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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 22nd 2016
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   Author: Mike Shulman
   Format: MarkdownItexWhat are some examples of $(\infty,1)$-categories that are complete and cocomplete but not locally presentable? For 1-categories there are plenty of examples, such as: * Any category [[topological concrete category|topological]] over $Set$ (or over any bicomplete category); for example, [[Top]]. * Some categories of algebras for inaccessible monads on $Set$ (or other bicomplete category); for example, [[Sup]]. Of course, these examples can also be regarded as $(\infty,1)$-categories, but I'd like some examples that are "honestly $\infty$". I don't know of direct $\infty$-analogues of categories like $Top$ and $Sup$.

   What are some examples of $(\infty,1)$-categories that are complete and cocomplete but not locally presentable? For 1-categories there are plenty of examples, such as:

   • Any category topological over $Set$ (or over any bicomplete category); for example, Top.
   • Some categories of algebras for inaccessible monads on $Set$ (or other bicomplete category); for example, Sup.

   Of course, these examples can also be regarded as $(\infty,1)$-categories, but I’d like some examples that are “honestly $\infty$”. I don’t know of direct $\infty$-analogues of categories like $Top$ and $Sup$.

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeFeb 23rd 2016
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   Author: Zhen Lin
   Format: MarkdownItexI am informed that the $(\infty, 1)$-category corresponding to $Top$ with the Hurewicz model structure is complete, cocomplete, and not locally presentable.

   I am informed that the $(\infty, 1)$-category corresponding to $Top$ with the Hurewicz model structure is complete, cocomplete, and not locally presentable.

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 23rd 2016
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   Author: Dmitri Pavlov
   Format: MarkdownItexWhat about the ∞-category of Grothendieck ∞-toposes? It is complete and cocomplete, but not locally presentable. I think it can also be seen as an ∞-version of Top: there is a functor from simplicial locales to Grothendieck ∞-toposes: given a simplicial locale L, take the ∞-colimit over Δ^op of the associated localic ∞-toposes. This functor is probably a localization of some sort (though not necessarily reflective or coreflective).

   What about the ∞-category of Grothendieck ∞-toposes? It is complete and cocomplete, but not locally presentable.

   I think it can also be seen as an ∞-version of Top: there is a functor from simplicial locales to Grothendieck ∞-toposes: given a simplicial locale L, take the ∞-colimit over Δ^op of the associated localic ∞-toposes. This functor is probably a localization of some sort (though not necessarily reflective or coreflective).

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 24th 2016
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   Author: Mike Shulman
   Format: MarkdownItexThanks, those are good examples. Are there any general families of examples? I suppose one fairly trivial one is "the opposite of any locally presentable $(\infty,1)$-category".

   Thanks, those are good examples. Are there any general families of examples?

   I suppose one fairly trivial one is “the opposite of any locally presentable $(\infty,1)$-category”.

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeFeb 24th 2016
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   Author: Marc Hoyois
   Format: MarkdownItexPro-objects in presentable ∞-categories is a family of locally small examples. Their opposites are not presentable either.

   Pro-objects in presentable ∞-categories is a family of locally small examples. Their opposites are not presentable either.

6. • CommentRowNumber6.
   • CommentAuthorZhen Lin
   • CommentTimeFeb 24th 2016
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   Author: Zhen Lin
   Format: MarkdownItexAt least, provided you don't take pro-objects in a small presentable category (a.k.a. complete lattice).

   At least, provided you don’t take pro-objects in a small presentable category (a.k.a. complete lattice).