Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
For a regular cardinal, an (∞,1)-category is called -compactly generated if it is -accessible and locally presentable.
Terminology The terms “-compactly generated (∞,1)-category” and “locally -presentable (∞,1)-category” have the same meaning. There are differences in usage, though.
If we “drop the ”, then a locally presentable (∞,1)-category is a an (∞,1)-category which is locally -presentable for some , but a compactly generated (∞,1)-category is an (∞,1)-category which is locally finitely presentable, i.e. locally -presentable for .
If we “leave the in”, the terms “-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, “-compactly generated (∞,1)-category” is preferred. Albeit, Lurie uses to denote the (∞,1)-category of -compactly generated (∞,1)-categories.
Compact generation for stable (∞,1)-categories is detected already on their triangulated homotopy category as discussed at compactly generated triangulated category
Jacob Lurie, section 5.5.7 of Higher Topos Theory
Jacob Lurie, section 1.4.4 of Higher Algebra
Last revised on March 18, 2024 at 22:13:02. See the history of this page for a list of all contributions to it.