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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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