned in the introduction, it was known early on that decomposition spaces were the same thing as unital 2-Segal spaces, and later on it was discovered that the unitality condition is automatic \cite{Feller_et_al:E2SSU}.

\begin{theorem}\label{thm unitality} 
A simplicial space $X$ is a decomposition space if and only if it is both upper and lower 2-Segal.
\end{theorem}

It is immediate that every decomposition space is 2-Segal, since the squares in \eqref{eq upper and lower 2-Segal} of \cref{def 2-segal} are active-inert pullback squares.
The converse is more interesting.
If $X$ is a decomposition space, then the following basic squares (for all $0 \leq i \leq n$) come from active-inert pushout squares in $\Delta$, so must be pullbacks:
\[ \begin{tikzcd}
X_{n+1} \rar{s_{i+1}} \dar[swap]{d_\bot} \ar[dr, phantom, "\lrcorner" very near start]  & X_{n+2} \dar{d_\bot} 
& &
X_{n+1} \rar{s_i} \dar[swap]{d_\top} \ar[dr, phantom, "\lrcorner" very near start]  & X_{n+2} \dar{d_\top} 
\\
X_n  \rar[swap]{s_i} & X_{n+1} 
& &
X_n  \rar[swap]{s_i} & X_{n+1}.
\end{tikzcd}
\]
However, the 2-Segal condition makes no mention of any squares involving degeneracies.
The main effort of the proof of \cref{thm unitality} is addressing this issue.
Most proofs in the rest of the paper rely on two basic tools.
Here is the first.

\begin{pastinglaw}
Suppose we have a diagram
\[ \begin{tikzcd}
\bullet \rar \dar & \bullet \rar \dar \ar[dr, phantom, "\lrcorner" very near start] & \bullet \dar \\
\bullet \rar & \bullet \rar & \bullet 
\end{tikzcd} \]
with right square a pullback.
Then the left square is a pullback if and only if the outer (composite) square is a pullback.
\end{pastinglaw}

For a 1-category $C$, such as that of sets, this law is traditionally left as an exercise, but see \cite[Proposition 2.5.9]{Borceux:HCA1} for a proof.
In an $(\infty,1)$-category $C$ (such as that of spaces), this law is sometimes called the \emph{prism lemma}, since the diagram above should be specified by a functor $\Delta^2 \times \Delta^1 \to C$ (see \cite[Lemma 4.4.2.1]{Lurie:HTT} or \cite[\href{https://kerodon.net/tag/03FZ}{Tag 03FZ}]{kerodon}).

As an illustration of the utility of the pasting law, we establish the following characterization of the (upper) 2-Segal condition from \cite[Lemma 3.6]{GKT1}.

\begin{proposition}\label{prop fewer squares}
The following are equivalent for a simplicial space $X$.
\begin{enumerate}
\item The simplicial space $X$ is upper 2-Segal.\label{item fewer upper}
\item For each $n\geq 2$, the left square of \eqref{eq upper and lower 2-Segal} is a pullback for some $0 < i < n$.\label{item fewer some}
\end{enumerate}
\end{proposition}
\begin{proof}
It is immediate that \eqref{item fewer upper} implies \eqref{item fewer some}.
To prove the converse, it is helpful to add a third equivalent condition:
\begin{enumerate}[start=3]
\item For each $n\geq 2$, the square below is a pullback.\label{item fewer composite}
\[ \begin{tikzcd}
X_{n+1} \rar{d_2^{n-1}} \dar[swap]{d_\bot} & X_2 \dar{d_\bot} \\
X_n \rar[swap]{d_1^{n-1}} & X_1
\end{tikzcd} \]
\end{enumerate}
The following diagram is used for the remaining implications:
\[ \begin{tikzcd}
X_{n+2} \rar{d_{i+1}} \dar[swap]{d_\bot} &
X_{n+1} \rar{d_2^{n-1}} \dar[swap]{d_\bot} \ar[dr, phantom, "\lrcorner" very near start] & X_2 \dar{d_\bot}
\\
X_{n+1} \rar[swap]{d_i} & 
X_n \rar[swap]{d_1^{n-1}} & X_1.
\end{tikzcd} \]
To prove that \eqref{item fewer some} implies \eqref{item fewer composite}, we assume inductively that the right square above is a pullback, and that for some $0 < i < n+1$ the left square is a pullback, which implies that the outer rectangle is also a pullback.
But $d_2^{n-1}d_{i+1} = d_2^n$ and $d_1^{n-1}d_i = d_1^n$, so this outer rectangle is the case of \eqref{item fewer composite} in the next level.
To show that \eqref{item fewer composite} implies \eqref{item fewer upper}, let $0 < i < n+1$ be arbitrary. 
Then the outer rectangle and the right square in the diagram are pullbacks, hence so is the left square.
Thus $X$ is upper 2-Segal.
\end{proof}

% A similar result holds for lower 2-Segal spaces using the square
% \[ \begin{tikzcd}
% X_{n+1} \rar{d_1^{n-1}} \dar[swap]{d_\top} & X_2 \dar{d_\top} \\
% X_n \rar[swap]{d_1^{n-1}} & X_1
% \end{tikzcd} \]
% instead. Formally, one can apply \cref{prop fewer squares} to $X^\op$.


\begin{lemma}\label{lem higher degen}
Suppose $X$ is an upper 2-Segal space.
If $n > 0$ and $0\leq i \leq n$, then the square
\[ \begin{tikzcd}
X_{n+1} \rar{s_{i+1}} \dar[swap]{d_\bot} \ar[dr, phantom, "\lrcorner" very near start]  & X_{n+2} \dar{d_\bot} \\
X_n  \rar[swap]{s_i} & X_{n+1}
\end{tikzcd} %\qquad
% \left(
% \text{resp.\ } 
% \begin{tikzcd}
% X_{n+1} \rar{s_i} \dar[swap]{d_\top} \ar[dr, phantom, "\lrcorner" very near start]  & X_{n+2} \dar{d_\top} \\
% X_n  \rar[swap]{s_i} & X_{n+1}
% \end{tikzcd}
% \right)
\]
is a pullback.
\end{lemma}
\begin{proof}
This proof is an application of the pasting law.
We can choose $j \in \{i,i+1 \}$ with $0 < j < n+1$, and then we have the following diagram:
\[
\begin{tikzcd}
X_{n+1} \ar[rr,bend left, "\id"'] \rar{s_{i+1}} \dar[swap]{d_\bot} & X_{n+2} \dar[swap]{d_\bot} \ar[dr, phantom, "\lrcorner" very near start] \rar{d_{j+1}} & X_{n+1} \dar{d_\bot} \\
X_n  \rar[swap]{s_i} \ar[rr, bend right, "\id"] & X_{n+1} \rar[swap]{d_j} & X_n.
\end{tikzcd}
\]
The right square is a pullback since $X$ is upper 2-Segal, and the outer square is a pullback.
Hence the left square is a pullback.
\end{proof}

Unfortunately, the proof of the lemma does not cover the $n=0$ case, for which we need our second basic tool.

\begin{retrstab}
Retracts of pullbacks are pullbacks.
\end{retrstab}
Being a pullback is a property of an object of $\fun(\Delta^1 \times \Delta^1, C)$, the category of commutative squares in $C$, and this property is stable under retracts.\phnote{May be confusing if not working in $\infty$-categories.}
See \cite[Appendix A]{Feller_et_al:E2SSU} for homotopy pullbacks in model categories; for quasi-categories use the duals of \cite[Lemma 5.1.6.3]{Lurie:HTT} or \cite[\href{https://kerodon.net/tag/05E6}{Tag 05E6}]{kerodon}.
The following is proved in \cite{Feller_et_al:E2SSU}.

\begin{lemma}\label{lem bot degen}
If $X$ is an upper 2-Segal space, then the square
\[ \begin{tikzcd}
X_1 \rar{s_1} \dar[swap]{d_0} \ar[dr, phantom, "\lrcorner" very near start]  & X_2 \dar{d_0} \\
X_0  \rar[swap]{s_0} & X_1
\end{tikzcd} 
\]
is a pullback.
\end{lemma}
\begin{proof}
The indicated square is a retract of 
\[
\begin{tikzcd}
X_2 \rar{s_2} \dar[swap]{d_\bot} \ar[dr, phantom, "\lrcorner" very near start]  & X_3 \dar{d_\bot} \\
X_1  \rar[swap]{s_1} & X_2
\end{tikzcd} 
\]
which is known to be a pullback by \cref{lem higher degen}.
This property is witnessed by the maps of squares
\[
\setlength\arraycolsep{1pt}
S = \,
\begin{matrix}
s_1 & \vline & s_1 \\
\hline
s_0 & \vline & s_0
\end{matrix}
\qquad
\text{and}
\qquad
D = 
\, \begin{matrix}
d_1 & \vline & d_1 \\
\hline
d_0 & \vline & d_0
\end{matrix}
\]
with $DS = \id$.
More explicitly, maps between squares are cubes, and the diagram
  \[
  \begin{tikzcd}[sep={30pt,between origins}]
      {X_1} \ar[rrr, "s_1"] \ar[dd, "d_0"] \ar[dr, "s_1"] & & & 
      {X_2} \ar[rrr, "d_1"] \ar[dd, "d_0"' very near start] \ar[dr, "s_2"] & & & 
      {X_1} \ar[dd, "d_0"' very near start] \ar[dr, "s_1"] & \\ &
      {X_2} \ar[rrr, pos=0.4, "s_1", crossing over]   & & & 
      {X_3} \ar[rrr, pos=0.4, "d_1", crossing over]   & & & 
      {X_2} \ar[dd,"d_0"] \\ 
      {X_0} \ar[rrr, pos=0.64, "s_0"] \ar[dr, "s_0"'] & & & 
      {X_1} \ar[rrr, pos=0.64, "d_0"] \ar[dr, "s_1"'] & & & 
      {X_0} \ar[dr, "s_0"] & \\ &
      {X_1} \ar[rrr, "s_0"] \ar[from=uu, crossing over, "d_0" very near start] & & &  
      {X_2} \ar[rrr, "d_0"] \ar[from=uu, crossing over, "d_0" very near start] & & &  
      {X_1}
  \end{tikzcd}
  \]
is commutative.
\end{proof}

Dually, if $X$ is a lower 2-Segal space then for $n\geq 0$ and $0\leq i \leq n$, the square 
\[
\begin{tikzcd}
X_{n+1} \rar{s_i} \dar[swap]{d_\top} \ar[dr, phantom, "\lrcorner" very near start]  & X_{n+2} \dar{d_\top} \\
X_n  \rar[swap]{s_i} & X_{n+1}
\end{tikzcd}
\]
is a pullback: apply \cref{lem higher degen} and \cref{lem bot degen} to $X^\op$.
To prove \cref{thm unitality}, one can now follow \cite[3.4]{GKT1}, or reason about the situation directly as follows.

\begin{proof}[Proof sketch for \cref{thm unitality}]
First consider active-inert pushout squares
\[ \begin{tikzcd}
{[n]} \rar[-act, "\alpha"] \dar[tail, "\delta^k"'] \ar[dr, phantom, "\ulcorner" very near end] & {[m]} \dar[tail, "\theta"] \\
{[n+1]} \rar[-act, "\phi"'] & {[m+1]}
\end{tikzcd} \]
where $k$ is either $0$ or $n+1$ (implying $\theta = \delta^\ell$ for $\ell$ either $0$ or $m+1$).
By factoring $\alpha$ into degeneracies and inner faces, we see that 
\begin{equation}\label{diag codim one inert}
\begin{tikzcd}
X_{m+1} \rar{\phi^*} \dar[swap]{d_\ell} 
\ar[dr, phantom, "\lrcorner" very near start]
& X_{n+1} \dar{d_k} 
\\
X_m \rar[swap]{\alpha^*} & X_n
\end{tikzcd}
\end{equation}
is a pullback using the pasting law and our known pullback squares.
Then, for an arbitrary active-inert pushout square as in \eqref{diag ai pushout} of \cref{lem ai pushout}, use induction on $k-n$ along with the squares \eqref{diag codim one inert} and the pasting law. 
\end{proof}

File created successfully at: /tmp/scratchpad/segal.py (file state is current in your context — no need to Read it back)

circle lower_odd k=1 (Segal), expect FAIL:
    S1 k=1 lower_odd m=2 I=(0, 2): FAIL ('no-filler', ((0, 1), (0, 1)))
    S1 k=1 lower_odd m=3 I=(0, 2): FAIL ('no-filler', ((0, 1, 0), (0, 1, 1)))
    S1 k=1 lower_odd m=3 I=(0, 3): FAIL ('no-filler', ((0, 0, 1), (0, 1, 1)))
    S1 k=1 lower_odd m=3 I=(1, 3): FAIL ('no-filler', ((0, 0, 1), (1, 0, 1)))
circle lower_even k=1 (lower 2-Segal), expect OK:
    S1 k=1 lower_even m=3 I=(1, 3): FAIL ('no-filler', ((0, 0, 1), (1, 0, 1)))
    S1 k=1 lower_even m=4 I=(1, 3): FAIL ('no-filler', ((0, 0, 1, 0), (1, 0, 1, 1)))
    S1 k=1 lower_even m=4 I=(1, 4): FAIL ('no-filler', ((0, 0, 0, 1), (1, 0, 1, 1)))
    S1 k=1 lower_even m=4 I=(2, 4): FAIL ('no-filler', ((0, 0, 0, 1), (1, 1, 0, 1)))
circle upper_even k=1, expect OK:
    S1 k=1 upper_even m=3 I=(0, 2): FAIL ('no-filler', ((0, 1, 0), (0, 1, 1)))
    S1 k=1 upper_even m=4 I=(0, 2): FAIL ('no-filler', ((0, 1, 0, 0), (0, 1, 1, 1)))
    S1 k=1 upper_even m=4 I=(0, 3): FAIL ('no-filler', ((0, 0, 1, 0), (0, 1, 1, 1)))
    S1 k=1 upper_even m=4 I=(1, 3): FAIL ('no-filler', ((0, 0, 1, 0), (1, 0, 1, 1)))
Shell cwd was reset to ~/twisted_arrow

circle lower_odd k=1 (Segal), expect FAIL:
    S1 k=1 lower_odd m=2 I=(0, 2): FAIL ('no-filler', ((0, 1), (0, 1)))
    S1 k=1 lower_odd m=3 I=(0, 2): OK
    S1 k=1 lower_odd m=3 I=(0, 3): FAIL ('no-filler', ((0, 0, 1), (0, 1, 1)))
    S1 k=1 lower_odd m=3 I=(1, 3): OK
circle lower_even k=1 (lower 2-Segal), expect OK:
    S1 k=1 lower_even m=3 I=(1, 3): OK
    S1 k=1 lower_even m=4 I=(1, 3): OK
    S1 k=1 lower_even m=4 I=(1, 4): OK
    S1 k=1 lower_even m=4 I=(2, 4): OK
circle upper_even k=1, expect OK:
    S1 k=1 upper_even m=3 I=(0, 2): OK
    S1 k=1 upper_even m=4 I=(0, 2): OK
    S1 k=1 upper_even m=4 I=(0, 3): OK
    S1 k=1 upper_even m=4 I=(1, 3): OK
Shell cwd was reset to ~/twisted_arrow

sd(S^1) lower_odd k=1 (Segal), expect OK:
    sdS1 k=1 lower_odd m=2 I=(0, 2): OK
    sdS1 k=1 lower_odd m=3 I=(0, 2): OK
    sdS1 k=1 lower_odd m=3 I=(0, 3): OK
    sdS1 k=1 lower_odd m=3 I=(1, 3): OK
 -> True

X=S^2 model. sanity: not lower 2-Segal (k=1), expect a FAIL:
    S2 k=1 lower_even m=3 I=(1, 3): FAIL ('non-unique', ((0, 1, 2), '*'), (0, 0, 1, 2), (0, 1, 1, 2))
    S2 k=1 lower_even m=4 I=(1, 3): FAIL ('no-filler', ((0, 0, 1, 2), (0, 1, 2, 2)))
    S2 k=1 lower_even m=4 I=(1, 4): FAIL ('non-unique', ((0, 1, 1, 2), '*'), (0, 0, 1, 1, 2), (0, 1, 1, 1, 2))
    S2 k=1 lower_even m=4 I=(2, 4): FAIL ('non-unique', ((0, 0, 1, 2), '*'), (0, 0, 0, 1, 2), (0, 0, 1, 1, 2))
 -> False
Shell cwd was reset to ~/twisted_arrow

=== X = Delta^1/bdry ===
 k=1 (cubes of dim 2), dims up to 4 for X; up to 4 for sd X
  X                low1:n  low2:Y  upp2:Y  upp3:Y
  sd X             low1:Y  low2:Y  upp2:Y  upp3:Y
 k=2 (cubes of dim 3), dims up to 6 for X; up to 6 for sd X
  X                low3:Y  low4:Y  upp4:Y  upp5:Y
  sd X             low3:Y  low4:Y  upp4:Y  upp5:Y
 k=3 (cubes of dim 4), dims up to 8 for X; up to 8 for sd X
  X                low5:Y  low6:Y  upp6:Y  upp7:Y
  sd X             low5:Y  low6:Y  upp6:Y  upp7:Y

=== X = Delta^2/bdry ===
 k=1 (cubes of dim 2), dims up to 4 for X; up to 4 for sd X
  X                low1:n  low2:n  upp2:n  upp3:n
  sd X             low1:n  low2:n  upp2:n  upp3:n
 k=2 (cubes of dim 3), dims up to 6 for X; up to 6 for sd X
  X                low3:n  low4:Y  upp4:Y  upp5:Y
  sd X             low3:Y  low4:Y  upp4:Y  upp5:Y
 k=3 (cubes of dim 4), dims up to 8 for X; up to 8 for sd X
  X                low5:Y  low6:Y  upp6:Y  upp7:Y
  sd X             low5:Y  low6:Y  upp6:Y  upp7:Y

=== X = Delta^3/bdry ===
 k=1 (cubes of dim 2), dims up to 4 for X; up to 4 for sd X
  X                low1:n  low2:n  upp2:n  upp3:n
  sd X             low1:n  low2:n  upp2:n  upp3:n
 k=2 (cubes of dim 3), dims up to 6 for X; up to 6 for sd X
  X                low3:n  low4:n  upp4:n  upp5:Y
  sd X             low3:n  low4:n  upp4:Y  upp5:Y
 k=3 (cubes of dim 4), dims up to 8 for X; up to 8 for sd X
  X                low5:n  low6:Y  upp6:Y  upp7:Y
  sd X             low5:Y  low6:Y  upp6:Y  upp7:Y
Shell cwd was reset to ~/twisted_arrow

D3/<012,123>                        k=1 X[...Y] sd[..YY] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<013,023>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<013,123>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[.Y.Y] sd[..YY] A:ok B:ok
D3/<023,123>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<0,1,2>                          k=1 X[.YYY] sd[YYYY] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<0,1,3>                          k=1 X[.YYY] sd[YYYY] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<0,1,23>                         k=1 X[.Y.Y] sd[..YY] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<0,2,3>                          k=1 X[.YYY] sd[YYYY] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<0,2,13>                         k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<0,3,12>                         k=1 X[...Y] sd[..YY] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<0,12,13>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<0,12,23>                        k=1 X[...Y] sd[..YY] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<0,13,23>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<1,2,3>                          k=1 X[.YYY] sd[YYYY] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<1,2,03>                         k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<1,3,02>                         k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<1,02,03>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<1,02,23>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<1,03,23>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<2,3,01>                         k=1 X[..YY] sd[..YY] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<2,01,03>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<2,01,13>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<2,03,13>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<3,01,02>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<3,01,12>                        k=1 X[...Y] sd[..YY] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<3,02,12>                        k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,02,03>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<01,02,12>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,02,13>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,02,23>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,02,123>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,03,12>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,03,13>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,03,23>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<01,03,123>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,12,13>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,12,23>                       k=1 X[...Y] sd[..YY] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,12,023>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,13,23>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<01,13,023>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<02,03,12>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<02,03,13>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<02,03,23>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<02,03,123>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<02,12,13>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<02,12,23>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<02,12,013>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<02,13,23>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<02,23,013>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<03,12,13>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<03,12,23>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<03,13,23>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[YYYY] sd[YYYY] A:ok B:ok
D3/<03,13,012>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<03,23,012>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<12,13,23>                       k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<12,13,023>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<12,23,013>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok
D3/<13,23,012>                      k=1 X[....] sd[....] A:ok B:ok  k=2 X[.YYY] sd[YYYY] A:ok B:ok

COUNTEREXAMPLES: 0
Shell cwd was reset to ~/twisted_arrow

D3/<0,1,23>    k=1  X:.Y.Y  X^op:..YY  sdX:..YY  sd(X^op):..YY  (sdX)^op:.Y.Y
D3/<0,1,23>    k=2  X:YYYY  X^op:YYYY  sdX:YYYY  sd(X^op):YYYY  (sdX)^op:YYYY
D3/<3,01,12>   k=1  X:...Y  X^op:...Y  sdX:..YY  sd(X^op):..YY  (sdX)^op:.Y.Y
D3/<3,01,12>   k=2  X:.YYY  X^op:.YYY  sdX:YYYY  sd(X^op):YYYY  (sdX)^op:YYYY
S^2            k=1  X:....  X^op:....  sdX:....  sd(X^op):....  (sdX)^op:....
S^2            k=2  X:.YYY  X^op:.YYY  sdX:YYYY  sd(X^op):YYYY  (sdX)^op:YYYY
S^3            k=1  X:....  X^op:....  sdX:....  sd(X^op):....  (sdX)^op:....
S^3            k=2  X:...Y  X^op:...Y  sdX:..YY  sd(X^op):..YY  (sdX)^op:.Y.Y
Shell cwd was reset to ~/twisted_arrow

S^4 k=2: X:.... sdX:....   A:ok B:ok  [0s]
S^4 k=3: X:.... sdX:....   A:ok B:ok  [0s]
Shell cwd was reset to ~/twisted_arrow

Command running in background with ID: b8kccs5dc. Output is being written to: /tmp/claude-session/tasks/b8kccs5dc.output. You will be notified when it completes. To check interim output, use Read on that file path.
Session cwd remains ~/twisted_arrow; directory changes made by the backgrounded command do not apply to subsequent commands.

X : .YYY 0s
---

File created successfully at: ~/twisted_arrow/notes/edgewise-higher.md (file state is current in your context — no need to Read it back)

X : .YYY 0s
sd: YYYY 65s
A: True  B: True

edgewise-higher.md
search.py
segal.py
?? notes/

The file ~/twisted_arrow/notes/edgewise-higher.md has been updated successfully. (file state is current in your context — no need to Read it back)

Switched to a new branch 'notes/edgewise-higher-segal'
bf4b43b Add notes and code on edgewise subdivision for higher Segal conditions
ad88309 Initialize repository

checked 22638 pairs (U subset V); failures: 0
with the naive choice t = min(S\U): 140 broken squares (e.g. ([0, 1, 2], (0,), set(), {0}, ('bot', None)))
Shell cwd was reset to ~/twisted_arrow

The file ~/twisted_arrow/notes/edgewise-higher.md has been updated successfully. (file state is current in your context — no need to Read it back)

8b54b9b Prove the retract lemma, completing both converse directions

File created successfully at: ~/twisted_arrow/notes/edgewise.tex (file state is current in your context — no need to Read it back)

/Library/TeX/texbin/pdflatex

exit=0
39

-rw-r--r--  1 user  staff  350924 Aug  1 07:07 edgewise.pdf
Output written on edgewise.pdf (6 pages, 350924 bytes).

f7033ff Write up both theorems in full
8b54b9b Prove the retract lemma, completing both converse directions
bf4b43b Add notes and code on edgewise subdivision for higher Segal conditions
?? .DS_Store