---
title: "Adequate subcategory"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Adequate_subcategory"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T13:38:46.918580+00:00"
instance: "kb-cron"
---

In category theory, a branch of mathematics, an adequate subcategory of a category X is an analog of a dense subspace in topology for presheaves: namely, a subcategory 
  
    
      
        i
        :
        A
        ↪
        X
      
    
    {\displaystyle i:A\hookrightarrow X}
  
 such that the restriction of the Yoneda embedding 
  
    
      
        X
        ↪
        
          P
        
        (
        X
        )
      
    
    {\displaystyle X\hookrightarrow \mathbf {P} (X)}
  
 along 
  
    
      
        i
      
    
    {\displaystyle i}
  
 is still fully faithful. The notion was introduced by Isbell in 1960. Note some authors use the term dense subcategory for this notion, although it can mean a different thing in other contexts.


== References ==

John Isbell, Adequate subcategories , Illinois J. Math. 4 (1960) pp. 541–552. [1]


== Further reading ==
Dense subcategory