---- > [!definition] Definition. ([[multi-object coslice category]]) > Let $\mathsf{C}$ be a [[category]] and $\{ A_{j} \}_{j \in J}$ an indexed family of objects in $\mathsf{C}$. We define a 'multi-object' generalization $\mathsf{C}^{\{ A_{j} \}_{j \in J}}$ of [[coslice category]] as follows. > > An object in $\mathsf{C}^{\{ A_{j} \}_{j \in J}}$ is an indexed family of maps $\{ A_{j} \to Z_{} \}_{j \in J}$. > > A morphism between objects $\{A_{j} \to Z_{1} \}_{j \in J}$ and $\{A_{j} \to Z_{2} \}_{j \in J}$ is a family of commutative diagrams > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZAJgBoAGAXVJADcBDAGwFcYkQBBAfQCsQBfUuky58hFAEZSxanSat2ALS4SBQkBmx4CRAMzTZDFm0QhlxNcK1ii5UhMPyTIADouAFswDmMN6EFWojooACz2jsbsbp4+bvxcwDwABG5YYEkAUvwCsjBQPggooABmAE4QALZIdiA4EEhSIIz0AEYwjAAKItriIKVYXu44liBlldU0dUjEAaPlVYiNU4hkcpGmbtheFfQgNM1tnd02pv2Dw-yU-EA > \begin{tikzcd} > & & A_j \arrow[ldd] \arrow[rdd] & & \\ > \huge\{ & & & & \huge\}_{j \in J} \\ > & Z_1 \arrow[rr, "\sigma"'] & & Z_2 & > \end{tikzcd} > \end{document} > ``` > > where the *same* $\sigma \in \text{Hom}_{\mathsf{C}}(Z_{1},Z_{2})$ makes every diagram in the family commute. ^definition > [!intuition] Motivation. >The key idea regarding morphisms in the [[two-object coslice category]] is not so much the diagrammatic nature in which they are depicted, but the fact that one finds a *single* morphism $\sigma$ that makes both 'triangles' in the resulting diagram commute. That is the interpretation that is generalized here. ---- #### ---- #### 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 > ```