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nLab > Latest Changes: cosmic cube

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 13th 2010
    • (edited Jul 13th 2010)

    Were we to have an entry on the cosmic cube, would people be happy with that name, or should we have something less dramatic?

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeJul 13th 2010

    I think that cosmic cube is all right to start with.

    But even John never published that name. He published no particularly meaningful name at all, so we still have ‘cosmic cube’ for now, but perhaps somebody will think of another name. If they publish that name and it’s a reasonable one, or if we just like it better even if they don’t publish it, then we can move our page.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeJul 13th 2010

    And also: the Cosmic Cube

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJul 13th 2010

    Ad 1: what are you talking about ? I did not find phrase “the cosmic cube” at the lihk provided.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 13th 2010

    Whoops. Fixed.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2010

    If I were to be really frank, I’d say I find it too dramatic. Just my gut feeling.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 14th 2010

    Nothing much comes to mind.

    Presumably it’s easy enough to change a name later on, so that we could link to it. E.g., we could do so from the beginning of nonabelian algebraic topology.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2010

    started adding some details to cosmic cube

    Notice that the term “stable nn-category” is ambiguous in this context, in view of stable (infinity,1)-categories. Should be “stably monoidal nn-category” or “stably monoidal and groupal nn-category”.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJul 14th 2010

    What about “symmetric monoidal n-category”?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2010

    Yes, I guess by “stable” we mean symmetric monoidal and groupal.

    • CommentRowNumber11.
    • CommentAuthorTobyBartels
    • CommentTimeJul 15th 2010

    Oh yes, it’s this thing! Yes, ‘stable’ in John’s cube does mean ‘stably monoidal’ or ‘symmetric monoidal’; we pretty much agreed never to just say ‘stable’.

    But not also groupoidal, that’s a different leg of the cube.

    • CommentRowNumber12.
    • CommentAuthorJohn Baez
    • CommentTimeJul 15th 2010
    • (edited Jul 15th 2010)

    Toby wrote:

    But not also groupoidal, that’s a different leg of the cube.

    Right. For me, stabilization is the process that pushes you down columns of the periodic table, making your nn-category (or ∞\infty-category) more “commutative”. For an nn-category this process is done when you’ve turned it into an (n+2)(n+2)-tuply monoidal nn-category. For an ∞\infty-category you need a limiting process.

    This is logically distinct from making morphisms invertible. It’s just an accident of history that stabilization is best understood in the case of nn-groupoids, where it’s the basis of a huge subject called stable homotopy theory.

    By the way, I only used the term “cosmic cube” as a kind of blog post joke. It seems a bit silly as terminology, even though (or perhaps because) Connes or somebody has a “cosmic Galois group”. There’s probably some more descriptive name for it.

    Also by the way: it’s too bad that the article on the cube doesn’t show a picture of the frigging cube! It’s like having an encyclopedia article on elephants that doesn’t show a picture of an elephant. Maybe someone smart can just take this picture and stuff it on the nnLab.

    • CommentRowNumber13.
    • CommentAuthorEric
    • CommentTimeJul 15th 2010
    • (edited Jul 15th 2010)

    Also by the way: it’s too bad that the article on the cube doesn’t show a picture of the frigging cube! It’s like having an encyclopedia article on elephants that doesn’t show a picture of an elephant. Maybe someone smart can just take this picture and stuff it on the nnLab.

    How’s that?

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 15th 2010

    Am I reading in spectrum correctly that we’re being taken round the other two edges of the the upper face of the cube? I.e., the the nerve operation of the Dold-Kan correspondence runs from chain complexes to connective spectra which are included in ∞\infty-groupoids.

    So essentially one either includes then takes nerve or takes nerve then includes. What processes run the other way? Adjoints?

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010
    • (edited Jul 15th 2010)

    When people traditionally say things like “stable model category” and “stable (∞,1)(\infty,1)-category” they do not mean “stably monoidal”. When these people say “stabilization” they do not mean “making symmetric monoidal” but “making symmetric monoidal and groupal”, i.e. making spectrum objects.

    So in order not to run into misunderstandings (especially in this entry where both meanings matter!), I vote for reserving the bare “stable” for the standard use and say “stably monoidal ∞\infty-category” or “symmetric monoidal ∞\infty-category” for that case.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010
    • (edited Jul 15th 2010)

    But my original point was supposed to be this: an nn-groupoid that is symmetric monoidal is not yet necessarily a loop space. For that it needs to be groupal symmetric monoidal: it needs to be an abelian infinity-group.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010
    • (edited Jul 15th 2010)

    I split off the remark about “types of homotopy theories” as a small subsection and expanded slightly.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010

    Am I reading in spectrum correctly that we’re being taken round the other two edges of the the upper face of the cube? I.e., the the nerve operation of the Dold-Kan correspondence runs from chain complexes to connective spectra which are included in ∞-groupoids.

    Yes.

    • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 15th 2010

    Tried to put this in here. Where should ’connective’ be used?

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010
    • (edited Jul 15th 2010)

    I have edited your paragraph a little.

    Where should ’connective’ be used?

    The image of infinite loop spaces under Σ ∞:Top→Spectra\Sigma^\infty : Top \to Spectra or equivalently of groupal symmetric monoidal ∞\infty-groupoids under Σ ∞:∞Grpd→Sp(∞Grpd)\Sigma^\infty : \infty Grpd \to Sp(\infty Grpd) is what is called the connective spectra.

    If these are modeled by chain complexes, then by chain complexes in non-negative degree. No homotopy/homology in negative degrees.

    So the cube is actually bigger: there is also unbounded chain complexes embedding as the strict but not-necessarily connective spectrum objects.

    • CommentRowNumber21.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 15th 2010

    Your hidden fourth dimension?

    Is there degeneracy, or are the four different things: stable/unstable strict/weak ℤ−groupoids\mathbb{Z}-groupoids?

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010

    Maybe it’s not a cubical thing in total? I am not sure that I believe in ℤ\mathbb{Z}-groupoids apart from spectra.

    But the “hidden” (currently) edge “strict spectra ↪\hookrightarrow spectra” is certainly important. There is a whole industry of people working on (just) strict spectra. Notably in Sheffield.

    • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeJul 15th 2010

    I agree with Urs’s proposal in #15.

    Re: #20, connective spectra are not the image of Σ ∞\Sigma^\infty, rather they are the image of the “delooping” functor which takes a symmetric monoidal ∞-groupoid and builds a spectrum from it. The functor Σ ∞\Sigma^\infty should be thought of as constructing the free spectrum on a (not necessarily symmetric monoidal) ∞-groupoid. I added “connective” and the chain-complex version “bounded below” in a few places where I think they go. I think the Z-version would just extend the “stablization” axis one step further, “∞-groupoid -> stable ∞-groupoid -> Z-groupoid”, so we could consider it a rectangular parallelpiped instead of a cube.

    What is a strict spectrum? And what does it mean for a non-strict ∞-groupoid to be “strictly stable,” as opposed to just “stable”?

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeJul 15th 2010

    Re: #15, I might go even further and suggest that we use “symmetric monoidal” in the n-category case, “symmetric groupal” in the n-groupoid case, and avoid the use of the bare word “stable” altogether.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2010
    • (edited Jul 15th 2010)

    Re: #20, connective spectra are not the image of Σ ∞\Sigma^\infty, rather they are the image of the “delooping” functor which takes a symmetric monoidal ∞-groupoid and builds a spectrum from it.

    But that’s what Σ ∞\Sigma^\infty does. I said connective spectra are the image of infinite loop spaces under Σ ∞\Sigma^\infty. Not the full image of Σ ∞\Sigma^\infty. Isn’t that right?

    What is a strict spectrum?

    Something represented by a chain complex.

    And what does it mean for a non-strict ∞-groupoid to be “strictly stable,” as opposed to just “stable”?

    I don’t know what it should mean for a non-strict ∞\infty-groupoid. Do we need that?

    • CommentRowNumber26.
    • CommentAuthorMike Shulman
    • CommentTimeJul 16th 2010

    No, that’s not what Σ ∞\Sigma^\infty does. The functor Σ ∞\Sigma^\infty knows nothing about whether or not its argument is an infinite loop space, and in particular cannot make any use of an infinite loop space structure if one exists. Therefore it is not the same as delooping. Consider, for instance, the based space ℤ/2\mathbb{Z}/2, which is an abelian group and therefore an infinite loop space. Its delooping is the Eilenberg-Mac Lane spectrum K(ℤ/2,0)K(\mathbb{Z}/2,0) (often written as H(ℤ/2)H(\mathbb{Z}/2)). But if you forget the infinite loop space structure, then ℤ/2\mathbb{Z}/2 is just the 0-sphere S 0S^0, and therefore Σ ∞\Sigma^\infty of it is the sphere spectrum – very different! You should think of Ω ∞\Omega^\infty acting on connective spectra as the forgetful functor from “symmetric groupal ∞-groupoids” to ∞-groupoids, and Σ ∞\Sigma^\infty is its left adjoint, which constructs the “free symmetric groupal ∞-groupoid” on an ∞-groupoid.

    • CommentRowNumber27.
    • CommentAuthorMike Shulman
    • CommentTimeJul 16th 2010

    If a strict spectrum is just a chain complex, why do we need a new word for it?

    I don’t think we need a notion of “strictly stable non-strict ∞-groupoid” either, but it’s there in the entry: “A strictly stable ∞-groupoid is modeled by a connective spectrum” at the very bottom.

    • CommentRowNumber28.
    • CommentAuthorJohn Baez
    • CommentTimeJul 16th 2010
    Eric wrote:

    > How's that?

    Brilliant!

    Thanks, Eric. An article about a cube should show a picture of a cube. A picture is worth a thousand words.
    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2010

    Thanks, Mike.

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeJul 16th 2010

    If a strict spectrum is just a chain complex, why do we need a new word for it?

    Well, I thought in the context of the cosmic cube it helps to make a point.

    By the way, I was thinking of dg-models for rational spectra as in Shipley’s An algebraic model for equivariant stable homotopy theory.

    • CommentRowNumber31.
    • CommentAuthorMike Shulman
    • CommentTimeJul 17th 2010

    Sure, I can see that.

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