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In 1-categorical algebra one knows that, in a locally presentable category, every algebra for a finite product theory is a colimit of free algebras. Is the same true for algebras of finite product theories in presentable $\infty$-categories? I would already be content to know this for $E_\infty$-algebras and modules over a fixed $E_\infty$-algebra in a presentable $\infty$-category. Thanks!

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    $\begingroup$ Going from "finite product theory" to "presentable $\infty$-category" is a vast leap. In the first place, without a fixed forgetful functor, there is no reasonable notion of "free", and even locally finitely presentable 1-categories do not come equipped with such a forgetful functor. $\endgroup$ – Zhen Lin Aug 29 '13 at 23:23
  • $\begingroup$ Oh sorry, I didn't note how this could be misunderstood. I wanted to know about an $\infty$-categorical version of the 1-categorical statement. I clarified the question accordingly... $\endgroup$ – Who Aug 30 '13 at 15:37
  • $\begingroup$ I'd say this would be better asked at MathOverflow. $\endgroup$ – user43208 Sep 11 '13 at 0:53

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