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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 18th 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea and definition Compactly assembled (∞,1)-categories are characterized among [[presentable (∞,1)-categories]] by the following equivalent conditions (due to Clausen and Lurie): * They are [[retracts]] in the (∞,1)-category of [[presentable (∞,1)-categories]] and [[left adjoints]] of [[compactly generated (∞,1)-categories]]. * Small [[(∞,1)-limits]] distribute over [[(∞,1)-filtered (∞,1)-colimits]]. * The [[colimit]] (∞,1)-functor $Ind(C)\to C$ admits a left adjoint. * See Theorem 2.2.11 in Nikolaus for a few other conditions. ## Related concepts * [[Efimov K-theory]] * [[compactly generated (∞,1)-category]] ## References * [[Thomas Nikolaus]], _Sheaves on manifolds_, [PDF](https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/sheaves-on-manifolds.pdf) <a href="https://ncatlab.org/nlab/revision/compactly+assembled+%28%E2%88%9E%2C1%29-category/1">v1</a>, <a href="https://ncatlab.org/nlab/show/compactly+assembled+%28%E2%88%9E%2C1%29-category">current</a>

   Created:

   Idea and definition

   Compactly assembled (∞,1)-categories are characterized among presentable (∞,1)-categories by the following equivalent conditions (due to Clausen and Lurie):

   • They are retracts in the (∞,1)-category of presentable (∞,1)-categories and left adjoints of compactly generated (∞,1)-categories.

   • Small (∞,1)-limits distribute over (∞,1)-filtered (∞,1)-colimits.

   • The colimit (∞,1)-functor $Ind(C)\to C$ admits a left adjoint.

   • See Theorem 2.2.11 in Nikolaus for a few other conditions.

   Related concepts

   • Efimov K-theory

   • compactly generated (∞,1)-category

   References

   • Thomas Nikolaus, Sheaves on manifolds, PDF

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 18th 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded an example. <a href="https://ncatlab.org/nlab/revision/compactly+assembled+%28%E2%88%9E%2C1%29-category/1">v1</a>, <a href="https://ncatlab.org/nlab/show/compactly+assembled+%28%E2%88%9E%2C1%29-category">current</a>

   Added an example.

   v1, current