nLab compactly generated (∞,1)-category

Definition

For $\kappa$ a regular cardinal, an (∞,1)-category is called $\kappa$-compactly generated if it is $\kappa$-accessible and locally presentable.

Terminology The terms “$\kappa$-compactly generated (∞,1)-category” and “locally $\kappa$-presentable (∞,1)-category” have the same meaning. There are differences in usage, though.

• If we “drop the $\kappa$”, then a locally presentable (∞,1)-category is a an (∞,1)-category which is locally $\kappa$-presentable for some $\kappa$, but a compactly generated (∞,1)-category is an (∞,1)-category which is locally finitely presentable, i.e. locally $\kappa$-presentable for $\kappa = \aleph_0$.

• If we “leave the $\kappa$ in”, the terms “$\kappa$-compactly generated (∞,1)-category” and “locally $\kappa$-presentable (∞,1)-category” have the same meaning. Some authors choose one term over the other. For example, in Higher Topos Theory, “$\kappa$-compactly generated (∞,1)-category” is preferred. Albeit, Lurie uses $\mathrm{Pr}_\kappa$ to denote the (∞,1)-category of $\kappa$-compactly generated (∞,1)-categories.

(HTT, def. 5.5.7.1)

Properties

Recognition for stable $(\infty,1)$-categories

Compact generation for stable (∞,1)-categories is detected already on their triangulated homotopy category as discussed at compactly generated triangulated category

(HA, remark 1.4.4.3))

Related concepts

• ind-object in an (∞,1)-category

• compact object in an (∞,1)-category

• compactly generated model category

• compactly generated triangulated category

• compactly assembled (∞,1)-category

References

• Jacob Lurie, section 5.5.7 of Higher Topos Theory

• Jacob Lurie, section 1.4.4 of Higher Algebra