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nLab > Latest Changes: stable factorization system

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2012
    • (edited Nov 28th 2012)

    added in an Examples-section to stable factorization system the statement that in an adhesive category, in particular in a topos, the (epi, mono)-factorization is stable.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2012

    ah, do I need to say “co-adhesive”? I should fix that, but my bus is about to arrive…

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2012

    So I fixed something here and there.

    Eventually I really wanted to put more examples there. But not right now.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeNov 29th 2012

    I missed something — why does an adhesive category even have (epi,mono) factorizations?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2012
    • (edited Nov 29th 2012)

    I had fixed it by #3.

    Here is a question: how known is it that the nn-connected/nn-truncated factorization system in an ∞\infty-topos is stable?

    (For n=−1n = -1 this is in HTT, of course. I suppose generally it follows directly from the fact that truncation is an idempotent monad on TypeType? )

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeNov 29th 2012

    The examples section at stable factorization system still says “In an a topos, epimorphism are stable under pullback and hence the (epi, mono) factorization system in an adhesive category is stable”.

    I think stability of the nn-connected/nn-truncated factorization system is HTT 6.5.1.16(6).

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2012

    still says

    Ah, sorry. Fixed now. (I should stop editing entries in a haste at a bus stop…)

    I think stability of the n-connected/n-truncated factorization system is HTT 6.5.1.16(6).

    Ah, great. I had missed that. Thanks!!

    But still my question: in the type theory nn-truncation is a reflection on TypeType. Does that not also directy imply the statement?

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeNov 30th 2012

    Yes, once you know also that dependent sums preserve n-truncated types.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2012
    • (edited Dec 18th 2012)

    Mike,

    what would be the preferred way to cite your result that generalizes the equivalence “stable” = “comes from lex reflector” for reflective factorization systems as stated at stable factorization system - stable reflective factorization systems, but now generalized from categories to ∞\infty-categories?

    Your blog notes? Our QFT writeup? Should we have an nnLab entry on it?

    In either case, I’d like to have a stable (though not necessarily reflective) way to cite this.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeDec 19th 2012

    I suppose that would be good. Maybe there should be an nLab entry. But I don’t really have time to write out any details right now…

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2012

    Okay, thanks. I have added a brief pointer to your HoTT posting to the entry.

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