# What is this symmetric simplex category, concretely?

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). 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 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.

$\Delta_+$ 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.
• @HeinrichD, Exactly. Maybe I should have recalled the general construction. Given a (single colored) symmetric operad $P$, the symmetric monoidal category generated from $P$ has as objects the finite sets and the morphisms from $I$ to $J$ are given by maps $f: I \to J$ together with a choice, for each $j \in J$, of a multi-operation $\varphi \in P(f^{-1}(j))$ (here we think of the underlying data of $P$ as a functor from the groupoid of finite sets to sets) . Composition is then defined using the composition structure of the operad. Commented Mar 19, 2017 at 7:42