Sure, I can see that.

]]>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.

]]>Thanks, Mike.

]]>> 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. ]]>

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.

]]>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.

]]>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?

]]>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.

]]>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”?

]]>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.

]]>Your hidden fourth dimension?

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

]]>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.

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

]]>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.

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

]]>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.

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.

]]>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?

]]>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?

]]>Toby wrote:

But not

alsogroupoidal, 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.

]]>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.

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

What about “symmetric monoidal n-category”?

]]>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”.

]]>