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• CommentRowNumber1.
• CommentAuthorJon Beardsley
• CommentTimeJul 21st 2016

Couldn’t find a latest changes discussion for symmetric sequence so I am just reporting that I added a little bit to that page. In particular, I added another slicker definition in the case that we are interested in a symmetric sequence for the sequence of symmetric groups.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJul 22nd 2016

Where you mention the definition of operads as monoids, I have added a pointer to Operad – Definition as monoids.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeJul 24th 2016

If the slicker definition mentioned in #1 refers to Definition 2, then I’m sorry, but that’s a mistake. As definition 1 indicates, the category of symmetric sequences is equivalent to the category of just plain functors out of $\Sigma$ (the groupoid coproduct of finite permutation groups $\Sigma_n$) or just plain functors out of the category $FB$ of finite sets and bijections.

Now if $C$ is symmetric monoidal, then symmetric monoidal functors $F: FB \to C$ might (for some authors) mean strong symmetric monoidal functors, where “strong” means that the structural maps $F(S) \otimes F(T) \to F(S \otimes T)$ and $I \to F(I)$ are isomorphisms. But since for $\Sigma$ or $FB$ each object is isomorphic to $1^{\otimes n}$ for some $n$, we get that each strong symmetric monoidal functor $F: FB \to C$ is uniquely (up to isomorphism) determined by the value $F(1)$. More precisely, the category of strong symmetric monoidal functors $\Sigma \to C$ and monoidal natural transformations between them is equivalent to $C$, which is clearly not what we want. Even better, we could say that $\Sigma$ or $FB$ is the free symmetric monoidal category on one object, and this is a starting point of Kelly’s elegant theory of operads.

Even if symmetric monoidal functor is meant in the lax sense (where the structural constraints need not be isos), we still get some notion which differs from the intended one.

• CommentRowNumber4.
• CommentAuthorJon Beardsley
• CommentTimeJul 24th 2016

Oh yeah you’re definitely right. This, and the thing about operads being commutative monoids, was some kind of weird confused fugue state I was having in my brain. I realized it and changed it in a few places, but appear to have forgotten all the places I had done it. I’ll change that.