Let $\Delta\_+$ denote the category of finite ordinal numbers with monotonic maps (the subscript indicates that $0$ is included, so this is the [augmented simplex category](https://ncatlab.org/nlab/show/simplex+category)). This has a monoidal structure (given by the sum), which is not symmetric. But we can make it symmetric in a universal way, see [here](https://mathoverflow.net/questions/117533/free-symmetric-monoidal-category-on-a-monoidal-category) for the general procedure. Let us denote this symmetric monoidal category by $(\Delta\_+)\_{\mathrm{sym}}$.


**Question.** What is a more "concrete" symmetric monoidal category which is equivalent to $(\Delta\_+)\_{\mathrm{sym}}$?


Notice that this is not the symmetric monoidal category $\mathcal{F}$ of finite sets. Whereas $(\Delta\_+)\_{\mathrm{sym}}$ classifies algebra objects in symmetric monoidal categories, $\mathcal{F}$ classifies commutative algebra objects in symmetric monoidal categories. Hence, there will be a strong symmetric monoidal functor $(\Delta\_+)\_{\mathrm{sym}} \to \mathcal{F}$, which is essentially surjective, but not fully faithful.



What is this symmetric simplex category, concretely?
$\Delta\_+$ is the monoidal category generated from the associative operad, considered as a non-symmetric operad. Similarly, $(\Delta\_+)\_{{\rm sym}}$ is the symmetric monoidal category generated from the associative operad, this time considered as a symmetric operad. This category can then be explicitly described as the category whose objects are finite sets and such that the morphisms from $I$ to $J$ are given by maps $f:I \to J$ together with, for each $j \in J$, a choice of a linear order on $f^{-1}(j)$. The symmetric monoidal structure is given by disjoint union. 


