---- > [!definition] Definition. ([[diagram]]) > Let $\mathsf{I}$ be an 'indexing category'. Let $\mathsf{C}$ be a [[category]]. A **($\mathsf{I}$-shaped)** **diagram** in $\mathsf{C}$ is a (covariant) [[covariant functor|functor]] $\mathsf{I} \to \mathsf{C}$. > The idea is that $\mathsf{I}$ 'indexes both morphisms and objects'. The actual objects and morphisms in $\mathsf{I}$ do not matter — only their interrelationships. > > For example, if $\mathsf{I}=\square$ is the [[category]] > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12AjJhhmHCAC+pdJlz5CKAIzkqtRizacefAcNEgM2PASJlp8+s1aIO3Xv0EixOyUVmHqxpWZWX1Q+TCgBzeESgAGYAThAAtkhkIDgQSNI2IKERUdSxSABMicmRiBlpcYgAzNlhubIxhSUUQkA > \begin{tikzcd} > \bullet \arrow[r] \arrow[d] & \bullet \arrow[d] \\ > \bullet \arrow[r] & \bullet > \end{tikzcd} > \end{document} > ``` > > then a [[covariant functor|diagram]] $\mathsf{\square} \to \mathsf{C}$ is precisely the data of a commutative square in $\mathsf{C}$. Sometimes the object of $\mathsf{C}$ to which $i$ in $\mathsf{I}$ gets assigned under the [[covariant functor|functor]] is denoted $A_{i}$. > [!basicexample] > A diagram $D:S \to \mathsf{C}$ indexed by a [[poset]] $S$ is an assignment of an object in $\mathsf{C}$ to each element of $S$ and a morphism in $\mathsf{C}$ to each relation $s \leq s'$. [[the category of diagrams on poset is equivalent to that of sheaves on its Alexandrov topological space|This is called a sheaf on the poset]] $S$. A common example would be a [[cellular sheaf]], such as a sheaf on a [[network|graph]]. > > If $S$ is in fact a [[filtered poset]], then $D$ is called a **directed system**. An **inverse system** would be the contravariant version of this: a functor $\mathsf{I}^{\text{op}} \to \mathsf{C}$. Note that the contexts where the term 'sheaf' is used — like in the case of graphs — rarely involve directed systems. For instance, a sheaf on a graph is a directed system iff the graph is fully connected. Directed systems common, though, and are friendly to [[categorical colimit|(co)]][[categorical limit|limits]].(Hmm but presheaves in general are prototypical inverse systems) ^basic-example ---- #### ---- #### 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 > ```