compactly generated (∞,1)-category


(,1)(\infty,1)-Category theory

Compact objects



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

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

  • If we “drop the κ\kappa”, then a locally presentable (∞,1)-category is a an (∞,1)-category which is locally κ\kappa-presentable for some κ\kappa, but a compactly generated (∞,1)-category is an (∞,1)-category which is locally finitely presentable, i.e. locally κ\kappa-presentable for κ= 0\kappa = \aleph_0.

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

(HTT, def.


Recognition for stable (,1)(\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


Last revised on March 22, 2016 at 19:56:26. See the history of this page for a list of all contributions to it.