It is well-known that the (augmented) simplex category is the universal monoidal category with a monoid object. What about a commutative analogue? Consider the category $\mathsf{FinSet}$ of finite sets. It is a symmetric monoidal category with tensor product $\coprod$ and unit $\emptyset$. It contains a commutative monoid $\{\star\}$, which seems to be the universal one: For every symmetric monoidal category $\mathcal{C}$ the assignment $F \mapsto F(\{\star\})$ provides an equivalences of categories


$\mathrm{Hom}\_{\otimes}(\mathsf{FinSet},\mathcal{C}) \cong \mathrm{CMon}(\mathcal{C}).$


Here, $\mathrm{Hom}\_{\otimes}$ denotes the category of strong symmetric monoidal functors (not assumed to be strict).


**Question 1.** This should be well-known, is there a reference in the literature?


**Question 2.** Is there any description of the *category* $\mathsf{FinSet}$ using generators and relations, i.e. an elementary description of $\mathrm{Hom}\_{\mathrm{Cat}}(\mathsf{FinSet},-)$? Since every map of finite sets is a bijection followed by a monotonic map, or vice versa, I expect that we need face maps, degeneracies and transpositions as generators. What are the relations? Is this written down in the literature?


**Question 3.** An answer to question 2 will also describe presheaves on $\mathsf{FinSet}$, which are simplicial sets with a certain extra structure. Do they have a geometric interpretation and are these geometric objects used somewhere? In some sense, this corrects the failure of the [join of simplicial sets](https://mathoverflow.net/questions/7134/joins-of-simplicial-sets) to be commutative.


***Edit*** after Eric's comment. Ok, Q3 was already answered [on MO](https://mathoverflow.net/questions/57370/analogue-of-simplicial-sets). The objects are called *symmetric simplicial sets*. Relevant papers are [Higher Fundamental Functors for Simplicial Sets](http://arxiv.org/pdf/math/0009004v3.pdf) by M. Grandis and *Toposes generated by codiscrete objects in combinatorial topology and functional analysis* by F. W. Lawvere and [Left-determined model categories and universal
homotopy theories](http://www.ams.org/journals/tran/2003-355-09/S0002-9947-03-03322-1/) (Section 3) by J. Rosicky and W. Tholen.



The symmetric monoidal category of finite sets
Marco Grandis has done some work on this, and you can extract answers for 1-3 from these papers


* *Finite Sets and Symmetric Simplicial Sets* - M Grandis - TAC ([pdf](http://www.emis.ams.org/journals/TAC/volumes/8/n8/n8.pdf))
* *Higher Fundamental Functors for Simplicial Sets* - M Grandis - CTGDC ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_2001__42_2/CTGDC_2001__42_2_101_0/CTGDC_2001__42_2_101_0.pdf))


See also


* *An Alternative Presentation of the Symmetric-Simplicial Category* - Eric R. Antokoletz - arxiv ([link](http://arxiv.org/abs/1012.1587))


Question 1 and 2
----------------


The first paper by Grandis above gives a nice detailed overview of all this, including 


1. a description of (a skeleton of) $\mathbf{FinSet}$ as the walking symmetric strict monoidal category with a commutative monoid
2. in terms of generators and relations, with generators faces + degeneracies + transpositions, and relations the standard ones for faces + degeneracies, the Moore ones for transpositions, and some compatibility rels between those.


About who first proved this kind of things, in the second paper, he acknowledges that:



> 
> In November 1998, at a PSSL meeting in Trieste, Bill
>  Lawvere suggested I might extend my study of the homotopy of simplicial
>  complexes to symmetric simplicial sets, on the basis of his draft [19] where the
>  fundamental groupoid of the latter is presented as a left adjoint. I would like to
>  express my gratitude for his kind encouragement and for helpful discussions.
> 
> 
> 


[19] is Lawvere's *Toposes generated by codiscrete objects, in combinatorial topology and functional analysis* from 1989, which I don't have access to; maybe there's some more info in there.


Question 3
----------


You could find more about this in the second paper by Grandis. The idea is that the classical homotopy theory of *simplicial complexes* can be extended to symmetric simplicial sets (presheaves on $\mathbf{FinSet}$), so that the edge-path groupoid of a simplicial complex can be identified with what Grandis calls the intrinsic fundamental groupoid, which is the left adjoint of a symmetric nerve (this goes back to Lawvere notes ref above). See also 


* *An intrinsic homotopy theory for simplicial complexes, with applications to image analysis* - M Grandis ([pdf](http://www.dima.unige.it/~grandis/Cmb.pdf))


