That sounds good. I have changed the page name to locally monoidal (infinity,1)-operad.

]]>Hmm, well, a *commutative theory*, or *monoidal monad*, is one whose algebras come with a tensor product. Since this is “an extra level up”, as Zoran says, maybe something like “locally commutative” or “locally monoidal”?

I don’t know, I am not aware of an explanation of the choice of term. Just some term. Make a suggestion for how to rather call it.

]]>P.S. why is it called “coherent” ?

That was my question too!

]]>It is amazing that the structure would induce the *tensor product* two levels up, not one level up, what was my first guess of the meaning. (first level algebras over operads, second algebras over algebras… so to speak) In Hopf algebras the coproduct on Hopf algebra induces rigid monoidal structure on the category of usual modules over Hopf algebras. So this thing here is very different…

P.S. why is it called “coherent” ?

]]>Sorry, what’s the question?

There is an operad $\mathcal{O}$, there is an $\mathcal{O}$-algebra $A$ and a category $A Mod$ of $A$-modules. Lurie calls $\mathcal{O}$ “coherent” if for all $A$ the $\infty$-category $A Mod$ is nice (has a well behaved tensor product)

]]>whose modules over $\mathcal{O}$-algebras

Modules over algebras over operad ??

]]>mentioned some of the examples.

]]>stub for coherent (infinity,1)-operad

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