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Atrium > Mathematics, Physics & Philosophy: coshep in realizability toposes

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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 4th 2012

    At presentation axiom (more fondly known to me as ’coshep’), there is the statement that the effective topos satisfies coshep, and I suppose this is true of any realizability topos. Is there a convenient place online where I could follow that up?

    • CommentRowNumber2.
    • CommentAuthorjim_stasheff
    • CommentTimeSep 4th 2012
    so coshep means coshe property?
    now what does coshe mean
    of course, we all know cosh 8-)
    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeSep 4th 2012

    FWIW, Mike added that remark.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 4th 2012

    Jim, you can click on the link to find out, but COSHEP stands for “Category Of Sets Has Enough Projectives” – it’s a fairly weak form of axiom of choice which allows you to do many things like construct projective resolutions, apply dependent choice, etc. but which holds in many toposes where the full axiom of choice fails badly.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2012

    do many things like construct projective resolutions

    Does anyone feel like adding a remark on that point to the (fairly new) entry projective resolution?

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeSep 4th 2012

    Sadly, as far as I know, there’s not much about realizability or the effective topos available free online.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 4th 2012

    @Urs #5: I wrote up something quickly at projective resolution. No time now to do anything more elaborate.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2012

    Thanks, Todd! Nice.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 4th 2012
    • (edited Sep 5th 2012)

    @Mike #6: well, then, I guess the nLab will have to fill in some gaps. :-) May have to roll up the sleeves here.

    I am reading the “Justification” section at COSHEP, and I sort of get it, but it reads sort of coyly allusive. Some pointers to the literature, even if not free online, would help.

    Regarding the fact that the effective topos satisfies COSHEP, I’m sort of muddling around here, but I’m guessing that if a (finitely complete) category has enough projectives, then so does its ex/lex completion. Does that seem right? Can that be taken as a key observation in proving said fact about the effective topos? I’ll try to think about this more myself; I suspect that the proofs are not going to be that difficult in the end.

    Edit: Ah, I see now the paragraph on “enough regular projectives” under Properties at exact completion, so this seems to make life very simple indeed. :-)

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeSep 5th 2012

    Yes, you’ve got it: any ex/lex completion has enough projectives. It requires a little thought to go from this “external” version of COSHEP to the appropriate “internal” version, but I think I convinced myself once upon a time that it does work.

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 5th 2012

    Thanks! To be continued in another thread.

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