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Were we to have an entry on the cosmic cube, would people be happy with that name, or should we have something less dramatic?
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.
And also: the Cosmic Cube
Ad 1: what are you talking about ? I did not find phrase “the cosmic cube” at the lihk provided.
Whoops. Fixed.
If I were to be really frank, I’d say I find it too dramatic. Just my gut feeling.
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.
started adding some details to cosmic cube
Notice that the term “stable $n$-category” is ambiguous in this context, in view of stable (infinity,1)-categories. Should be “stably monoidal $n$-category” or “stably monoidal and groupal $n$-category”.
What about “symmetric monoidal n-category”?
Yes, I guess by “stable” we mean symmetric monoidal and groupal.
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.
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 $n$-category (or $\infty$-category) more “commutative”. For an $n$-category this process is done when you’ve turned it into an $(n+2)$-tuply monoidal $n$-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 $n$-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 $n$Lab.
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 $n$Lab.
How’s that?
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?
When people traditionally say things like “stable model category” and “stable $(\infty,1)$-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.
But my original point was supposed to be this: an $n$-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.
I split off the remark about “types of homotopy theories” as a small subsection and expanded slightly.
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.
Tried to put this in here. Where should ’connective’ be used?
I have edited your paragraph a little.
Where should ’connective’ be used?
The image of infinite loop spaces under $\Sigma^\infty : Top \to Spectra$ or equivalently of groupal symmetric monoidal $\infty$-groupoids under $\Sigma^\infty : \infty Grpd \to Sp(\infty Grpd)$ 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.
Your hidden fourth dimension?
Is there degeneracy, or are the four different things: stable/unstable strict/weak $\mathbb{Z}-groupoids$?
Maybe it’s not a cubical thing in total? I am not sure that I believe in $\mathbb{Z}$-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.
I agree with Urs’s proposal in #15.
Re: #20, connective spectra are not the image of $\Sigma^\infty$, rather they are the image of the “delooping” functor which takes a symmetric monoidal ∞-groupoid and builds a spectrum from it. The functor $\Sigma^\infty$ 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”?
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.
Re: #20, connective spectra are not the image of $\Sigma^\infty$, 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 $\Sigma^\infty$ does. I said connective spectra are the image of infinite loop spaces under $\Sigma^\infty$. Not the full image of $\Sigma^\infty$. 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?
No, that’s not what $\Sigma^\infty$ does. The functor $\Sigma^\infty$ 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 $\mathbb{Z}/2$, which is an abelian group and therefore an infinite loop space. Its delooping is the Eilenberg-Mac Lane spectrum $K(\mathbb{Z}/2,0)$ (often written as $H(\mathbb{Z}/2)$). But if you forget the infinite loop space structure, then $\mathbb{Z}/2$ is just the 0-sphere $S^0$, and therefore $\Sigma^\infty$ of it is the sphere spectrum – very different! You should think of $\Omega^\infty$ acting on connective spectra as the forgetful functor from “symmetric groupal ∞-groupoids” to ∞-groupoids, and $\Sigma^\infty$ is its left adjoint, which constructs the “free symmetric groupal ∞-groupoid” on an ∞-groupoid.
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.
Thanks, Mike.
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.
Sure, I can see that.
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