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    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 7th 2013
    • (edited Nov 7th 2013)

    Hi all,

    I decided to release the short paper WISC may fail in the category of sets, as I feel it’s a nice construction, even if it didn’t achieve my original goal of showing WISC independent of ZF (and Karagila beat me to it anyway). Rather, I get the result that WISC can fail in a well-pointed topos with nno (and assuming at most ETCS as base set theory). In fact I suspect my proof is constructive, but I’d like some outside perspective on that (Mike? Toby?)

    I still need to fix a couple of references (chapter and verse of an example from SGA IV, a specific statement of Mike and Benno vdB’s paper with Moerdijk rather than his unpublished note), but it feels reasonably close to final. However, I wouldn’t mind comments (any sort) if people are so inclined. EDIT: reading it again now that I’ve exposed it to the world has made me pick up on a few things that I want to change. This is precisely the sort of response I wanted from myself, at least.

    Also, I’m terrible at choosing journals (I’d be inclined to stick everything I can in TAC, but I get the feeling this is not, for better or worse, the best career move), so if anyone has any suggestions, I’m all ears.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 7th 2013

    OK, so I found a stupid inconsequential error rendering my key definition wrong, bear with me (cleaned things up a bit too much from my earlier draft). I’ll have to update tomorrow, my time, but for now any comments are still fine.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 8th 2013

    All sorted. Apologies for premature announcement.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2013

    David, could you remind me what your motivating question is? Is this still about defining internal groupoids/internal categories?

    And is the issue now how to do it in non-Grothendieck toposes?

    because there is this statement discussed at 2-topos – In terms of internal categories which seems to work entirely generally for the Grothendieck case.

    Maybe just remind me what your motivating question is. I mean, if there still is one, maybe you are now all motivated by set theory questions in themselves?

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 8th 2013

    No, this note is purely about foundational issues.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2013

    I know, but I am wondering what motivates you. You dó briefly mention a motivation from Rob12. I would just like to understand that better. You seemed to come to consider WISC in order to construct a 2-category of internal categories. Now you are considering WISC in a topos. And I am just wondering if/why one would really need that.

    I am just wondering if you have an answer to that. Or maybe you now don’t care about that anymore?

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 8th 2013

    Even though WISC arose in the context of sites, interpreted in the canonical singleton pretopology of a well-pointed topos it takes on a foundational character. The trick is then to consider WISC in the internal logic of a general topos - it then ceases to have much to do with the topos qua canonical site and more to do with the topos as a kind of set theory.

    The reason for considering WISC for internal categories etc is for local smallness of the localisation at what from the higher topos perspective are the stalkwise equivalences. It’s just a condition to add to the hypotheses of the main theorem of Rob12. It’s not so much interesting thing in it’s own right.

    Does that answer your question? If it confuses the issue, I can relegate the backstory to a footnote or a passing reference to Rob12 as the original published definition (although really it’s due to Mike and Toby, I believe).

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2013

    thanks. I am not trying to bug you. I am just trying to get myself interested in this. Currently I have trouble. If I were more interested in this, I might have more interesting discussion with you about it! :-)

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeNov 8th 2013

    One reason WISC is interesting as a foundational axiom is that it seems to be the minimal natural hypothesis for various constructions, and is in turn preserved under passage to sheaves, realizability, and exact completion. See for instance this paper, which calls WISC “AMC”.

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 8th 2013

    (Note Mike is talking about internal WISC).

    I should probably mention those facts in my intro :-)

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2013

    I am aware of the axiom of multiple choice and its relation to predicative toposes.

    What I still don’t fully see is what motivates David’s question in his note. But if it’s just me, then never mind.

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 10th 2013

    What motivated Freyd to give a topos not satisfying IAC? My result is admittedly somewhat lesser, but I hope interesting to a certain type of foundationally curious person.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeNov 11th 2013

    Whenever we consider a new foundational axiom, we should investigate whether it follows from other axioms!

    • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 11th 2013

    Oh, and Mike, I have to add in that if I take ETCS+R as my base topos, then I get my new topos autological (as it’s cocomplete and locally small), so I should have that the negation of WISC is compatible with structural Replacement, for what it’s worth.

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 13th 2013

    For what it’s worth, the article is done and will appear on the arXiv in the next posting, looking something like this. A slow train coming, indeed.