Abstract: Left and right completeness of a derived category D(A) with respect to its standard t-structure have been introduced by Neeman [3], specializing a more general (and more precise) notion for (∞,1)-categories given by Lurie [2, Chapter 2]. Roughly speaking, D(A) is left-complete if any of its objects X is the homotopy limit of its truncations. In this language, one of the main results of [1] can be reinterpreted by saying that a weak exactness of products in A implies the left-completeness of D(A). 
Given a stable derivator D and a t-structure t=(U,V) on its base D(1), we will show how to construct a left (resp., right, two-sided) completion of D with respect to t and we will say that D is left (resp., right, two-sided) complete if it is equivalent to its completion. We will show how this relates to the notions for derived and (∞,1)-categories, and we will show that a weak closure of U under products is sufficient to ensure left completeness, thus extending, and putting in a new framework, the results in [1]. 
This talk is based on the resent preprints [4] and [5].

[1] Hogadi, A., Xu, C. Products, homotopy limits and applications. arXiv:0902.4016 (2009).
[2] Lurie, J. Higher Algebra. Available at http://www.math.harvard.edu/~lurie/
[3] Neeman, A. Non-left-complete derived categories. Mathematical Research Letters 18(5), 827–832 (2011)
[4] Saorín, M., Šťovíček, J., Virili, S. t-Structures on stable derivators and Grothendieck hearts. arXiv:1708.07540 (2017).
[5] Virili, S. Morita theory for stable derivators. arXiv:1807.01505 (2018).