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Added a reference
Presumably the equivariant derived category approach suffers from all the deficiencies of derived categories (as in sec 2.1.1 of Weil’s Conjecture for Function Fields I) and the modal cohesive approach points a better way.
for example by definition […] such as recalled in the above reference presumably
That’s a way of putting it. :-)
But I guess David was meaning to wonder about the relation to equivariant stable homotopy theory.
The answer to that is roughly that by “the” derived/stable-$\infty$ category associated with a ($G$-)space/scheme/whatnot one means the homotopy/$\infty$-category of all sheaves of spectra over it (or often just the linear ones over some ground field). Any single object in here may be regarded as representing a twisted equivariant Whitehead-generalized cohomology theory over the given $G$-space/scheme, in that its global (co-)sections are its twisted equivariant generalized (co)homology groups.
The key issue here will be, as usual, if the equivariance on the spectra is implemented Borel-ly (“naively”) or Bredon-ly (less naively) or “genuinely” (with transfers) and this will depend on what authors do. People who say “the derived/stable-$\infty$ category” of a scheme typically care about it as a motivic reflection of that scheme, not as the home of its twisted cohomologies and may have attitudes towards equivariance that the stable homotopy theorists, in their infinite wisdom, have declared to be “naive”.
I suspect/presume this is the case for the thesis that David pointed to, but I haven’t dived into it.
Sorry, it was a bit of a throw away comment. There was the Gaitsgory and Lurie point
The theory of derived categories is a very useful tool in homological algebra, but has a number of limitations. [the derived category] is not very well-behaved from a categorical point of view.
And then, as Urs writes, which form of equivariance.
Just to be clear, the issue with “derived categories” is simply that they are just the homotopy categories of full stable $\infty$-categories, and as such have lost all of the higher homotopy information. This is an old hat, being the source of a plethora of models for “enhanced triangulated categories” where people tried to add more higher homotopy information back in. The modern (i.e. Lurie’s) notion of stable $\infty$-category is in a way the culmination point of this search for the full structure of which derived categories are the 1-truncated shadows.
This used to be a point that occupied people’s energy, but it has now long been settled to the extent that it’s regarded as a triviality, or at least as a basic fact of the field, as witnessed in the guest comment in #3.
Just to add that there are many situations where it’s perfectly sensible to restrict attention to the homotopy category of an $\infty$-category, hence to a “derived category” or “triangulated category” when the corresponding $\infty$-category is stable. Even more: there are many (though less) situations where one just needs to know the set of equivalence classes of an $\infty$-category. Nothing wrong with that notion, one just needs to know that it’s a faint shadow of something richer.
$\overset{ { pre\, K } \atop { group } }{ \tau_0(\mathcal{C}) } \overset{\;\;\;\;}{\longleftarrow} \overset{ { {triangulated} \atop {(derived)} } \atop {category} }{ \tau_1(\mathcal{C}) } \overset{\;\;\;\;}{\longleftarrow} \cdots \overset{\;\;\;\;}{\longleftarrow} \overset{ { stable } \atop \infty\text{-}category }{ \mathcal{C} }$1 to 6 of 6