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Picard $n$-groupoids are expected to model stable homotopy $n$-types. So far this has been proved for $n=1$ in

Niles Johnson, Angélica M. Osorno, Modeling stable one-types. Theory Appl. Categ. 26 (2012), No. 20, 520–537; arXiv:1201.2686.

and for $n=2$ in

Nick Gurski, Niles Johnson, Angélica M. Osorno, The $2$-dimensional stable homotopy hypothesis. Journal of Pure and Applied Algebra, Volume 223, Issue 10, 2019, Pages 4348-4383. arXiv:1712.07218.

In particular, for $n=1$, Johnson and Osorno have also shown that the free Picard groupoid on one object $\mathbb{S}$ models the $1$-truncation of the sphere spectrum. In detail, this is the Picard groupoid where

  • The objects of $\mathbb{S}$ are the integers;
  • For $m,n\in\mathrm{Obj}(\mathbb{S})$, we have $$ \mathrm{Hom}_{\mathbb{S}}(m,n) \overset{\mathrm{def}}{=} \begin{cases} \emptyset &\text{if $m\neq n$,}\\ \mathbb{Z}_{2} &\text{if $m=n$;} \end{cases} $$
  • The monoidal structure $\oplus$ on $\mathbb{S}$ is given by addition of integers;
  • The symmetry of $\mathbb{S}$ at $(A,B)$ is the morphism $\beta^{\mathbb{S},\oplus}_{m,n}\colon m\oplus n\to n\oplus m$ defined by $$ \beta^{\mathbb{S},\oplus}_{m,n} \overset{\mathrm{def}}{=} \begin{cases} 0 &\text{if $mn$ is even,}\\ \eta_{m+n} &\text{if $mn$ is odd,} \end{cases} $$ where $\eta_{m+n}$ is the unique non-zero element of $\mathrm{Hom}_{\mathbb{S}}(n,n)$.

In section 3 of the same paper, they also show that there is a symmetric monoidal functor $$\xi\colon\mathbb{F}\longrightarrow\mathbb{S}$$ from the symmetric monoidal category of finite sets and bijections $\mathbb{F}$, the categorification of the monoid of natural numbers $\mathbb{N}$, defined on objects by the inclusion $\mathbb{N}\hookrightarrow\mathbb{Z}$ and on morphisms by the sign map $\mathrm{sgn}\colon\Sigma_{n}\to\mathbb{Z}_{2}$.

Questions. Here are some questions around these topics:

  • Free Picard $n$-groupoids on one-object are supposed to model the $n$-truncations of the sphere spectrum. Is there some sense in which free symmetric monoidal $n$-categories on one object model the $n$-truncations of the "directed sphere spectrum"?
  • The zeroth and first truncations of the sphere spectrum are given respectively by the abelian group of integers $\mathbb{Z}$ and by the Picard groupoid $\mathbb{S}$ above. What are the second and third truncations of the sphere spectrum, or, equivalently, how can we explicit describe the free Picard $2$- and $3$-groupoids on one object?
  • The free symmetric monoidal $n$-categories on one object for $n=0,1,2$ are given by the commutative monoid $\mathbb{N}$ of natural numbers, the symmetric monoidal category $\mathbb{F}$, and the discrete monoidal bicategory on $\mathbb{F}$ (see arXiv:1210.1174, Corollary 1.11). What is an explicit description of the free symmetric monoidal tricategory on one object?
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    $\begingroup$ Regarding your first question, what's the "directed sphere spectrum"? Unless this concept has been otherwise defined, you're free to define it as the free symmetric monoidal n-category on one object. However, I believe this will be vastly less interesting than the free symmetric monoidal n-groupoid on one object: inverses are what make things complicated here. $\endgroup$
    – John Baez
    Jul 24 at 21:25
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    $\begingroup$ A preprint from last year proves the stable homotopy hypothesis for all n! $\endgroup$ Jul 24 at 21:39
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    $\begingroup$ @Théo - if you work out an easy example like the free braided monoidal groupoid on one object $x$, you'll see how $\pi_3(S^2) = \mathbb{Z}$ arises from the braiding $B_{x,x}: x \otimes x \to x \otimes x$ together with the isomorphism $i: 1 \to x \otimes x$ and its inverse $e: x \otimes x \to 1$. $\endgroup$
    – John Baez
    Jul 25 at 5:16
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    $\begingroup$ @Théo - what you've described sounds like the free braided 2-group on one object. A braided 2-group is a braided monoidal groupoid where every object $x$ has an "inverse", meaning an object $y$ such that $x \otimes y \cong y \otimes x \cong I$, where $I$ is the unit for the tensor product. The free braided monoidal groupoid on one object has objects $\mathbb{N}$ and $\mathrm{Hom}(n,m)$ being empty if $n \ne m$ and the braid group $B_n$ if $n = m$. $\endgroup$
    – John Baez
    Aug 2 at 23:26
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    $\begingroup$ When you take $S^2$ viewed as an $\infty$-groupoid in the manner you describe, and truncate it to a trigroupoid, and deloop it twice, you get the free braided "groupal groupoid" on one object, not the free braided monoidal category on one object. A "groupal groupoid", more commonly called a 2-group, is a monoidal category where all objects and morphisms are invertible. The objects of the free braided monoidal category on one object (namely $\mathbb{B}$) are not invertible! Those objects - more precisely isomorphism classes of those objects - form the monoid $\mathbb{N}$. So it's wrong. $\endgroup$
    – John Baez
    Aug 8 at 21:30
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I don't understand what you mean about the "directed sphere" so will focus on the other questions.

The free Picard $n$-category on one object has a description as a bordism $n$-category. Specifically it has:

  • Objects are stably framed 0-manifolds;
  • 1-Morphisms are stably framed 1-dimensional bordisms;
  • 2-morphisms are stably framed 2-dimensional bordism between bordisms;

etc

until level $n$ where we take equivalence classes of stably framed $n$-dimensional bordisms between bordisms between ... and the equivalence relation is up to one more layer of stable bordism.

Using the Pontyragin-Thom construction, one can see that this is just the same as the fundamental $n$-groupoid of $\Omega^\infty S^\infty$.

The free (non-Picard) symmetric monoidal $n$-category is always just $\mathbb{F}$ when $n\geq 1$. This is explained in my answer to you other recent question. https://mathoverflow.net/a/398384/184

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