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?

defineit 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. $\endgroup$monoidal categoryon 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$14more comments