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

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?.