From cc7edc1ce5f1ddaca315e81e756318cc249f6f20 Mon Sep 17 00:00:00 2001 From: ykawase5048 Date: Thu, 9 Apr 2026 19:54:59 +0900 Subject: [PATCH 1/2] The walking parallel pair has sifted colimits --- .../004_property-assignments/walking_pair.sql | 6 ++++++ .../walking_parallel_pair_sifted_colimit.pdf | Bin 0 -> 150805 bytes 2 files changed, 6 insertions(+) create mode 100644 static/pdf/walking_parallel_pair_sifted_colimit.pdf diff --git a/database/data/004_property-assignments/walking_pair.sql b/database/data/004_property-assignments/walking_pair.sql index 3fad7543..19991150 100644 --- a/database/data/004_property-assignments/walking_pair.sql +++ b/database/data/004_property-assignments/walking_pair.sql @@ -70,4 +70,10 @@ VALUES 'pullbacks', FALSE, 'The two morphisms $a,b : 0 \rightrightarrows 1$ have no pullback, since it would have to consist of identities $0 \leftarrow 0 \rightarrow 0$, but $a \neq b$.' +), +( + 'walking_pair', + 'sifted colimits', + TRUE, + 'A proof can be found 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mfIy;v%K!U+oA}>8fGuhb9Gu)7j7{KJSvcV+D8%H&;r<^Y3bp3| literal 0 HcmV?d00001 From ce46c37ca5d13295adf2b1adb3e2b6ad81ccc012 Mon Sep 17 00:00:00 2001 From: ykawase5048 Date: Thu, 9 Apr 2026 20:07:03 +0900 Subject: [PATCH 2/2] Add TeX source --- .../walking_parallel_pair_sifted_colimit.tex | 112 ++++++++++++++++++ 1 file changed, 112 insertions(+) create mode 100644 static/pdf/walking_parallel_pair_sifted_colimit.tex diff --git a/static/pdf/walking_parallel_pair_sifted_colimit.tex b/static/pdf/walking_parallel_pair_sifted_colimit.tex new file mode 100644 index 00000000..9bfb0fbc --- /dev/null +++ b/static/pdf/walking_parallel_pair_sifted_colimit.tex @@ -0,0 +1,112 @@ +\documentclass[a4paper,12pt,reqno]{amsart} +\usepackage[utf8]{inputenc} +\usepackage[top=25truemm,bottom=25truemm,left=20truemm,right=20truemm]{geometry} + +\usepackage{amsmath, amssymb, amsfonts, mathtools} +\usepackage{tikz-cd} + +\title{The walking parallel pair has sifted colimits} +\author{Yuto Kawase} +\date{\today} + +\begin{document} +\maketitle + +Let $D\colon\mathcal{C}\to \{u,v\colon 0 \rightrightarrows 1\}$ be a functor from a sifted category. +If the object $1$ is not contained in the image under $D$, the object $0$ gives a colimit of $D$ because the sifted category $\mathcal{C}$ is connected. +In what follows, we suppose that there is an object $c_0\in\mathcal{C}$ such that $D(c_0)=1$. + +We first claim that for every object $c\in\mathcal{C}$ such that $D(c)=0$, there is a morphism $f\colon c \to x$ with $D(x)=1$; moreover, which of $u$ and $v$ such a morphism is sent to by $D$ is independent of the choice of $f$. +The existence of $f$ is easy. +Indeed, since $\mathcal{C}$ is sifted, there is a cospan $c \rightarrow x \leftarrow c_0$, and $D(x)=1$ follows from $D(c_0)=1$. + +To show the independence of the value of $D(f)$, suppose that there are morphisms $f\colon c \to x$ and $g\colon c\to y$ such that $D(f)=u$ and $D(g)=v$. +Since $\mathcal{C}$ is sifted, there is a cospan consisting of $p\colon x \rightarrow z$ and $q\colon z \leftarrow y$. +Since $\mathcal{C}$ is sifted again, two cospans $(p\circ f, q\circ g)$ and $(\mathrm{id}_c, \mathrm{id}_c)$ are connected to each other, that is, there are a zigzag consisting of $s_i\colon d_{i-1} \rightarrow e_i$ and $t_i\colon e_i \leftarrow d_i$ $(1 \le i \le n)$ and parallel pairs $(l_i,r_i)\colon c \rightrightarrows d_i$ $(0 \le i \le n)$ such that $d_0=z$, $l_0=p\circ f$, $r_0=q\circ g$, $d_n=c$, $l_n=r_n=\mathrm{id}_c$, $s_i \circ l_{i-1} = t_i \circ l_i$, and $s_i \circ r_{i-1} = t_i \circ r_i$ $(1\le i\le n)$. +\begin{equation*} + \begin{tikzcd}[column sep = 1.6em] + &[-1.2em] + & + & + & + & + c + \ar[dllll,bend right=25,shift right=1,"l_0 = p\circ f"',pos=0.8] + \ar[dllll,bend right=25,shift left=1,"r_0 = q\circ g",pos=0.8] + \ar[dll,bend right=10,shift right=1,"l_1"'] + \ar[dll,bend right=10,shift left=1,"r_1"] + \ar[d,shift right=1,"l_2"'] + \ar[d,shift left=1,"r_2"] + \ar[drrrr,bend left=25,equal] + \ar[drr,bend left=10,shift right=1,"l_{n-1}"{below left=-2}] + \ar[drr,bend left=10,shift left=1,"r_{n-1}"] + & + & + & + & + &[-1.2em] + \\[30pt] + z + \ar[r,equal] + & + d_0 + & + & + d_1 + & + & + d_2 + & + \cdots + & + d_{n-1} + & + & + d_n + & + c + \ar[l,equal] + \\ + & + & + e_1 + \ar[ul,leftarrow,"s_1"] + \ar[ur,leftarrow,"t_1"'] + & + & + e_2 + \ar[ul,leftarrow,"s_2"] + \ar[ur,leftarrow,"t_2"'] + & + & + & + & + e_n + \ar[ul,leftarrow,"s_n"] + \ar[ur,leftarrow,"t_n"'] + & + & + \end{tikzcd} +\end{equation*} +Then, the equality $D(t_1) \circ D(l_1) = D(s_1)\circ D(l_0) = u$ implies that either $D(l_1)=u$ or $D(t_1)=u$ holds, while $D(t_1) \circ D(r_1) = D(s_1)\circ D(r_0) = v$ implies that either $D(r_1)=v$ or $D(t_1)=v$ holds. +However, the only possible combination is $D(l_1)=u$ together with $D(r_1)=v$, and by repeating this argument, we have $D(l_n)=u$ and $D(r_n)=v$, which is a contradiction. + +By the claim, each object $c\in\mathcal{C}$ can be classified exclusively into the following three cases: +\begin{enumerate} + \item + $D(c)=1$; + \item + $D(c)=0$ and there is a morphism from itself sent to $u$ by $D$; + \item + $D(c)=0$ and there is a morphism from itself sent to $v$ by $D$. +\end{enumerate} +Now, we have a cocone $(\alpha_c\colon D(c) \to 1)_{c\in\mathcal{C}}$ over $D$ by letting $\alpha_c\coloneqq \mathrm{id}_1$ if $c$ is classified into the first case, $\alpha_c\coloneqq u$ for the second case, and $\alpha_c\coloneqq v$ for the third case. +Moreover, this is a unique cocone over $D$: +If $\beta$ is another cocone, its vertex should be $1$ by the existence of $c_0$. +If $c\in\mathcal{C}$ is classified into the first case, $\beta_c$ should be the identity. +For the second case, taking a morphism $f\colon c\to x$ such that $D(f)=u$, we can obtain $\beta_c = \beta_x \circ D(f) = D(f) = u$. +Similarly, we have $\beta_c = v$ for the third case. +This concludes $\beta=\alpha$, and since there is no non-trivial endomorphism on the vertex $1$, $\alpha$ gives a colimit. + + +\end{document}