Pro-object

Summary

A pro-object in a category $\mathcal{C}$ is a formal cofiltered limit of objects of $\mathcal{C}$. Pro-objects form the category $\mathrm{Pro}(\mathcal{C})$, which is the dual of the Ind-completion of the opposite category. They appear in algebraic geometry, in shape theory and in the étale homotopy type. There a limit that does not exist in $\mathcal{C}$ is kept as a formal system. The category of profinite sets, that is of Stone spaces, is equivalent to the pro-objects in finite sets.