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First of all, let me see if i got the 1-categorical version right:

• Let $\mathcal F:C\to Cat $ be a (pseudo-) functor. The 2-colimit $\mathrm{colim}_C\mathcal F$ is then given by the grothendieck construction $\int_C \mathcal F$ and the 2-limit is given by the category of cartesian sections of the fibration $\int_C \mathcal F\to C$, right?

Can this be transported to the setting of dg$_k$-categories? So:

1. Is there a notion of fibration of dg categories? I would imagine them to be algebras for a dg-monad $(\mathrm{id}_C,-)$ arising from forming dg-comma categories with the identity-span on $C$.
2. What about a grothendieck construction for functors $\mathcal F: C\to \mathrm{dgCat}$ from a category $C$ to dg-Categories?
3. Cartesian sections should then be defined as algebra-morphisms from the identity on $C$ to $\int_C \mathcal F$.

I think there are some problems with what i just said: What are dg-comma categories? What are the right functors $C\to\mathrm{dgCat}$? (I guess one should build a dg-category $C'$ out of $C$ by taking the free $k$-category and then consider it as a dg-category, concentrated in degree 0, and then consider $\mathrm{dgCat}(C',\mathrm{dgCat})$) Same goes for the definition of an algebra morphism: What are the coherences to consider?

So i guess the core question is:

• What is the right notion of limit for dg categories? (And why?)

First of all, let me see if I got the 1-categorical version right:

• Let $\mathcal F:C\to Cat $ be a (pseudo-) functor. The 2-colimit $\mathrm{colim}_C\mathcal F$ is then given by the Grothendieck construction $\int_C \mathcal F$ and the 2-limit is given by the category of Cartesian sections of the fibration $\int_C \mathcal F\to C$, right?

Can this be transported to the setting of dg$_k$-categories? So:

1. Is there a notion of fibration of dg categories? I would imagine them to be algebras for a dg-monad $(\mathrm{id}_C,-)$ arising from forming dg-comma categories with the identity-span on $C$.
2. What about a Grothendieck construction for functors $\mathcal F: C\to \mathrm{dgCat}$ from a category $C$ to dg-Categories?
3. Cartesian sections should then be defined as algebra-morphisms from the identity on $C$ to $\int_C \mathcal F$.

I think there are some problems with what i just said: What are dg-comma categories? What are the right functors $C\to\mathrm{dgCat}$? (I guess one should build a dg-category $C'$ out of $C$ by taking the free $k$-category and then consider it as a dg-category, concentrated in degree 0, and then consider $\mathrm{dgCat}(C',\mathrm{dgCat})$) Same goes for the definition of an algebra morphism: What are the coherences to consider?

So I guess the core question is:

• What is the right notion of limit for dg categories? (And why?)

First of all, let me see if i got the 1-categorical version right:

• Let $\mathcal F:C\to Cat $ be a (pseudo-) functor. The 2-colimit $\mathrm{colim}_C\mathcal F$ is then given by the grothendieck construction $\int_C \mathcal F$ and the 2-limit is given by the category of cartesian sections of the fibration $\int_C \mathcal F\to C$, right?

Can this be transported to the setting of dg-categories? So:

1. Is there a notion of fibration of dg categories? I would immagine them to be algebras for a dg-monad $(\mathrm{id}_C,-)$ arising from forming dg-comma categories with the identity-span on $C$.
2. What about a grothendieck construction for functors $\mathcal F: C\to \mathrm{dgCat}$ from a category $C$ to dg-Categories?
3. Cartesian sections should then be defined as algebra-morphisms from the identity on $C$ to $\int_C \mathcal F$.

I think there are some problems with what i just said: What are dg-comma categories? What are the right functors $C\to\mathrm{dgCat}$? (I guess one should build a free dg-category out of $C$ ) Same goes for the definition of an algebra morphism: What are the coherences to consider?

So i guess the core question is:

• What is the right notion of limit for dg categories? (And why?)

First of all, let me see if i got the 1-categorical version right:

• Let $\mathcal F:C\to Cat $ be a (pseudo-) functor. The 2-colimit $\mathrm{colim}_C\mathcal F$ is then given by the grothendieck construction $\int_C \mathcal F$ and the 2-limit is given by the category of cartesian sections of the fibration $\int_C \mathcal F\to C$, right?

Can this be transported to the setting of dg$_k$-categories? So:

1. Is there a notion of fibration of dg categories? I would imagine them to be algebras for a dg-monad $(\mathrm{id}_C,-)$ arising from forming dg-comma categories with the identity-span on $C$.
2. What about a grothendieck construction for functors $\mathcal F: C\to \mathrm{dgCat}$ from a category $C$ to dg-Categories?
3. Cartesian sections should then be defined as algebra-morphisms from the identity on $C$ to $\int_C \mathcal F$.

I think there are some problems with what i just said: What are dg-comma categories? What are the right functors $C\to\mathrm{dgCat}$? (I guess one should build a dg-category $C'$ out of $C$ by taking the free $k$-category and then consider it as a dg-category, concentrated in degree 0, and then consider $\mathrm{dgCat}(C',\mathrm{dgCat})$) Same goes for the definition of an algebra morphism: What are the coherences to consider?

So i guess the core question is:

• What is the right notion of limit for dg categories? (And why?)