nLab compactly assembled (∞,1)-category

Idea and definition

Compactly assembled (∞,1)-categories are characterized among presentable (∞,1)-categories by the following equivalent conditions (due to Clausen and Lurie):

• They are retracts in the (∞,1)-category of presentable (∞,1)-categories and left adjoints of compactly generated (∞,1)-categories.

• Small (∞,1)-limits distribute over (∞,1)-filtered (∞,1)-colimits.

• The colimit (∞,1)-functor $Ind(C)\to C$ admits a left adjoint.

• See Krause & Nikolaus 2024, Thm 2.2.11 for a few other conditions.

Examples

(∞,1)-sheaves of ∞-groupoids on a locally compact Hausdorff topological space form a compactly assembled (∞,1)-category.

Related concepts

• Efimov K-theory

• compactly generated (∞,1)-category

References

• Achim Krause, Thomas Nikolaus, Sheaves on manifolds (2024) [pdf, pdf]