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



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> Niles Johnson, Angélica M. Osorno, *Modeling stable one-types*. Theory Appl. Categ. 26 (2012), No. 20, 520–537; [arXiv:1201.2686](https://arxiv.org/abs/1201.2686).
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and for $n=2$ in



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> 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](https://arxiv.org/abs/1712.07218).
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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](https://arxiv.org/abs/1210.1174), Corollary 1.11). What is an explicit description of the free symmetric monoidal tricategory on one object?



Categorical models for truncations of the sphere spectrum
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>


