---
tag: GAFQDGF6FM
kind: concept
name: Pro-object
msc: [18A30]
sources:
  - url: https://en.wikipedia.org/wiki/Ind-completion
---

## 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|1DKFWGWHH5]] 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.

## Explanations

### L1 human

A pro-object is a system of ever finer approximations, kept as a whole instead of being replaced by its limit.
The finite groups $\Z / p^n \Z$, linked by the reduction maps, form a pro-object in finite groups.
Its limit is the group of $p$-adic integers, but the pro-object remembers the finite stages themselves.