Slice category

Summary

The slice category $\mathcal{C}/X$ of a category $\mathcal{C}$ over an object $X$ has the morphisms into $X$ as objects. Its morphisms are the commuting triangles over $X$. It is the comma category $(1_{\mathcal{C}} \downarrow X)$. Slice categories describe objects "over a base": families of sets indexed by $X$, spaces over a base space, and schemes over a base scheme. The dual notion is the coslice category $X/\mathcal{C}$ of morphisms out of $X$, such as the category of pointed sets.