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1\documentclass[12pt,a4paper]{amsart}
2\usepackage[tt=false]{libertine}
3\usepackage{microtype}
4\usepackage{amsmath,amssymb,amsfonts,mathtools}
5\usepackage{tikz-cd}
6\usetikzlibrary{arrows.meta}
7
8\title{The synthetic Sierpi\'nski cone}
9\author{Jonathan Sterling}
10\thanks{This is an extended version of the conference paper, \emph{When is the partial map classifier a Sierpi\'nski cone?} by Pugh and Sterling, which was presented at LICS 2025. Relative to \emph{op.\ cit.}, I present some new results including a strengthening of the Sierpi\'nski completeness theorem to apply to completely untruncated spaces.}
11
12\NewDocumentCommand\Spx{m}{\Delta^{#1}}
13\NewDocumentCommand\Ord{m}{\mathbf{#1}}
14\NewDocumentCommand\One{}{\Ord{1}}
15\NewDocumentCommand\Zero{}{\Ord{0}}
16
17\NewDocumentCommand\II{}{\mathbb{I}}
18\NewDocumentCommand\XX{}{\mathbb{X}}
19
20\begin{document}
21\maketitle
22\tableofcontents
23
24\tikzcdset{
25 open immersion/.style={
26 -{Triangle[open]}, hook
27 },
28 closed immersion/.style={
29 -{Triangle[fill=black]}, hook
30 }
31}
32
33\section{Introduction}
34
35In both category theory and domain theory, there is a remarkable coincidence involving the Sierpi\'nski object $\II\equiv \{ 0\hookrightarrow 1 \}$. A figure $\II\xrightarrow{\alpha}\mathbb{C}$ represents an arrow $\alpha \colon \alpha(0)\to \alpha(1)$ in $\mathbb{C}$; on the other hand, an arrow $\mathbb{C}\xrightarrow{\chi}\II$ represents an ``open'' subspace of $\mathbb{C}$ given by the preimage of $1\in \II$.
36
37The two-handedness of $\II$ extends to an identification between the \emph{Sierpi\'nski cone} and the \emph{partial map classifier} of any space $\XX$, which we illuminate below. The Sierpi\'nski cone $\XX_\bot$ of a space $\XX$ is the following co-comma square, which we may compute by means of a pushout:
38\[
39 \begin{tikzcd}
40 \XX\ar[r,equals]\ar[d] & \XX \ar[d,open immersion,"\iota_\XX"]\\
41 \One\arrow[r,swap,closed immersion,"\bot_\XX"]\ar[ur,phantom,"{\Uparrow}\gamma_\XX"] & \XX\mathrlap{_\bot}
42 \end{tikzcd}
43 \qquad
44 \begin{tikzcd}
45 \XX\ar[r,closed immersion,"{(0,\XX)}"]\ar[d] & \II\times\XX \ar[d,"\gamma_\XX"description] & \XX\ar[l,open immersion,swap,"{(1,\XX)}"]\ar[dl,sloped,swap,open immersion,"\iota_\XX"] \\
46 \One\ar[r,swap,closed immersion,"\bot_\XX"] & \XX\mathrlap{_\bot}\ar[ul,phantom,very near start,"\ulcorner"]
47 \end{tikzcd}
48\]
49
50The (open) partial map classifier $\eta\colon \XX\hookrightarrow L(\XX)$ is, on the other hand, the partial product of the open embedding $\{1\}\hookrightarrow \XX$ and classifies spans
51\[
52 \begin{tikzcd}
53 \XX & U\ar[l]\ar[r,open immersion]& \mathbb{Y}
54 \end{tikzcd}
55\]
56where $U$ is an \emph{open} subspace of $\mathbb{Y}$, \emph{i.e.}\ a pullback of $\{1\}\hookrightarrow \II$ in the such that every such span arises in the context of a unique pullback square:
57\[
58 \begin{tikzcd}
59 U\ar[r]\ar[d,open immersion]\ar[dr,phantom,very near start,"\lrcorner"] & \XX\ar[d,open immersion,"\eta_\XX"]\\
60 \mathbb{Y}\ar[r,densely dashed,swap,"\exists!"] & L(\XX)
61 \end{tikzcd}
62\]
63
64Now, it so happens that we may define a comparison map $\sigma_\XX\colon \XX_\bot\to L(\XX)$ using \emph{either} the universal property of the Sierpi\'nski cone or the universal property of the partial map classifier. In the former case, we construct a lax square (left) and in the latter case we observe that $\iota_\XX\colon \XX\hookrightarrow \XX_\bot$ is an open embedding and construct a partial map (right):
65
66\[
67 \begin{tikzcd}
68 \XX\ar[r,equals]\ar[d] & \XX \ar[d,open immersion,"\eta_\XX"]\\
69 \One\arrow[r,swap,closed immersion,"\bot_\XX"]\ar[ur,phantom,"{\Uparrow}"] & L(X)
70 \\
71 |[gray]|\varnothing \ar[u,gray,open immersion]\ar[ur,gray,phantom,very near start,"\urcorner"] \ar[r,gray,-{Triangle[fill=gray]}, hook] & |[gray]|\XX \ar[u,swap,gray,open immersion,"\eta_\XX"]
72 \end{tikzcd}
73 \qquad
74 \begin{tikzcd}
75 \XX\ar[r,equals]\ar[d,open immersion,swap,"\iota_\XX"]\ar[dr,phantom,very near start,"\lrcorner"] & \XX\ar[d,open immersion,"\eta_\XX"]\\
76 \XX_\bot\ar[r,densely dashed,swap,"\exists!"] & L(\XX)
77 \end{tikzcd}
78\]
79
80The promised coincidence is that the induced comparison map $\XX_\bot\to L(\XX)$ is invertible. This property also extends to models of higher categories: when $L(X)$ is the open partial map classifier of a simplicial set,\footnote{We mean, $L(X)$ classifies partial maps whose support is classified by the nerve of the open embedding $\{1\}\hookrightarrow \{0\to 1\}$.} we actually do have $L(X) \cong \Spx{\Zero}\star X \equiv X_\bot$. It is therefore reasonable to say that the correspondence between classifying partial maps and representing mapping cylinders is part of the core logic--geometry duality shared by (higher) category theory, domain theory, topology, \emph{etc.}
81
82Surprisingly, the situation changes considerably when passing to synthetic notions of space---as in synthetic domain theory, synthetic topology, and synthetic (higher) category theory.
83
84
85\subsection{Foundations and preliminaries}
86
87In what follows, we work in the vernacular of univalent foundations---though, it is worth noting, none of our results depends on univalent universes. Unlike many works of synthetic mathematics, we do not globally assume any axioms aside from function extensionality; instead we locally assume structures satisfying various axioms where needed.
88
89\end{document}