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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 13th 2016

    Started lift.

    weak factorization system has redirects from: lifting property, right lifting property, left lifting property, lifting problem, lifting problems.

    Would it be better to have these redirect to lift?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2016

    Thanks. True, the entire paragraph “Preliminaries” would deserve to have a dedicated entry. You could copy that over to “lift” and redirect the redirects accordingly.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 14th 2016

    Ok, have done that, but no doubt there’s a better arrangement at lift.

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTimeDec 6th 2018

    The lift of a morphism f:YBf: Y\to B along an epimorphism (or more general map) p:XBp:X\to B is a morphism f˜:YX\tilde{f}: Y\to X such that f=pf˜f = p\circ\tilde{f}.

    I changed “the” here to “a”.

    diff, v12, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 6th 2018
    • (edited Dec 6th 2018)

    Looking at extension where it has dually

    The extension of a morphism…

    it doesn’t seem so clearly wrong. I guess there’s ambiguity between ’extension’ as the act of extending and as an instance of so extending, as you see in the second paragraph of lift

    The dual problem is the extension of a morphism…

    I’ll change ’the’ to ’a’ at extension.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 7th 2021

    Following discussion in another thread here, I have created a new section

    and filled it with material that, up to some superficial first polishing by me, has kindly been written by the Anonymous Guest from that thread (copied over from Sandbox rev 2402, where it was first compiled as Sandbox rev 2401).

    As I write in that other thread, I think there is much room to polish this material up and make it more readable and more use-friendly, but I think it’s great to have this considerable list of examples now. Maybe it inspires more readers here to join in and try their hand at some editing.

    diff, v15, current

    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeDec 14th 2021
    • (edited Dec 14th 2021)

    At present the Idea section starts

    Intuitively, the lifting property is a negation in category theory:

    This seems strange to me. That does not fit my intuition at all! No doubt there are examples of lifting properties that are somehow a bit like some form of negation, and a lot of others a bit like ’orthogonality or ’being perpendicular to’ but it does seem a bit strange to start like this. There are also important examples that do not correspond intuitively to a negation at all.

    • CommentRowNumber8.
    • CommentAuthorvarkor
    • CommentTimeDec 14th 2021

    Removed the first paragraph, which was duplicated in the third paragraph.

    diff, v24, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 14th 2021

    I have tried to streamline the Idea-paragraph on qualitative Quillen negation a little (here).

    diff, v26, current

    • CommentRowNumber10.
    • CommentAuthorTim_Porter
    • CommentTimeDec 14th 2021

    Excellent. That looks much clearer and I now understand where the ‘negation’ idea was coming from.

    • CommentRowNumber11.
    • CommentAuthorRichard Williamson
    • CommentTimeDec 14th 2021
    • (edited Dec 14th 2021)

    I found myself musing for a moment how general this ’negation’ point of view is, for instance whether it holds for the standard model structures on simplicial and cubical sets…I think it does! One can I think reasonably argue that horn inclusions are the correct analogues of *\emptyset \rightarrow * in higher dimensions, so it works out correctly that Kan fibrations are the ’negations of archetypical cofibrations’, and I think it is also true that cofibrations are the ’negations of archetypical Kan fibrations’, though I’ve never seen this expressed before: I think the obvious analogues of the map from a two point set to a one point set in higher dimensions would be the fold maps n n n\square^{n} \sqcup \square^{n} \rightarrow \square^{n} for all nn (for cubical sets; the obvious analogue for simplicial sets), and, as far as I see, a) these are indeed Kan fibrations b) cofibrations (i.e. monomorphisms) are exactly those morphisms of cubical/simplicial sets with the right lifting property with respect to the set of these fold maps ranging over n0n \geq 0. I wonder if this observation is good for anything!

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2021

    Yes, I gather such observations are the reason for Misha Gavrilovich’s choice of terminology “Quillen negation” (though that entry remains a stub and the respective comments in the article listed there are brief).

    The whole development of separation axioms in terms of lifting properties is driven, I feel, by a sense of wonder that some basic but neat observations about lifting properties have remained underappreciated.

    • CommentRowNumber13.
    • CommentAuthorHurkyl
    • CommentTimeDec 15th 2021
    • (edited Dec 15th 2021)

    This reminds me a lot of Categories, Allegories where Freyd and Scedrov formulate a version of first-order logic of diagrams that was centered around looking at how a diagram J 0CJ_0 \to C extends along a functor J 0J 1J_0 \to J_1. For example, 1C1 \to C is an initial object iff every extension along 11+11 \to 1+1 (inserting into the left term) has a unique extension along 1+1[1]1+1 \to [1] (inserting as source and target).

    (I’m more reacting to “formulate things as lifting properties” in general, rather than an idea of ’negation’)

  1. Perhaps Misha will see this thread and care to comment. If there were some way to construct a model structure using these kind of observations, or use the characterisation of the cofibrations that I gave in #11 for some interesting purpose, that would be very nice.

  2. Added a characterisation of contractability (for finite CW complexes) via weak orthogonals of a map of finite spaces, and a conjecture asking for a similar characterisation of trivial fibrations (among “nice” maps).


    diff, v27, current

    • CommentRowNumber16.
    • CommentAuthorSeth Chaiken
    • CommentTimeSep 7th 2022

    This page defines the left lifting property of morphism i with respect to a morphism p. However, projective+object uses the concept of left lifting property of object p (with respect to epimorphisms) to define p being a projective object, and then it links to here.

    This page does not define what it means for an object to have the left lifting property with respect to a class of morphisms.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2022

    Added a remark (here) making explicit what lifting property of an objects means.

    Also touched the formatting on the following pragraphs on Quillen negation.

    diff, v31, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2022

    While I was at it, I have re-typeset the definition of “left/right orthigonal class”/”Quillen negation” (here)

    And I am moving the Example that followed from here to Quillen negation.

    diff, v31, current

  3. Correct me if I'm wrong, but under the modules portion of the elementary examples section, I believe the characterization of projective modules being in {0 → R}^(•lr) is incorrect, and that instead should be either {0 → R}^(•ll) or {0→R}^(•rl), right?
    The first option works because {0 → R}^(•l) = {0 → R}^(•r) = surjective morphisms.
    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2023

    Looking over it now, it seems logically correct to me; but the problem may be that the notation is ambiguous and not what the previous definition 2.3 actually introduces.

    i am trying to fix it…

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2023

    I have reworded and re-typeset the Example-section on modules: here

    (Did the same to the previous section on sets but not yet to the following sections on Groups etc., which still suffer from using undefined notation for orthogonality classes)

    diff, v34, current

  4. Yes, that follows what I was thinking! Looks good now :)
  5. I was interpreting the notation as "repeated application of class transform operations", where the notation C^lr is read as (C^l)^r
    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2023

    Glad you like the edit. If you are energetic, please feel invited to look into fixing the notation in the following section (Examples - Groups), too! :-)

    diff, v37, current