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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJan 28th 2015
   • (edited Jan 28th 2015)
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   Author: Dmitri Pavlov
   Format: MarkdownItexHovey in Proposition 2.4.2 of his book “Model categories” shows that compact topological spaces are compact objects with respect to closed T_1-inclusions. Furthermore, as explained in the errata to the book, the indiscrete space consisting of two points is not a small object in the category of topological spaces. Clearly, both of these statements need subtle separation properties (or absence thereof), e.g., T_1 or indiscreteness, etc. Thus it is not unreasonable to expect different results for the case of locales. For example, do we know what the compact objects in the category of locales are? What about small objects? Do we know that compact locales (in the sense of finite subcovers) are compact objects with respect to monomorphisms of locales? The category of locales is not locally presentable, which implies that small objects don't generate all objects under small colimits.

   Hovey in Proposition 2.4.2 of his book “Model categories” shows that compact topological spaces are compact objects with respect to closed T_1-inclusions.

   Furthermore, as explained in the errata to the book, the indiscrete space consisting of two points is not a small object in the category of topological spaces.

   Clearly, both of these statements need subtle separation properties (or absence thereof), e.g., T_1 or indiscreteness, etc.

   Thus it is not unreasonable to expect different results for the case of locales.

   For example, do we know what the compact objects in the category of locales are? What about small objects?

   Do we know that compact locales (in the sense of finite subcovers) are compact objects with respect to monomorphisms of locales?

   The category of locales is not locally presentable, which implies that small objects don’t generate all objects under small colimits.

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeJan 28th 2015
   • (edited Jan 28th 2015)
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   Author: Zhen Lin
   Format: MarkdownItexWho says the category of frames is locally presentable? It's monadic over $\mathbf{Set}$, but I see no reason for the monad to be accessible.

   Who says the category of frames is locally presentable? It’s monadic over $\mathbf{Set}$, but I see no reason for the monad to be accessible.

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJan 28th 2015
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   Author: Dmitri Pavlov
   Format: MarkdownItex@ZhenLin: I thought that frames are locally presentable because then can be specified using an algebraic theory and algebras over an algebraic theory form a locally presentable category.

   @ZhenLin: I thought that frames are locally presentable because then can be specified using an algebraic theory and algebras over an algebraic theory form a locally presentable category.

4. • CommentRowNumber4.
   • CommentAuthorThomas Holder
   • CommentTimeJan 28th 2015
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   Author: Thomas Holder
   Format: MarkdownItexIn Johnstone's _Stone spaces_ there is a proof (p.57) that **Loc** is not well-powered hence **Frm** can't be locally presentable (Adamek-Rosicky p. 46). **Frm** has an infinitary operation hence is not algebraic for a (finitary) Lawvere theory though it is equationally presentable and has a free functor $Set\to$**Frm**.

   In Johnstone’s Stone spaces there is a proof (p.57) that Loc is not well-powered hence Frm can’t be locally presentable (Adamek-Rosicky p. 46). Frm has an infinitary operation hence is not algebraic for a (finitary) Lawvere theory though it is equationally presentable and has a free functor $Set\to$Frm.

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJan 28th 2015
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   Author: Mike Shulman
   Format: MarkdownItexRight, neither Frm nor Loc are locally presentable. The definition of compact locale ought to imply that they are small wrt *open* inclusions of locales, but I don't know any more than that. My guess based on nothing at all is that there are not many compact objects in Loc.

   Right, neither Frm nor Loc are locally presentable. The definition of compact locale ought to imply that they are small wrt open inclusions of locales, but I don’t know any more than that. My guess based on nothing at all is that there are not many compact objects in Loc.

6. • CommentRowNumber6.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJan 28th 2015
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   Author: Dmitri Pavlov
   Format: MarkdownItex@ThomasHolder: Categories of algebras over infinitary algebraic theories with at most a set of operations are also locally presentable. I guess in this case there is a proper class of operations, which explains the failure of local presentability.

   @ThomasHolder: Categories of algebras over infinitary algebraic theories with at most a set of operations are also locally presentable. I guess in this case there is a proper class of operations, which explains the failure of local presentability.

7. • CommentRowNumber7.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJan 28th 2015
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexI removed the wrong claim about local presentability of frames.

   I removed the wrong claim about local presentability of frames.

8. • CommentRowNumber8.
   • CommentAuthorThomas Holder
   • CommentTimeJan 28th 2015
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   Author: Thomas Holder
   Format: Text@DmitriPavlov: indeed simply blaming the non-finitariness of the theory was thoughtless of me.

   @DmitriPavlov: indeed simply blaming the non-finitariness of the theory was thoughtless of me.