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].
References:
[1] Hogadi, A.,
Xu, C. Products, homotopy limits and applications. arXiv:0902.4016 (2009).
[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).