## January 29, 2018

### The Stable Homotopy Hypothesis and Categorified Abelian Groups

#### Posted by Tom Leinster

guest post by Nick Gurski

In December, Niles Johnson, Angélica Osorno, and I posted a proof of the stable homotopy hypothesis in dimension two on the arxiv. This is the fourth paper we have written together on this project (one also joint with Marc Stephan), and since proving the SHH in dimension two was the first step in a long line of research which you might call “an introduction to twice-categorified abelian groups,” I thought now was a good time to try to explain what we have been up to and why.

## The homotopy hypothesis

The homotopy hypothesis of Grothendieck is the idea that homotopy $n$-types (spaces with trivial homotopy groups above dimension $n$) should be modeled by weak $n$-groupoids. The really interesting feature of this is not the equivalence at the level of objects, or even morphisms, but taking into account the weak equivalences, or which morphisms should be invertible. Traditional homotopy theory inverts weak homotopy equivalences, or maps inducing an isomorphism on all homotopy groups. For $n$-types, we only go up to $\pi_n$.

On the categorical side and in the special case of groupoids, the homotopy hypothesis predicts that weak homotopy equivalences correspond to those functors which are equivalences in the naive sense, i.e., those with an inverse up to weakly invertible transformations or those which have the $n$-dimensional analogue of essentially surjective plus full and faithful. Thus we may interpret the homotopy hypothesis as saying that, in a weak $n$-groupoid, the $k$-morphisms really give you all the $k$-dimensional homotopical information, mediated by the $(k+1)$-morphisms.

## Stabilization and symmetric monoidal structure

Moving to the stable world means replacing spaces with spectra, or for a first approximation, only considering spaces $X$ equipped with an infinite sequence of deloopings:

$X \simeq \Omega Y_1 \simeq \Omega^2 Y_2 \cdots.$

Loops can be concatenated, so $X$ has a multiplication which is unital and associative up to coherent higher homotopies. More generally, $n$-fold loops end up being more and more commutative as $n$ gets large, and in the case of infinite deloopings we have a system of coherent homotopies exhibiting as much commutativity as you like for any given situation. When moving to the categorical world, it seems reasonable to expect that the corresponding higher groupoid would have a symmetric monoidal structure.

Loops also have up-to-homotopy inverses, so we should also have all our objects invertible, which is to say that for any object $x$ there is another object $y$ such that both $x \otimes y$ and $y \otimes x$ are internally equivalent to the unit object. We call a weak $n$-groupoid equipped with a symmetric monoidal structure such that all objects are invertible a Picard $n$-category. The stable homotopy hypothesis is that stable homotopy $n$-types are modeled by Picard $n$-categories, where we invert the stable $n$-equivalences on the topological side (maps which induce isomorphisms of stable homotopy groups up through dimension $n$) and symmetric monoidal $n$-equivalences on the categorical side (so the symmetric monoidal version of the same equivalences from the regular old homotopy hypothesis). The 1-dimensional stable homotopy hypothesis was kicked around in the lore for a while, with Niles and Angélica writing down a bona fide proof in [JO12]; our goal was to do $n=2$.

## Questions

This kind of theorem prompts some interesting questions, like how can you read off topological invariants from categorical ones? In the 1-dimensional case, there are only three topological invariants to know: two different homotopy groups, and a single $k$-invariant which is a map $\pi_0 \otimes \mathbb{Z}/2 \to \pi_1$. The groups are the two obvious ones: $\pi_0$ is the group of isomorphism classes of objects, and $\pi_1$ is the group of automorphisms of the unit object. In [JO12] they show the $k$-invariant is given by the symmetry, suitably interpreted (which is how everyone thought it should go, but of course that is why we write proofs down).

This question leads to ideas like: write down a simple categorical model for the 1-type of the sphere spectrum. The Barratt-Priddy-Quillen Theorem says that the free symmetric monoidal category generated by one object models the whole sphere spectrum, but it doesn’t have invertible objects, so we have to “Picardify” it to get the Picard 1-category of invertible $\mathbb{Z}$-graded abelian groups and isomorphisms between them. Checking that this does the trick isn’t entirely straightforward, and moving up to $n=2$ it gets tougher.

A number of people (myself included) have speculated that the 2-type of the sphere is the “obvious” Picard 2-category where you just put the right stable homotopy groups in the right dimensions and put in some appropriate negative signs for the $k$-invariants. But it turns out you can’t even make this structure into a Picard 2-category at all. Pinning down those details led to the paper [GJOS17] with Marc Stephan, where we show that in dimension two (and thus higher) having strictly invertible objects instead of weakly invertible ones causes serious problems for the $k$-invariants.

## Interesting phenomena

Along the way we have encountered some interesting, and unexpected, phenomena. One was a nice definition of a semi-strict form of symmetric monoidal 2-category we call permutative Gray-monoid. These appeared in [SP11] as quasi-strict symmetric monoidal 2-categories, but we repackaged his definition quite neatly using the Gray tensor product in [GJO17a]. We also found that working with certain kinds of lax maps simplified strategies significantly, prompting the writing of [GJO17b] in which we gave a general method for showing that homotopy theories defined via lax maps often coincide with those defined on the subcategory of strict maps.

We still have a lot of material that hasn’t made its way into a paper yet, like a correct construction of the 2-type of the sphere spectrum using some homological algebra of Picard 1-categories, a 2-categorical version of Quillen’s $S^{-1}S$ construction, and a semi-strict notion of symmetric monoidal Gray-category that arises from cofibers of permutative Gray-monoids. Progress continues, so stay tuned!

## Image Credits

These are some of the images one can find with an internet search for the keywords two, dimensional, stable, homotopy, hypothesis.

Posted at January 29, 2018 9:37 PM UTC

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### Re: The Stable Homotopy Hypothesis and Categorified Abelian Groups

Thanks for this nice summary! And thanks in particular for pointing out the definition of permutative Gray-monoid in [GJO17a], which I don’t think I noticed before (the abstract refers only obliquely to “strictfication results of independent interest for symmetric monoidal bicategories”).

It’s very interesting that symmetric monoidal bicategories can be strictified so far. I guess one could summarize the result by saying that the only obstacles to their strictification are the known ones: interchange for the tensor product (encapsulated in the Gray tensor product) and the action of symmetric groups on $X^{\otimes n}$ (encapsulated in permutativity): there’s no “interaction” or “higher commutativity” to worry about.

In particular, I think this (or, I suppose, more precisely Schommer-Pries’s original strictification theorem) is an answer to a version of this question for the symmetric case. Do you know anything about the merely braided (or sylleptic) case? If not, do you have any guesses?

Posted by: Mike Shulman on January 30, 2018 11:22 AM | Permalink | Reply to this

### Re: The Stable Homotopy Hypothesis and Categorified Abelian Groups

I think the braided case should turn out like the symmetric one since the operad in question is also made out of $K(G,1)$’s. That line of thinking seems correct to me, whether the calculations I did in my first paper on coherence for braided monoidal bicategories already suffice is not something I have tried working through. The sylleptic case seems very interesting, I have no good intuition for what should happen there. These are things with an $E_3$ structure from the operadic point of view, so not an operad made out of Eilenberg-Mac Lane spaces. I have tried selling the sylleptic case to grad students before under the guise of knotted surfaces, but so far no one has been interested.

Posted by: Nick Gurski on January 30, 2018 1:55 PM | Permalink | Reply to this

### Re: The Stable Homotopy Hypothesis and Categorified Abelian Groups

I also meant to add that the larger question of strictifying two structures at once that you bring up in your MO discussion is one that does not seem to have a solution in the literature. A specific instance of this that I have wanted to answer for a long time is that of cocomplete symmetric monoidal bicategories, where cocomplete means in the bicategorical sense. There is a theorem of John Power that strictifies finite bilimits to finite flexible limits, but it is not at all obvious you can do that sort of thing plus strictify the symmetric monoidal structure at the same time.

Posted by: Nick Gurski on January 30, 2018 6:48 PM | Permalink | Reply to this

### Re: The Stable Homotopy Hypothesis and Categorified Abelian Groups

That’s a really interesting question. There are certainly cases in which two structures can’t be simultaneously strictified, right? For instance, if you regard a braided monoidal category as a category with two interchanging monoidal structures, then each of those monoidal structures can be independently strictified, but the interchange can’t be. But I’m not sure offhand whether or not this is quite the same sort of “two structures at once”.

Posted by: Mike Shulman on January 30, 2018 11:13 PM | Permalink | Reply to this

### Re: The Stable Homotopy Hypothesis and Categorified Abelian Groups

On the face of it, one might guess that braided structures are this kind of problem, but I’m not actually sure they are. I think what you have here is a pseudodistributive law of a 2-monad over itself. You can strictify both of the algebra structures (I think, since they are actually the same up to isomorphism), but what you definitely can’t do is strictify the distributivity part, and I think that is the interchange. I started thinking about this a couple years ago in this exact framework (some 2-monads satisfying the hypotheses of an abstract coherence theorem, plus a pseudodistributive law) but never really made progress, so if you or any other reader finds the problem interesting I would love to have some help.

Posted by: Nick Gurski on February 1, 2018 8:48 PM | Permalink | Reply to this

### Re: The Stable Homotopy Hypothesis and Categorified Abelian Groups

It does sound like a really interesting problem. I think the counterexample doesn’t depend on the pseudo-ness of the distributive law either: in arXiv:1005.1520 I used a strict distributive law between 2-monads, each of which I believe should satisfy the strictification theorem, to construct a composite 2-monad that doesn’t satisfy the strictification theorem.

Do you have any examples of a (strict or pseudo) distributive law where you can say definitely that the composite pseudoalgebras do satisfy a strictification theorem?

Posted by: Mike Shulman on February 2, 2018 12:23 AM | Permalink | Reply to this

### Re: The Stable Homotopy Hypothesis and Categorified Abelian Groups

Good example! The most restrictive version of abstract coherence in which you conclude that the unit is an equivalence is when the 2-monad preserves codescent objects, and having a strict distributive law between two of those will definitely get you a new composite with the same property. Having a pseudodistributive law would disrupt this argument, as would having your monads preserve the left part of two different enhanced factorization systems (even if the distributive law is strict). I don’t have an example when I know it does work sadly.

Posted by: Nick Gurski on February 2, 2018 6:54 PM | Permalink | Reply to this

### Re: The Stable Homotopy Hypothesis and Categorified Abelian Groups

Indeed, thanks Nick! It’s great fun to hear about this work from a totally different point of view :) One other nice piece of algebra that appears “below the fold” in our work is the strictification via 2-monads in section 4 of the Postnikov data paper. This is an application of theory that is certainly familiar to this crowd, but some of you may still appreciate it as a worked example and application. From the introduction to section 4:

Our focus is on various strictification results for algebras and pseudoalgebras over 2-monads, and how strictification can often be expressed as a 2-adjunction with good properties. In Section 4.2 we apply this to construct a strictification of pseudodiagrams as a left 2-adjoint. The material in this section is largely standard 2-category theory, but we did not know a single reference which collected it all in one place.

The formalism of this section aids the proof of Theorem 3.11 in two ways. First, it allows us to produce strict diagrams of 2-categories by working with diagrams which are weaker (e.g., whose arrows take values in pseudofunctors) but more straightforward to define. This occurs in Section 5.1. Second, it allows us to construct strict equivalences of strict diagrams by working instead with pseudonatural equivalences between them. This occurs in Section 5.2.

Theorem 3.11 mentioned there is the result that the 1-object permutative Gray monoid $\Sigma C$ built from a permutative category $C$ is indeed a correct model for the stable delooping (a.k.a. suspension) of the spectrum corresponding to $C$.

Posted by: Niles Johnson on January 30, 2018 7:12 PM | Permalink | Reply to this

### Re: The Stable Homotopy Hypothesis and Categorified Abelian Groups

Yes, that’s a nice example! Thanks for mentioning it.

I do feel compelled to point out that the wording of the proof of Proposition 4.12 in that paper might lead a careless reader to believe that being finitary is a sufficient condition for all of Theorem 4.3 to hold. In fact finitary-ness is a sufficient “assumption” but not a sufficient “further assumption”, i.e. it implies the existence of the left adjoints but not that the adjunction unit consists of equivalences. Steve conjectured in [Lac02] that it would imply the latter as well, but I found a counterexample in arXiv:1005.1520.

Posted by: Mike Shulman on February 2, 2018 8:48 AM | Permalink | Reply to this

### Re: The Stable Homotopy Hypothesis and Categorified Abelian Groups

I hadn’t known about the paper Stable Postnikov data of Picard 2-categories. It’s nice! Could you summarize in simple terms what you now know about the Postnikov data required to specify a stable homotopy 2-type? Something like “abelian groups $\pi_0, \pi_1, \pi_2$, a stable quadratic map $\pi_0 \to \pi_1$, and…”

This story could help find stable 2-types lurking in various algebraic situations, much as Nora Gantner built categorical groups from lattices such as the Leech lattice in her paper Categorical tori. (At the bottom I believe these are braided categorical groups with $\pi_0 = \mathbb{Z}^n$ and $\pi_1 = \mathbb{Z}$, though she deloops them and gets gerbes.)

Posted by: John Baez on February 7, 2018 4:11 PM | Permalink | Reply to this

### Re: The Stable Homotopy Hypothesis and Categorified Abelian Groups

There are, I think, two more pieces of data needed, but we don’t (yet) fully understand how these correspond to the categorical models. The two pieces are

• another stable quadratic map $\pi_1 \rightarrow \pi_2$
• some kind of partially defined map from $\pi_0$ to $\pi_2$ that shifts stable dimension by 3

We talk about that first one in the Postnikov data paper – it’s the Postnikov invariant for the Picard category of 1- and 2-cells over the unit object in a strict Picard 2-category (that is, a permutative Gray-monoid which is Picard). If you’re looking at the paper, that’s Lemma 3.12; its proof requires the stable delooping theorem (3.11) that I mentioned above.

The second piece of data is the homotopical data that glues $\pi_2$ on to $\pi_0$. In a Postnikov tower, it corresponds to a map defined on the kernel of the bottom Postnikov invariant, lifting up the first stage in the tower (a fibration), and then composing with the next Postnikov invariant (the top one for a 2-type) – this is the maneuver which defines the $d_3$ differential in the spectral sequence you get by mapping into a Postnikov tower, and if your tower only has 2 stages, it’s the longest nontrivial differential.

That much is “well-known” – it appears, for example, in Mosher and Tangora’s Cohomology Operations book. I bet that the algebra of such things is explained in Eilenberg and Mac Lane’s On the groups $H(\Pi,n)$ II, Methods of computation (1954, Ann. of Math. vol. 60). But we don’t understand it yet. One of our key motivations for getting a Picard 2-category which models the truncated sphere is so that we can start to understand where in a Picard 2-category that data lies, in the same way that we understand that the two stable quadratic maps come from the symmetry of the multiplication.

As a guess, this second piece of data might be defined on the objects $x$ of a Picard 2-category for whom the symmetry $\beta_{x,x}$ is trivial, and it might have something to do with Massey products. If you’ve seen such a thing in the wild, we would love to know!

Posted by: Niles Johnson on February 8, 2018 8:17 PM | Permalink | Reply to this

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