nLab prestable (∞,1)-category

Idea

A prestable ∞-category axiomatizes the properties of the connective part of a t-structure on a stable ∞-category.

Definition

A prestable ∞-category is a pointed finitely cocomplete ∞-category $C$ with a fully faithful suspension functor such that the base change of any morphism $Y\to\Sigma Z$ along a map $0\to\Sigma Z$ exists and the resulting pullback square is also a pushout square.

The fully faithfulness condition can be rephrased by saying that the functor $C\to SW(C)$ is fully faithful, and then the base change condition can be reformulated by saying that the image of $C\to SW(C)$ is closed under extensions. Here $SW(C)$ is the Spanier-Whitehead category of $C$.

Properties

An ∞-category is prestable if and only if it is a full subcategory of a stable ∞-category closed under finite colimits and extensions.

Examples

Any stable ∞-category is prestable.

If $C$ is a stable ∞-category with a t-structure $(C_{\ge0},C_{\le0})$, then $C_{\ge0}$ is prestable. Any finitely complete prestable ∞-category arises in such a fashion, and there are two canonical choices for $C$: the Spanier-Whitehead category and the category of spectrum objects. In fact, any other choice can be squeezed in between these two.

Grothendieck prestable ∞-categories and a Gabriel-Popescu theorem

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?.

References

Lurie, Spectral Algebraic Geometry, §A.3.