Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
I think Proposition 3.1.5 in Leinster's "Homotopy algebras for operads" is relevant. Namely, $\Gamma^{op}$ is the Kleisli category for the monad $1\times -$ on finite sets (and Fin is the prop for the commutative monoid operad, as you say). Similarly, $\Delta^{op}$ is the Kleisli category for the monad $1\times - \times 1$ on finite ordinals. This passage was studied more systematically by Barwick in "From operator categories to topological operads".
Sorry for the edits: I've been confused by this. I'm pretty convinced that what I wrote at the end is true. I also now realize that this is the statement underlying what Groth wrote. However, I still wonder if what Leinster was thinking is true, i.e. that special $\Gamma$-categories are equivalent to symmetric monoidal categories.
Perhaps you should first observe that chain complexes (over a commutative ring) are tensored and cotensored over modules. If you're into algebraic topology, you'll see that any reasonable category of spectra is tensored and cotensored over pointed spaces (or pointed simplicial sets).
Nice example! One could say that the "monoid axiom", which holds for chain complexes, guarantees that the monoids get a model category structure, but it doesn't mean that the category of monoids itself has to be a monoidal model category (for monoidal model categories, the tensor product of two cofibrant objects is cofibrant).
I think you misread: he says that, for $R$ a $\mathbb{Q}$-algebra, $\psi$ is an equivalence (without connectivity restrictions), and $\phi$ is an equivalence onto its essential image which consists of the bounded below ("connective") chain complexes. $$$$ In the DAG reference given by Marc above, he goes directly from the simplicial into the spectra world. To see a comparison of $E_\infty$ $R$-algebras with commutative dgas, see Higher Algebra, Proposition 7.1.4.11.
