Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has algebro-geometric application in mind they seem more attractive then stable $(\infty,1)-$categories which seem to carry data in a slightly more convoluted way (which I realize makes for more powerful techniques and generalizations).

So far though I've only seen a very limited amount of how actual algebraic geometry looks from a DG point of view and most of the stuff I read about DG categories was either definitions or general theory (papers by Toen for example). Here are several questions I have in mind:

• What is the "correct" DG category associated to a scheme/algebraic space/stack?

 
• Can the different possiblities here be organized as different "DG-stacks" (of certain dg-categories of sheaves) on the relevant site?

 
• How can I see the classical "category" of derived categories as some kind of "category" of homotopy categories of dg-categories? (I'm putting category in brackets since i'm not sure that there's such an object, what I really want is to really understand the link between all the classical theory of derived categories and dg-categories). In particular the six functor formalism.

I realize that these question might not have a straight yes/no answer and so what I'm looking for is a kind of roadmap to the relevant litrature where the application and formalization of the place of dg-categories in algebraic geometry is discussed.

Main question: What are some relevant articles/notes/books which establish and discuss the details of the formalism of DG-categories in the algebro-geometric world?

Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has algebro-geometric application in mind they seem more attractive then stable $(\infty,1)-$categories which seem to carry data in a slightly more convoluted way (which I realize makes for more powerful techniques and generalizations).

So far though I've only seen a very limited amount of how actual algebraic geometry looks from a DG point of view and most of the stuff I read about DG categories was either definitions or general theory (papers by Toen for example). Here are several questions I have in mind:

• What is the "correct" DG category associated to a scheme/algebraic space/stack?

• Can the different possiblities here be organized as different "DG-stacks" (of certain dg-categories of sheaves) on the relevant site?

• How can I see the classical "category" of derived categories as some kind of "category" of homotopy categories of dg-categories? (I'm putting category in brackets since i'm not sure that there's such an object, what I really want is to really understand the link between all the classical theory of derived categories and dg-categories). In particular the six functor formalism.

I realize that these question might not have a straight yes/no answer and so what I'm looking for is a kind of roadmap to the relevant litrature where the application and formalization of the place of dg-categories in algebraic geometry is discussed.

Main question: What are some relevant articles/notes/books which establish and discuss the details of the formalism of DG-categories in the algebro-geometric world?

Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has algebro-geometric application in mind they seem more attractive then stable $(\infty,1)-$categories which seem to carry data in a slightly more convoluted way (which I realize makes for more powerful techniques and generalizations).

So far though I've only seen a very limited amount of how actual algebraic geometry looks from a DG point of view and most of the stuff I read about DG categories was either definitions or general theory (papers by Toen for example). Here are several questions I have in mind:

• What is the "correct" DG category associated to a scheme/algebraic space/stack?

• Can the different possiblities here be organized as different "DG-stacks" (of certain dg-categories of sheaves) on the relevant site?

• How can I see the classical "category" of derived categories as some kind of "category" of homotopy categories of dg-categories? (I'm putting category in brackets since i'm not sure that there's such an object, what I really want is to really understand the link between all the classical theory of derived categories and dg-categories).

I realize that these question might not have a straight yes/no answer and so what I'm looking for is a kind of roadmap to the relevant litrature where the application and formalization of the place of dg-categories in algebraic geometry is discussed.

Main question: What are some relevant articles/notes/books which establish and discuss the details of the formalism of DG-categories in the algebro-geometric world?

Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has algebro-geometric application in mind they seem more attractive then stable $(\infty,1)-$categories which seem to carry data in a slightly more convoluted way (which I realize makes for more powerful techniques and generalizations).

So far though I've only seen a very limited amount of how actual algebraic geometry looks from a DG point of view and most of the stuff I read about DG categories was either definitions or general theory (papers by Toen for example). Here are several questions I have in mind:

• What is the "correct" DG category associated to a scheme/algebraic space/stack?

• Can the different possiblities here be organized as different "DG-stacks" (of certain dg-categories of sheaves) on the relevant site?

• How can I see the classical "category" of derived categories as some kind of "category" of homotopy categories of dg-categories? (I'm putting category in brackets since i'm not sure that there's such an object, what I really want is to really understand the link between all the classical theory of derived categories and dg-categories). In particular the six functor formalism.

I realize that these question might not have a straight yes/no answer and so what I'm looking for is a kind of roadmap to the relevant litrature where the application and formalization of the place of dg-categories in algebraic geometry is discussed.

Main question: What are some relevant articles/notes/books which establish and discuss the details of the formalism of DG-categories in the algebro-geometric world?