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prestable (∞,1)-category

Prestable ∞-category

Prestable ∞-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 CC with a fully faithful suspension functor such that the base change of any morphism YΣZY\to\Sigma Z along a map 0ΣZ0\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 CSW(C)C\to SW(C) is fully faithful, and then the base change condition can be reformulated by saying that the image of CSW(C)C\to SW(C) is closed under extensions. Here SW(C)SW(C) is the Spanier-Whitehead category of CC.

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 CC is a stable ∞-category with a t-structure (C 0,C 0)(C_{\ge0},C_{\le0}), then C 0C_{\ge0} is prestable. Any finitely complete prestable ∞-category arises in such a fashion, and there are two canonical choices for CC: 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.

Last revised on February 24, 2016 at 10:07:20. See the history of this page for a list of all contributions to it.