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].

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.[4] Saorín, M., Šťovíček, J., Virili, S. t-Structures on stable derivators and Grothendieck hearts.