A prestable ∞-category axiomatizes the properties of the connective part of a t-structure on a stable ∞-category.
A prestable ∞-category is a pointed finitely cocomplete ∞-category with a fully faithful suspension functor such that the base change of any morphism along a map exists and the resulting pullback square is also a pushout square.
The fully faithfulness condition can be rephrased by saying that the functor is fully faithful, and then the base change condition can be reformulated by saying that the image of is closed under extensions. Here is the Spanier-Whitehead category of .
An ∞-category is prestable if and only if it is a full subcategory of a stable ∞-category closed under finite colimits and extensions.
Any stable ∞-category is prestable.
If is a stable ∞-category with a t-structure , then is prestable. Any finitely complete prestable ∞-category arises in such a fashion, and there are two canonical choices for : the Spanier-Whitehead category and the category of spectrum objects. In fact, any other choice can be squeezed in between these two.
A prestable ∞-category is Grothendieck if it is presentable and filtered colimits are left exact.
There is a Gabriel-Popescu theorem for prestable ∞-categories: the class of Grothendieck prestable ∞-categories coincides with the class of accessible left exact localizations of connective modules over a connective E_1-ring?.
Lurie, Spectral Algebraic Geometry, §A.3.
Last revised on February 24, 2016 at 15:07:20. See the history of this page for a list of all contributions to it.