### The Stable Homotopy Hypothesis and Categorified Abelian Groups

#### Posted by Tom Leinster

*guest post by Nick Gurski*

In December, Niles Johnson, AngÃ©lica 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 AngÃ©lica 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!

## References

[GJO17a] N. Gurski, N. Johnson, and A. M. Osorno, K-theory for 2-categories, Advances in Mathematics 322 (2017), 378-472. doi:10.1016/j.aim.2017.10.011

[GJO17b] N. Gurski, N. Johnson, and A. M. Osorno, Extending homotopy theories across adjunctions, Homology, Homotopy and Applications 19 (2017), no. 2, 89-110. doi:10.4310/HHA.2017.v19.n2.a6

[GJO17c] N. Gurski, N. Johnson, and A. M. Osorno, The 2-dimensional stable homotopy hypothesis, 2017. arXiv:1712.07218

[GJOS17] N. Gurski, N. Johnson, A. M. Osorno, and M. Stephan, Stable Postnikov data of Picard 2-categories, Algebr. Geom. Topol. 17 (2017), 2763-2806. doi:10.2140/agt.2017.17.2763

[JO12] N. Johnson, A. M. Osorno, Modeling stable one-types, Theory and Applications of Categories. 26 No. 20 (2012), pp 520-537.

[SP11] C. Schommer-Pries, The classification of two-dimensional extended topological field theories, 2011. arXiv:1112.1000v2

## Image Credits

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

- “Pears” by Edouard Manet, 1880. National Gallery of Art, Washington DC.
- WikiHow, Build a Model Bridge out of Skewers. https://www.wikihow.com/Build-a-Model-Bridge-out-of-Skewers.

## 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?