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It is known that differential graded categories (or, also, $A_\infty$-categories) are 'incarnations', in some sense, of (stable) $(\infty,1)$-categories. I'm not used to the theory of $(\infty,1)$-categories, but I've read that one has a notion of $(\infty,1)$-(co)limit in an $(\infty,1)$-category, for example as explained in this answer. Moreover, I found a paper by D. Gaitsgory which speaks about continuous and cocontinuous (quasi-)functors between dg-categories. Also, cones of closed degree $0$ morphisms in a pretriangulated dg-category are clearly an example of (homotopy) colimit.

So, I believe there is a way to define "dg-(co)limits" in a dg-category. What is a possible pattern? Is there any reference about that?

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This discussion on Kan extensions in simplicially enriched categories should translate directly to the dg-setting: ncatlab.org/nlab/show/homotopy+Kan+extension#InKanCplCat One then gets (co)limits as a special case. Alternatively, "most" dg-categories are actually dg-enriched model categories, and one can use the notion of homotopy (co)limit there. –  Adeel Aug 7 at 11:27
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Of course, as with Kan-enriched categories, one must first start with an adequate theory of homotopy limits in the base category (in this case, dg-modules). –  Zhen Lin Aug 7 at 11:30

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In your linked paper, $\mathrm{Funct}_{\mathrm{cont}}(\mathbf{C}_1,\mathbf{C}_2)$ is the full subcategory of $\mathrm{Funct}(\mathbf{C}_1,\mathbf{C}_2)$, of $k$-linear functors, spanned by those functors that commute with infinite direct sums, so infinite direct sums are like (homotopy) colimits. Lemma 1.1.1. of the same paper defines cocontinuous functors as well, and it's not hard to see what they commute with. It's also easy to see that this is a special case of the Kan extension discussion at the nlab which Adeel linked to in the comments.

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