---
tag: V3BFGKBJF7
kind: concept
name: Full subcategory
msc: [18A05]
specializes: [D8C0EYD9ED]
sources:
  - url: https://en.wikipedia.org/wiki/Subcategory
---

## Summary

A full subcategory of a [[category|A2DE66937D]] $\mathcal{C}$ is a [[subcategory|D8C0EYD9ED]] that keeps every morphism of $\mathcal{C}$ between any two of its objects.
So it is fixed by its class of objects alone.
Its inclusion functor is [[fully faithful|GRG8WEBSH5]].
Most categories of "special" objects, such as abelian groups among groups or Hausdorff spaces among topological spaces, are full subcategories.

## Explanations

### L1 human

To form a full subcategory, choose some of the objects and keep all arrows between them.
Nothing about the arrows is decided: once the objects are chosen, the arrows come along.
For example, the finite sets with all functions between them form a full subcategory of the category of sets.
Keeping all sets but only the injective functions gives a subcategory that is not full.