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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 to be commutative.

Edit after Eric's comment. Ok, Q3 was already answered on MO. The objects are called symmetric simplicial sets. Relevant papers are Higher Fundamental Functors for Simplicial Sets 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 (Section 3) by J. Rosicky and W. Tholen.

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Interesting observation. I was just looking at the Grothendieck–Serre correspondence the other day and noticed that Grothendieck refers to functors $\textbf{FinSet}^\textrm{op} \to \mathcal{C}$ as "simplicial objects in $\mathcal{C}$", so I also wondered whether there was any connection with simplicial objects as we understand them today. –  Zhen Lin Jan 3 '13 at 10:17
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For question 3, see mathoverflow.net/questions/57370/analogue-of-simplicial-sets –  Eric Wofsey Jan 3 '13 at 10:19
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2 Answers

up vote 11 down vote accepted

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)
  • Higher Fundamental Functors for Simplicial Sets - M Grandis - CTGDC (pdf)

See also

  • An Alternative Presentation of the Symmetric-Simplicial Category - Eric R. Antokoletz - arxiv (link)

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)
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Thank you, this is exactly what I was looking for. –  Martin Brandenburg Jan 3 '13 at 13:12
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Re Question 1, one place where this appears is in Example (d) on p.23 of my paper Homotopy algebras for operads (arXiv:math.QA/0002180). There I used $\mathrm{CMon}$ to denote the terminal symmetric operad (whose algebras are commutative monoids) and $\widehat{\mathrm{CMon}}$ to denote the free symmetric monoidal category containing a $\mathrm{CMon}$-algebra (a commutative monoid). The statement made there is that $\widehat{\mathrm{CMon}}$ is a skeleton of the category of finite sets.

I'm absolutely sure that this wasn't original to me, but I don't know what the correct reference is.

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