---
tag: 1DKFWGWHH5
kind: concept
name: Ind-completion
msc: [18A30, 18C35]
sources:
  - url: https://en.wikipedia.org/wiki/Ind-completion
same_as:
  wikidata: Q134469743
  wikipedia: Ind-completion
---

## Summary

The Ind-completion $\mathrm{Ind}(\mathcal{C})$ of a category $\mathcal{C}$ is the category of formal [[filtered|975PMY4FG8]] colimits of objects of $\mathcal{C}$.
Its objects are called ind-objects.
It is the free way to add filtered colimits to $\mathcal{C}$, and it can be realized inside the category of [[presheaves|AND2WHZC8P]] on $\mathcal{C}$.
For example, the Ind-completion of finite sets is the category of all sets.

## Explanations

### L1 human

Some objects are best seen as limits of growing chains of simpler objects.
An infinite set is the union of its finite subsets, and an infinite-dimensional vector space is the union of its finite-dimensional subspaces.
The Ind-completion adds such unions formally, even when the original category does not contain them.
So starting from finite-dimensional vector spaces, the Ind-completion gives all vector spaces.