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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 31st 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Definition A [[subcategory]] $C$ of an [[accessible category]] $D$ is __accessible__ if $C$ is an [[accessible category]] and the [[inclusion functor]] $C\to D$ is an [[accessible functor]]. Some authors, e.g., [[Lurie]] in [[Higher Topos Theory]] and [[Adámek]]–[[Rosický]], require accessible subcategories to be [[full|full subcategory]]. Some authors, e.g., [[Adámek]]–[[Rosický]] in [[Locally Presentable and Accessible Categories]] merely require $C$ to be accessible, referring to the stronger notion as an __accessibly embedded accessible subcategory__. ## Properties Accessible subcategories are [[idempotent complete]] and are closed under set-indexed intersections. ## Related concepts * [[accessible category]], [[accessible functor]] ## References See, for example, Definition 5.4.7.8 in * [[Jacob Lurie]], [[Higher Topos Theory]]. <a href="https://ncatlab.org/nlab/revision/accessible+subcategory/1">v1</a>, <a href="https://ncatlab.org/nlab/show/accessible+subcategory">current</a>

   Created:

   Definition

   A subcategory of an accessible category is accessible if is an accessible category and the inclusion functor is an accessible functor.

   Some authors, e.g., Lurie in Higher Topos Theory and Adámek–Rosický, require accessible subcategories to be full subcategory.

   Some authors, e.g., Adámek–Rosický in Locally Presentable and Accessible Categories merely require to be accessible, referring to the stronger notion as an accessibly embedded accessible subcategory.

   Properties

   Accessible subcategories are idempotent complete and are closed under set-indexed intersections.

   Related concepts

   • accessible category, accessible functor

   References

   See, for example, Definition 5.4.7.8 in

   • Jacob Lurie, Higher Topos Theory.

   v1, current