> Let $\sim$ be an [[equivalence relation]] on a set $A$. > [!theorem]+ Theorem. ([[universal property of quotient sets]]) > The [[quotient set]] $A / \sim$ is [[universal property|universal]] with respect to the property of mapping $A$ to a set in such a way that equivalent elements have the same image. > > *Explicitly*, if we take the [[coslice category|coslice]] [[subcategory]] $\mathsf{C}$ whose general object is a set-function $A \xrightarrow{\varphi}Z$ satisfying $a' \sim a'' \implies \varphi(a')=\varphi(a'')$ and whose morphisms are commutative diagrams [^1] > > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRQAGUsKq1GLNgC0A+sW68QGbHgJEATGIn1mrRCHmbuEmFADm8IqABmAJwgBbJKJA4ISMpP1sAOn709mgAFlgKSnaOLohuHkjaIAx0AEYwDAAK-OpCIPZYFiE4INR60oYBQaHhJjxRzl7U8YiJZQYgAdgWTnSmXEA > \begin{tikzcd} > Z_1 \arrow[rr, "\sigma"] & & Z_2 \\ > & A \arrow[lu, "\varphi_1"] \arrow[ru, "\varphi_2"'] & > \end{tikzcd} > \end{document} > ``` > then the [[quotient set|canonical projection]] $A \xrightarrow{\pi} A / \sim$ is an [[terminal object|initial object]] in $\mathsf{C}$. > i.e., there is a unique commutative diagram >```tikz \usepackage{tikz-cd} \begin{document} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEB6AHW+wFsQAX1LpMufIRQAmclVqMWbAFrDRIDNjwEiARlK759Zq0Qdh8mFADm8IqABmAJwiDEskDghIyIBnQAjGAYABXFtKT8YBxwQamMl140LDVHFzcPLyR9BRM2XnonNAALVOp-INDwyTYnLGsS2JF01x9qbMRchNMQXggaGCcGLDAYYEK6YrKhCyEgA \begin{tikzcd} A/\sim \arrow[rr, "\overline{\varphi}"] & & Z \\ & A \arrow[lu, "\pi"] \arrow[ru, "\varphi"'] & \end{tikzcd} \end{document} >``` > that is, a unique function $\overline{\varphi}$ making this diagram commute. $\overline{\varphi}$ is (well-)defined $\overline{\varphi}([a])=\varphi(a)$. ^theorem > [!note]+ Remark. > Observe the various implicitries in the original assertion. It did not say what the [[category]] is, nor that we should pay attention specifically to [[terminal object|initial objects]] rather than [[terminal object|final objects]]. Moreover, we also now know that the solution to the universal problem is not *really* $A / \sim$, but actually $A \xrightarrow{\pi} A / \sim$. > > But these are all clear from context. The canonical projection is really the only sensible set-morphism to define from $A$ to $A / \sim$. And the final object in $\mathsf{C}$ is the empty function (?), so surely that is not what is being referred to. ^note > [!proof]+ Proof. ([[universal property of quotient sets]]) > Let $[a] \in A / \sim$ be arbitrary. If the diagram is indeed going to commute, then necessarily $\overline{\varphi}([a])=\varphi(a);$ > this tells us that $\overline{\varphi}$ is unique so long as it is [[well-defined]] in the sense that $[y]=[x]$ $\implies$ $\overline{\varphi}([y])=\overline{\varphi}([x])$. > > This it certainly is, for $[y]=[x] \implies y \sim x \implies \varphi(y)=\varphi(x)$ > by definition of objects $\varphi$ in $\mathsf{C}$. ^proof #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch02