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nLab > Latest Changes: Cartesian morphism

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  1.  
    • CommentRowNumber1.
    • CommentAuthorHurkyl
    • CommentTimeNov 9th 2021

    I felt like the (∞,1) section should give an abstract description rather than a model specific one, so I’ve done so, and proved in the abstract the equivalence between the hom-space and slice-category characterizations.

    It feels like cheating to invoke the Grothendieck construction for it; can the argument be made just as cleanly without it?

    … I’m having trouble with the formatting, so I’m going to do some bisection to track down the issue….

    diff, v38, current

    • CommentRowNumber2.
    • CommentAuthorHurkyl
    • CommentTimeNov 9th 2021

    Okay, I think I got it

    diff, v38, current

    • CommentRowNumber3.
    • CommentAuthorHurkyl
    • CommentTimeNov 9th 2021

    It’s weird that you can’t do tikzcd inbetween dollar-sign pairs, and you have to actually use the centre environment thing.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 10th 2021

    Thanks for the additions.

    Yes, the tikzcd-functionality is not part of Instiki (which is the software platform the nnLab rendering engine is using) but is an external add-on that Richard kindly added a while back, so that we can have decent diagram typesetting at all.

    My understanding is that there are plans to improve on this and other aspects, but I gather the priority right now is to get the nnLab migrated from its server at CMU (which is running out) to a cloud provider.

    As usual, if anyone is interested in lending Richard a hand with coding desired functionality for the nnLab, drop him a note and he may tell you what you could do!

    • CommentRowNumber5.
    • CommentAuthorHurkyl
    • CommentTimeNov 10th 2021

    Changed some of the language in my additions to make the overall presentation flow more smoothly.

    diff, v40, current

    • CommentRowNumber6.
    • CommentAuthorJem Lord
    • CommentTimeJul 16th 2022

    Added alternative equivalent slightly more explicit definition.

    diff, v43, current

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeApr 23rd 2023

    Correction: a Cartesian morphism is the special case of a strictly final lift of a structured sink when the sink is a singleton.

    diff, v45, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 24th 2023

    Removed an old discussion:

    Variants

    David Roberts: There would surely be an anafunctor version of this, that would require no choices whatsoever. It is unlikely that I would be able to find time to write this up, so my plea goes out to those in the know…

    I imagine that there would then be an (∞,1)(\infty,1)-version using whatever passes as anafunctors in that setting (dratted memory, failing at the first gate)

    Mike Shulman: Yes, there would surely be such a version. (-: The simplest way would be to take the specifications |f *||f^*| for the anafunctor f *f^* to be the cartesian morphisms over ff, with domain and codomain giving the functions σ\sigma and τ\tau. Unique factorization would give you the values of morphisms.

    David Roberts: just stumbled on this old comment - I’m reading Makkai more closely, and I’m convinced that basically anything defined by a universal property is given by a saturated anafunctor. So this is a heads up for posterity, that a map is a fibration iff the fairly obvious span of functors defines a saturated anafunctor.

    diff, v46, current

  9. Fixed discrepancy between ’slightly more explicit’ definition and accompanying diagram

    Simon Dima

    diff, v48, current

  10. Fixed discrepancy between ’slightly more explicit’ definition and accompanying diagram

    Joseph Martin

    diff, v49, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2023

    I have moved the paragraphs on weak Cartesian morphisms out of the middle of the definition of Cartesian morphisms and gave them their own Remark-environment, now here.

    diff, v51, current

1 to 11 of 11

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