Diagram

Summary

A diagram of shape $J$ in a category $\mathcal{C}$ is a functor from an index category $J$ to $\mathcal{C}$. It picks out a family of objects and morphisms of $\mathcal{C}$, arranged in the pattern given by $J$, with the compositions that $J$ prescribes. Diagrams are the input of limits and colimits: products, equalizers, pullbacks and their duals are limits and colimits of diagrams of particular shapes.