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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2012
    • (edited Apr 26th 2012)

    I have done a fair bit of editing to décalage.

    1. expanded the Definition-section to make it contain explcitly the three different perspectives and their relation:

      a) in components, b) as a restriction of ordinary décalage, c) in terms of joins and cones;

    2. changed notation from “DecDec” to “Dec 0Dec_0”, not to collide with the notation for total décalage;

    3. finally added the explicit discussion and proof that for XX a Kan complex, Dec 0XXDec_0 X \to X is a Kan fibration replacement of the canonical effective epimorphism X 0XX_0 \to X. (you may remember – but hopefully not – that we had a bit of confused discussion about this a few years back, which I don’t want to keep from falling into oblivion by linking to it again… ;-)

    4. added more references with some of the lemmas that I am citing, mostly by Danny Stevenson, and also a technical result in Joyal’s lectures.

  1. In definition 3 I think \square is defined on sSet×sSetsSet\times sSet instead of on sSetsSet?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2012

    Yes, thanks! I have fixed it now.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2012

    Stephan,

    I have further expanded the Idea-section at décalage. When you have a minute, could you look over it and tell me if this is the kind of introduction that you would find useful? (Or would have found useful, had I mentioned it earlier ;-)

  2. :-) Yes, absolutely. This would have spared me a couple of hours laboring with an inconvenient fibration replacement.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeApr 26th 2012

    I have rearranged things a bit in that introductory section, saying informally what the construction is before going into what it does.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2012

    Thanks, Tim, that does look better. I have just broken the long sentence in half, where it leads over to my previous sentence.

    • CommentRowNumber8.
    • CommentAuthorTim_Porter
    • CommentTimeApr 26th 2012
    • (edited Apr 26th 2012)

    I wonder if the type of use I made of decalage in that Topology paper might not interpret nicely in other contexts, e.g. in simplicial sets. I think Simona Paoli looked at that but cannot remember what her conclusions were.

    • CommentRowNumber9.
    • CommentAuthorStephan A Spahn
    • CommentTimeApr 27th 2012
    • (edited Apr 27th 2012)

    The statement of Proposition 2 implicitly assumes that XX is connected, right? (Since the proof uses corollary 2 to lemma 1 which assumes this.)

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2012
    • (edited Apr 27th 2012)

    No, XX can be arbitrary. We are just using that the standard nn-simplex is connected.

    So we want to know what the nn-cells

    Δ[n]Dec 0X \Delta[n] \to Dec_0 X

    are. Since CC is defined to denote the left adjoint of Dec 0:sSetsSetDec_0 : sSet \to sSet, this are equivalently the morphisms

    C(Δ[n])X. C(\Delta[n]) \to X \,.

    To evaluate this we look at corollary 1 and find that C(Δ[1])=Δ[1]Δ[0]C(\Delta[1]) = \Delta[1] \star \Delta[0].

    Ah, I see what may be confusing, the variable in corollary 1 is called “XX”, but here it takes the value Δ[n]\Delta[n]. I’ll change the name of that variable in corollary 1 .

    [edit: okay, I have renamed it to “SS”]

  3. Ok, I see.

  4. Written in a slightly misleading way. It’s not clear what the functor Δ opΔ op\Delta^{op}\to \Delta^{op} being referred to was - the identity? I don’t see how you get from the identity functor to the comonad.

    Patrick N.

    diff, v35, current

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTime6 days ago
    • (edited 6 days ago)

    Slightly misleading!? I’m not sure what you mean; more precisely, I don’t know where on the page you are looking.

    Conceptually, I find it simplest to work in the algebraists’ category of simplices, which is the category of finite ordinals and order-preserving maps. This is a monoidal category under ordinal sum ++. There the terminal object [0][0] is automatically a monoid. Hence [0][0] seen in the opposite category is a comonoid, which induces a comonad Δ opΔ op\Delta^{op} \to \Delta^{op} by [0]+()[0] +(-). This restricts to a comonad on the topologists’ category of simplices, i.e., the category of finite nonempty ordinals. That’s the decalage comonad.

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTime6 days ago
    • (edited 6 days ago)

    Anyway, I wish you had asked before performing such a substantial edit, which not only erased the flow of reasoning which had been there (which was not IMO misleading, even though the notation doesn’t quite match the notation introduced in edits that came after revision #5, the point in history where that section was first written), but introduces other infelicities (such as “natural functor” and “(co)monoidal product”, among others). I think the section will simply have to be rewritten again.

    Since this appears to be your first edit, let me say that except for minor typos, the general informal rule is not to excise other people’s work to such an extent, before discussing the issue here first.

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTime5 days ago

    Extensive rewrite of the comonad section.

    diff, v37, current

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