---- > [!definition] Definition. ([[Cartesian square]]) > We say a commutative [[diagram]] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRQBGclVqMWbAELdeIDNjwEiZYePrNWiEAGE5fJYKKj11TVJ0ARbuJhQA5vCKgAZgCcIAWyRkQOCCRhHjdPH0Q-AKQAJhCQD28Y6ijEAGY4hPDRf0C0rgouIA > \begin{tikzcd} > A \arrow[r] \arrow[d] & B \arrow[d] \\ > C \arrow[r] & D > \end{tikzcd} > \end{document} > ``` > is **Cartesian** if $A$ is the [[categorical pullback|pullback]] of $C \to D \leftarrow B$: $A \cong B \times_{D} C.$ In other words, a diagram is Cartesian if $B \times_{D} C$ exists and the [[universal property|induced]] morphism $A \xrightarrow{\ex !} B \times _D C$ is an [[isomorphism]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY 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 > ```