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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 20th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexIn the article [[nice category of spaces]] (which I largely wrote, so any mistakes are probably mine), it is blithely asserted that the category of locales is extensive. I'm not saying I particularly doubt it, but is it true? If so, is there a reference for that?

   In the article nice category of spaces (which I largely wrote, so any mistakes are probably mine), it is blithely asserted that the category of locales is extensive. I’m not saying I particularly doubt it, but is it true? If so, is there a reference for that?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeAug 20th 2015
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   Author: Mike Shulman
   Format: MarkdownItexGood question; I thought it was true too, but I don't remember a reference. Maybe *Stone spaces*? I don't have it with me now to look.

   Good question; I thought it was true too, but I don’t remember a reference. Maybe Stone spaces? I don’t have it with me now to look.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 23rd 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI don't have Stone Spaces to hand either, but at least one person on the internet says it's true ([section 9](http://www.mta.ca/~cat-dist/catlist/1999/abstralggeo)).

   I don’t have Stone Spaces to hand either, but at least one person on the internet says it’s true (section 9).

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 23rd 2015
   • (edited Aug 23rd 2015)
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   Author: DavidRoberts
   Format: MarkdownItexWe should really have an entry [[Zariski category]]! Zhen Lin has extracted the definition [in this MO answer](http://mathoverflow.net/questions/185168/characterize-the-category-of-rings): ---- A **Zariski category** is a category $\mathcal{A}$ satisfying the following conditions: * $\mathcal{A}$ is cocomplete. * $\mathcal{A}$ has a strong generating set whose objects are finitely presentable and flatly codisjunctable. * Regular epimorphisms are universal i.e. stable under pullbacks. * The terminal object of $\mathcal{A}$ is finitely presentable and has no proper subobject. * Binary products of objects are co-universal i.e. stable under pushouts. * For any finite sequence of codisjunctable congruences $r_1, \ldots, r_n$ on any object with respect codisjunctors $d_1, \ldots, d_n$, we have $$ r_1 \vee^c \cdots \vee^c r_n = \mathrm{id}_{A \times A} \implies d_1 \vee \cdots \vee d_n = \mathrm{id}_A $$ where $\vee^c$ denotes the join in the lattice of congruences on $A$, while $\vee$ denotes the co-union of quotient objects of $A$.

   We should really have an entry Zariski category! Zhen Lin has extracted the definition in this MO answer:

   ---

   A Zariski category is a category $\mathcal{A}$ satisfying the following conditions:

   • $\mathcal{A}$ is cocomplete.
   • $\mathcal{A}$ has a strong generating set whose objects are finitely presentable and flatly codisjunctable.
   • Regular epimorphisms are universal i.e. stable under pullbacks.
   • The terminal object of $\mathcal{A}$ is finitely presentable and has no proper subobject.
   • Binary products of objects are co-universal i.e. stable under pushouts.

   • For any finite sequence of codisjunctable congruences $r_1, \ldots, r_n$ on any object with respect codisjunctors $d_1, \ldots, d_n$, we have

     $$r_1 \vee^c \cdots \vee^c r_n = \mathrm{id}_{A \times A} \implies d_1 \vee \cdots \vee d_n = \mathrm{id}_A$$

     where $\vee^c$ denotes the join in the lattice of congruences on $A$, while $\vee$ denotes the co-union of quotient objects of $A$.

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 23rd 2015
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexAnd the author of that series of categories mailing list posts has a [website](http://www.geometry.net/cg/index.html) that feels vaguely familiar in style.

   And the author of that series of categories mailing list posts has a website that feels vaguely familiar in style.