---
tag: FC7VB4WSG0
kind: concept
name: Reflective subcategory
msc: [18A40]
sources:
  - url: https://en.wikipedia.org/wiki/Reflective_subcategory
same_as:
  wikidata: Q6101219
  wikipedia: Reflective subcategory
---

## Summary

A reflective subcategory is a [[full subcategory|V3BFGKBJF7]] whose inclusion functor has a left [[adjoint|TZHF6ZM92S]], called the reflector.
The reflector sends each object to its best approximation inside the subcategory.
Abelianization, [[sheafification|2HYK3KBQNA]] and the [[Stone–Čech compactification|D9MWP5C85W]] are reflectors.
Reflective subcategories inherit limits from the ambient category, and colimits through the reflector.

## Explanations

### L1 human

Abelian groups form a part of the world of all groups.
Every group $H$ has a best abelian approximation, the quotient $H/[H, H]$ by its commutator subgroup.
Every homomorphism from $H$ to an abelian group passes through this quotient in exactly one way.
A reflective subcategory is a part of a category where every object has such a best approximation.