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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeNov 5th 2009
    • (edited Nov 5th 2009)

    Started writing regular and exact completion.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeNov 6th 2009

    Mike has added many interesting properties of these completions now.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeNov 6th 2009

    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) Set_{ex/lex} 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).

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeNov 6th 2009

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

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeNov 8th 2009

    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.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 3rd 2010
    • (edited Jan 3rd 2010)

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

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJan 3rd 2010

    Good questions, I've answered them.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 4th 2010