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I have created an entry model structure on dg-modules in order to record some references and facts.
I think using this I now have one version of the statement at derived critical locus (schreiber) that is fully precise. But I am still trying to see a better way. This is fiddly, because
contrary to what one might expect, thre is not much at all in the literature on general properties homotopy limits/colimits in dg-geometry;
and large parts of the standard toolset of homotopy theory of oo-algebras does not apply:
the fact that we are dealing with commutative dg-algebras makes all Schwede-Shipley theory not applicable,
the fact that we are dealing with oo-algebras in chain complexes makes all Berger-Moerdijk theory not apply;
and finally the fact that we are dealing with dg-algebras under another dg-algebra makes Hinich’s theory not apply!
That doesn’t leave many tools to fall back to.
I do not understand the last statement. Are you talking about undercategories or algebra in categories of modules over dg algebra ?
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