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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeNov 5th 2009
   • (edited Nov 5th 2009)
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownStarted writing [[regular and exact completion]].

   Started writing regular and exact completion.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeNov 6th 2009
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   Author: TobyBartels
   Format: MarkdownMike has added many interesting properties of these completions now.

   Mike has added many interesting properties of these completions now.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeNov 6th 2009
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownDo you know, I find myself thinking that an interesting argument *against* the powerset axiom is that unlike basically *all* the other logical/set-theoretic properties of categories, it is not preserved by the free ex/lex completion. Given that (in the absence of choice) <latex>Set_{ex/lex}</latex> is not necessarily a topos, and in fact isn't even well-powered relative to the original category Set, it becomes philosophically less self-evident to me that Set itself should be a topos. Especially since from the type-theoretic defined-equality point of view, Set is itself an ex/lex completion (of the category of presets).

   Do you know, I find myself thinking that an interesting argument against the powerset axiom is that unlike basically all the other logical/set-theoretic properties of categories, it is not preserved by the free ex/lex completion. Given that (in the absence of choice) is not necessarily a topos, and in fact isn't even well-powered relative to the original category Set, it becomes philosophically less self-evident to me that Set itself should be a topos. Especially since from the type-theoretic defined-equality point of view, Set is itself an ex/lex completion (of the category of presets).

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeNov 6th 2009
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   Author: TobyBartels
   Format: MarkdownHa ha, we'll make a predicativist out of you yet!

   Ha ha, we'll make a predicativist out of you yet!

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeNov 8th 2009
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownI added some more properties of [[regular and exact completions]], including that the ex/lex and reg/lex completions can never stabilize unless they do so in one step, and that they are Boolean if and only if the original category satisfies AC. I also added a higher-categorical perspective, and deleted the query box at [[regular category]] since its content was basically subsumed by my recent additions, and probably [[regular and exact completions]] would be a better place to continue it anyway.

   I added some more properties of regular and exact completions, including that the ex/lex and reg/lex completions can never stabilize unless they do so in one step, and that they are Boolean if and only if the original category satisfies AC. I also added a higher-categorical perspective, and deleted the query box at regular category since its content was basically subsumed by my recent additions, and probably regular and exact completions would be a better place to continue it anyway.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 3rd 2010
   • (edited Jan 3rd 2010)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownI have some questions for Mike at [[regular and exact completions]].

   I have some questions for Mike at regular and exact completions.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeJan 3rd 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownGood questions, I've answered them.

   Good questions, I've answered them.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 4th 2010
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   Author: Todd_Trimble
   Format: MarkdownThanks at [[regular and exact completions]].

   Thanks at regular and exact completions.