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