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1. • CommentRowNumber1.
   • CommentAuthorPaoloPerrone
   • CommentTimeJul 2nd 2019
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   Author: PaoloPerrone
   Format: MarkdownItexAre there results about the existence of [[2-limit#lax|lax and oplax limits and colimits]] in [[(0,1)-category |locally posetal categories]] (such as [[Pos]] and [[Rel]])? I would for example imagine that all those limits exist in Pos. But what about, for example, the [[Loc|category of locales]] or [[Top|the one of topological spaces]], with their canonical 2-cells? Also, are there lax/oplax analogues of the facts that right adjoint preserve colimits, that monadic functors create them, and so on? Which results carry over? If this has been done, does anyone of you know where I can find this? (Is all of this obvious?)

   Are there results about the existence of lax and oplax limits and colimits in locally posetal categories (such as Pos and Rel)? I would for example imagine that all those limits exist in Pos. But what about, for example, the category of locales or the one of topological spaces, with their canonical 2-cells?

   Also, are there lax/oplax analogues of the facts that right adjoint preserve colimits, that monadic functors create them, and so on? Which results carry over?

   If this has been done, does anyone of you know where I can find this? (Is all of this obvious?)

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJul 4th 2019
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   Author: Mike Shulman
   Format: MarkdownItexLax and oplax limits in 2-categories (including (2,1)-categories) are actually special cases of [[weighted limits]]; see [[2-limit#lax]]. Therefore, all of enriched category theory applies to them, in particular: 1. Because $Pos$ is a complete and cocomplete closed monoidal category, it has all $Pos$-weighted limits and colimits, hence *a fortiori* all lax and colax ones. 2. $Pos$-enriched right adjoints preserve weighted limits, hence *a fortiori* lax and colax ones. 3. Similarly, $Pos$-enriched monadic functors create weighted limits, hence *a fortiori* lax and colax ones. And so on. The existence of lax and colax (and other weighted) limits and colimits in particular other examples like $Rel$, $Loc$, and $Top$ is a rather different kind of question. I believe $Loc$ is $Pos$-enriched complete and cocomplete, because its opposite $Frm$ is essentially a category of infinitary algebras. I am not sure about $Top$; the category of sober spaces is coreflective in $Loc$ and so probably inherits limits (although one would have to check that the adjunction is enriched), but for arbitrary spaces I'm not sure, although I wouldn't be surprised if it had them. However, $Rel$ doesn't I think have very many limits and colimits, although it sits inside $Pos$-$Prof$ which has rather more (e.g. [[collages]] therein are a lax colimit).

   Lax and oplax limits in 2-categories (including (2,1)-categories) are actually special cases of weighted limits; see 2-limit#lax. Therefore, all of enriched category theory applies to them, in particular:

   1. Because $Pos$ is a complete and cocomplete closed monoidal category, it has all $Pos$-weighted limits and colimits, hence a fortiori all lax and colax ones.
   2. $Pos$-enriched right adjoints preserve weighted limits, hence a fortiori lax and colax ones.
   3. Similarly, $Pos$-enriched monadic functors create weighted limits, hence a fortiori lax and colax ones.

   And so on.

   The existence of lax and colax (and other weighted) limits and colimits in particular other examples like $Rel$, $Loc$, and $Top$ is a rather different kind of question. I believe $Loc$ is $Pos$-enriched complete and cocomplete, because its opposite $Frm$ is essentially a category of infinitary algebras. I am not sure about $Top$; the category of sober spaces is coreflective in $Loc$ and so probably inherits limits (although one would have to check that the adjunction is enriched), but for arbitrary spaces I’m not sure, although I wouldn’t be surprised if it had them. However, $Rel$ doesn’t I think have very many limits and colimits, although it sits inside $Pos$-$Prof$ which has rather more (e.g. collages therein are a lax colimit).

3. • CommentRowNumber3.
   • CommentAuthorPaoloPerrone
   • CommentTimeJul 5th 2019
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   Author: PaoloPerrone
   Format: MarkdownItexThank you, this is really helpful!

   Thank you, this is really helpful!