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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.

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$\Delta_+$ 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.

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    $\begingroup$ I'll add that this is usually called the category of "noncommutative sets". $\endgroup$
    – Denis Nardin
    Commented Mar 19, 2017 at 0:44
  • $\begingroup$ @DenisNardin: Thanks! Can you give a reference for this terminology? $\endgroup$
    – HeinrichD
    Commented Mar 19, 2017 at 7:30
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    $\begingroup$ @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. $\endgroup$
    – Yonatan Harpaz
    Commented Mar 19, 2017 at 7:42
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    $\begingroup$ @HeinrichD The category itself was introduced by Fiedorowicz and Loday in "Crossed simplicial groups and their associated homology". The name seems to be first used by Pirashvili in "On the PROP corresponding to bialgebras"; see also "Hochschild and Cyclic Homology via Functor Homology". $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 19, 2017 at 7:53
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    $\begingroup$ @HeinrichD Actually I've re-read Pirashvili's paper, and he points to "Additive K-theory" by Feigin and Tsygan, page 191 for the first appearance of the category in question (they call it "Symmetric category" there) $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 19, 2017 at 9:59

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