Anyone know of a reference for the statement (I think it's true?) that an essentially surjective and symmetric monoidal functor of ∞-categories F:C→D induces an essentially surjective functor of module ∞-categories LMod_{A}(C)→LMod_{F(A)}(D)?
@JonathanBeardsley I don't think it's true... what about something like Set-->Span(Set)? This is symmetric monoidal for cartesian product and essentially surjective, it also takes free monoids to free monoids. A module over the natural numbers N is just a set equipped with an endomorphism; but there are way more endomorphisms in the category of spans.
On the other hand, it'll be true if F preserves sifted colimits (and C and D have sifted colimits)
Consider the projective and injective model structures on chain complexes -- they present the same $\infty$-category via the same ordinary category, but with different fibrant-cofibrant objects. Is there anything enlightening to say about such a situation from an $\infty$-categorical perspective? Presenting via the projective model structure seems closely related to the intrinsically-defined $P_\Sigma(Free_{fg}(R))$, but I'm not sure how to motivate the injective model structure intriniscallly.
And I'm even more in the dark about what to say about the relationship between the two model structures $\infty$-categorically.
I suppose I'm even confused about the relationship between $P_\Sigma(Free_{fg}(R))$ and the projective model structure. Is there a conceptual reason why taking simplicial objects in the free sifted-colimit completion of an ordinary category should give you a category which admits a model structure presenting the $\infty$-categorical version of this construction?
@ufabao It looks like $T$ was assumed to be a finite spectrum and hence dualizable in the category of spectra. So this equation holds in the category of spectra. I think you're asking about the third line of page 21, so the question is why this holds in C/J. I would assume the appropriate functor involved is strong symmetric monoidal and so preserves dualizable objects. But I would have to chase down the definitions.
