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By homotopy pullback/pushout, I will take any construction that is equivalent to the derived pullback/pushout constructed using the projective/injective model structures on diagrams in a combinatorial model category modelling $(\infty,n)$-categories.
If I understand this right, you can take the multicolored operad coming out of your equivalence and take a sub singly colored operad which is weakly equivalent to it. This would give the desired simplicial operad.
You and Gijs have definitely convinced me that things work out if I allow the essential image of singly-colored operads in the multi-colored operads. I would like to know if this fattening is necessary (since many existing questions concern the smaller category only).
Right Urs. The question involves a subtlety about what kind of weak equivalences I am allowing and I think I have been vague (mostly due to conflicting nomenclature for operads). I'm allowing arbitrary equivalences of $\infty$-operads, but only weak equivalences between (singly-colored) simplicial operads. The point being if $\infty$-operads is indeed a framework where one can study classical (singly-colored) operads then I should be able to take my operad, turn it into an $\infty$-operad, and obtain a weakly equivalent (singly-colored) operad.
Yes welcome Gijs! Thanks for the answer to Q2, I do mean up to isomorphism. Regarding Q1, I just want to be clear: If I take an $\infty$-operad with a single equivalence class of objects, is there some construction which will take that $\infty$-operad and produce a simplicial (singly-colored) operad (not something weakly equivalent to a (singly-colored) operad)? Of course, this operad should have the property that if I apply the above construction I obtain an $\infty$-operad weakly equivalent to the one I started with.
Hi Urs, thank you for the answer, but since the category of simplicial operads you are talking about is necessarily bigger than the classical category of simplicial operads (the one you mention models simplicial multicategories I believe, while the one I'm talking about is still the one object version), I think this just pushes the same question into a different framework. Namely these adjunctions will give me a simplicial multicategory corresponding to an $\infty$-operad. Does that multicategory have one object, i.e., is it an actual simplicial operad?
Yes I believe Zhen is talking about the multisorted case and you will get the single-sorted case by taking a single object (a small projective generator). Note the algebras over a multi/singly-sorted theory correspond to cocomplete algebraic categories with a set of projective generators/a single projective generator in the sense of Quillen.
@David: The tensor product with a fixed commutative algebra preserves finite products of modules since they are also finite coproducts, so it preserves finite products of commutative algebras which can be calculated as products in modules. Now we just need to check that it preserves equalizers which can be calculated as the kernel of the difference of the maps, since our fixed commutative algebra is \emph{flat} (since we are working over the reals) tensoring with our algebra preserves kernels.
@Thomas: I don't localize at the inclusion $S^0\rightarrow QS^0$ but the map from $\mathbb{P}S^0\rightarrow QS^0$ adjoint to that map in the category of $E^\infty$ spaces. Here $\mathbb{P}S^0$ is the free $E_\infty$ space on the pointed space $S^0$.
