Presheaf

Summary

A presheaf on a category $\mathcal{C}$ is a contravariant functor from $\mathcal{C}$ to the category of sets. Presheaves on $\mathcal{C}$ and the natural transformations between them form a category that contains $\mathcal{C}$ and has all limits and colimits. The name comes from topology, where a presheaf on a space assigns data to open sets with restriction maps to smaller open sets.