Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundle bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2021

    am finally giving W¯G\overline{W}G its own entry, for ease of hyperlinking to it

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 5th 2021

    The functor LL (denoted there by GG) was introduced by Kan in §7 of

    • Daniel M. Kan, A combinatorial definition of homotopy groups, Annals of Mathematics 67:2 (1958), 282–312. doi.

    The functor W¯\bar W is essentially due to:

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 5th 2021
    • (edited Jun 5th 2021)

    The article coincides with the article simplicial loop space. Should they be merged?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2021

    Not sure I am following. You write in #2 as if pointing out omissions:

    The functor W¯\bar W is essentially due to: …

    But that’s exactly the references I gave. In fact you seem to have copied them from what I wrote, including my choice of anchors and uploaded pdf-s.

    The reference to Kan you added I moved further down, since (a) it’s not the topic of this entry, (b) it comes four years after the original references relevant to this entry; so that it seems weird to have it as the first reference item.

    In this vein, the entries should not be merged. Just as the entries on “classifying space” and “loop space” should not be merged! If they appear too close at the moment, that’s because they are waiting for somebody to spell out the definitions and discuss more of the properties.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2021

    added pointer to:

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2021
    • (edited Jun 5th 2021)

    added the component-definition (following Goerss-Jardine);

    made explicit the example for GG an ordinary group (constant simplicial group);

    added brief statement of the abstract description via total simplicial sets;

    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2021

    added brief statement of various properties, with pointers to the literature:

    • Kan fibrancy,

    • classification of simplicial principal bundles

    • relation to GG-actions by slicing

    diff, v5, current

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 5th 2021

    You write in #2 as if pointing out omissions:

    The functor W¯\bar W is essentially due to: … But that’s exactly the references I gave. In fact you seem to have copied them from what I wrote, including my choice of anchors and uploaded pdf-s.

    I think the whole adjunction deserves to be mentioned right away.

    Also, W¯\bar W is not actually defined in the Eilenberg–MacLane article as far as I can see.

    Hence, a slight adjustment in the description (“the functor W¯\bar W is essentially due to…”), which is also how Kan describes it in his article.

    Kan’s article is the first one to spell W¯\bar W completely explicitly, as far as I can see.

    This was the point of my edit.

    In this vein, the entries should not be merged. Just as the entries on “classifying space” and “loop space” should not be merged! If they appear too close at the moment, that’s because they are waiting for somebody to spell out the definitions and discuss more of the properties.

    Okay, I can certainly imagine having two separate entries on these two functors, but they are adjoint functors and currently the entries on these two adjoint functors are not even cross-linked.

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 5th 2021

    Related concepts

    diff, v7, current

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 5th 2021

    The adjunction was explicitly spelled out by Kan in

    • {#Kan58} Daniel Kan, Sections 10-11 in: On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958), 38-53 (jstor:1970042)

    Kan’s paper also appears to be the first reference where W¯\bar W is defined explicitly.

    diff, v7, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2021

    Oh, I see, you want to sort out the fine-print of the attribution. Then let’s add the explicit pointers to the original definition:

    Kan’s paper also appears to be the first reference where W¯\bar W is defined explicitly.

    The explicit definition does appear on p. 3 of MacLane 54 – only that MacLane insists on using the product in a simplicial ring instead of the product in a simplicial group.

    So after the component-definition in the entry, I have added pointer to p. 3 in MacLan54 and to Def. 10.3 in Kan 58.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2021

    but they are adjoint functors and currently the entries on these two adjoint functors are not even cross-linked.

    Not that it matters much, but allow me to say that the stub of the entry that I had yesterday (rev 1) contained essentially nothing else but the mentioning of this adjunction, with hyperlink. :-)

    Anyway, it’s not important, we both agree on what needs to be done to improve the entries.

    And I gather now that what you send through the announcement mechanism are not edit-logs but straight copies of the material that you edited! That caused the confusion in #2: I read this as a message to me/us (which is how we all usually use the nForum, no?) while you meant it to be the uncommented snippet of the entry that you had re-arranged.

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 5th 2021

    And I gather now that what you send through the announcement mechanism are not edit-logs but straight copies of the material that you edited! That caused the confusion in #2: I read this as a message to me/us (which is how we all usually use the nForum, no?) while you meant it to be the uncommented snippet of the entry that you had re-arranged.

    Yes. One advantage of this is that it is immediately clear what exactly has been changed, so one doesn’t have to actually look at the article.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2021
    • (edited Jun 5th 2021)

    That’s a good point. For that reason I often add pointers to anchors in the entry where the edit took place. When I feel I need to include actual snippets of edits in the nForum logs, then I usually put them in between horizontal lines after a line announcing an edit, like this:

      I have touched the following bit of the entry:
    
      ***
    
      ... Entry text goes here, 
    
      which might say things that acquire an unintended meaning 
    
      if they'd appear un-escaped in a discussion forum ...
    
      ***
    
    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 5th 2021

    The explicit definition does appear on p. 3 of MacLane 54 – only that MacLane insists on using the product in a simplicial ring instead of the product in a simplicial group.

    Yes, and MacLane uses tensor products of abelian groups, not cartesian products.

    Also, he gives a references to an earlier paper by Eilenberg and MacLane, which I am going to add now.

    • CommentRowNumber16.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 5th 2021

    Added more historical details:


    The idea of constructing W¯\overline{W} using the bar construction is due to Eilenberg and MacLane, who apply it to simplicial rings with the usual tensor product operation:

    This was also later discussed in

    The first reference where W¯\bar W is defined explicitly for simplicial groups and the adjunction between simplicial groups and reduced simplicial sets is explicitly spelled out is

    • {#Kan58} Daniel Kan, Sections 10-11 in: On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958), 38-53 (jstor:1970042)

    The left adjoint simplicial loop space functor LL is also discussed by Kan (there denoted “GG”) in

    • Daniel M. Kan, §7 of: A combinatorial definition of homotopy groups, Annals of Mathematics 67:2 (1958), 282–312. doi.

    The Quillen equivalence was established in

    • {#Quillen69} Dan Quillen, Section 2 of: Rational homotopy theory, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (jstor:1970725)

    diff, v11, current

    • CommentRowNumber17.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 5th 2021

    Mentioned that GG is the standard notation for the simplicial loop space.

    diff, v11, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2021

    added the statement that WGW G is contractible, with pointer to GJ V4.6.

    diff, v13, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2021

    For what it’s worth, I have added a rendering of the generic 2-simplex in WGW G: here

    diff, v14, current

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 24th 2021

    At the risk of self-promotion, I think it might be worth adding a reference to

    somewhere, as there’s a reasonable amount of discussion of WGWG. But I’m not sure where to fit it in the article, since the page is meant to be about W¯G\overline{W}G. I can work something up later, if it’s deemed reasonable to insert.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2021

    As long as you are not actually promoting yourself, but your results: that’s what the nnLab is for!

    There is already an entry essentially geared towards your result, it’s groupal model for universal principal infinity-bundles.

    But if you feel the simplicial aspect should be amplified further, there would be room for an entry universal simplicial principal bundle.

    Incidentally, the reason why the present entry, which a priori is about W¯G{\overline W} G, talks so much about WGW G is that the special property of the traditional WGW G is that it’s the natural intermediate step for obtaining/understanding W¯G\overline{W}G. The point being that for WGW G the left GG-action is most simple.

    This is in contrast to your W grpGW_{grp} G, which gets its simplicial group structure at the expense of the left GG-action having a more complicated component expression. (Nothing wrong with that, just pointing it out in response to you saying in #20 that you are not sure how things fit together.)

    By the way, in your Def. 1: Isn’t the case disctinction at i=0i = 0 unnecessary, while the necessary case distinction at i=ni = n is missing?

    • CommentRowNumber22.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 27th 2021

    The i=0i=0 case for the face maps of the standard WGWG being included is debatable, sure, but I included it for clarity. But the i=ni=n case … whoops! I think I was trying to parallel the case of W grGW_{gr}G, where that division of the cases makes more sense.

    Thanks for the hint of where it might fit in.

    I was hoping, way back then, that people who worked on rigidifying \infty-/weak models for Lawvere theories might be able to do something with the results of this short paper, but that never eventuated. I don’t know anything myself, but it seemed it might be possible.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJul 3rd 2021

    fixed a typo (here) in the subscripts in the formula for the degeneracy maps (the same typo is on Goerss&Jardine’s p. 269, so I guess I had copied it from there…)

    diff, v16, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJul 3rd 2021
    • (edited Jul 3rd 2021)

    have now spelled out (here) the face and degeneracy maps of W¯G\overline{W}G itself

    diff, v17, current

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeJul 3rd 2021

    added a remark on décalage (here)

    WG=Dec 0(W¯G). W G \;=\; Dec^0\big( \overline{W}G \big) \,.

    but see the comment in the thread there

    diff, v18, current

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2021

    to the statement (here) that every W¯G\overline{W}G is Kan fibrant I added an abstract argument to see this from lemmas elsewhere on the nnLab (which amounts to stating the proof as given in Goerss & Jardine)

    diff, v21, current

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2021

    added (here) statement and proof of π n+1(W¯G)π n(G)\pi_{n+1}(\overline{W}G) \simeq \pi_n(G)

    diff, v22, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeJul 4th 2021

    after the statement of WGW¯GW G \to \overline{W}G being a Kan fibration, I added the statement (here) that, more generally, (WG×X)/GW¯G(W G \times X)/G \to \overline{W}G is a Kan fibration for XX a Kan complex with simplicial G-action

    diff, v23, current

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2021
    • (edited Jul 5th 2021)

    made the statement that WGW¯GW G \to \overline{W}G is Kan fibrant (here) more explicitly a corollary of the fact that (WG×X)/GW¯G(W G \times X)/G \to \overline{W}G is Kan fibrant for fibrant simplicial group actions XX.

    diff, v25, current

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2021
    • (edited Jul 5th 2021)

    added also the following statement (here):


    Let 𝒢 1ϕ𝒢 2\mathcal{G}_1 \xrightarrow{\phi} \mathcal{G}_2 be a homomorphism of simplicial groups which is a Kan fibration. Then the induced morphism of simplicial classifying spaces W¯𝒢 1W¯(ϕ)W¯𝒢 2\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2 is a Kan fibration if and only if ϕ\phi is a surjection on connected components: π 0(ϕ):π 0(𝒢 1)π 0(𝒢 1)\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1).


    diff, v25, current

  1. Typo in index

    Anonymous

    diff, v26, current

    • CommentRowNumber32.
    • CommentAuthorTim_Porter
    • CommentTimeDec 17th 2021
    • (edited Dec 17th 2021)

    I am getting some strange behaviour with simplicial classifying space. I clicked on Changes from previous revision and goot an edit page with a load of code above the edit box. Does any one else get this?

    I also tried clicking on the Previous Revision tab at the bottom and got 500 Internal Server Error

    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

    added pointer to:

    for the generalization to simplicial groupoids

    diff, v27, current

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeApr 20th 2023

    added (here) a minimum indication of the generalization of W¯\overline{W} to simplicial groupoids

    diff, v29, current