The word “locally” has several different meanings in category theory.

A category $C$ may be called “locally $P$” (for some property $P$) if each slice category $C/X$ is $P$. Examples include locally cartesian closed categories.

A 2-category $K$ (or more generally enriched category) may be called “locally $P$ (for some property $P$) if each hom-category (or more generally hom-object) $K[X, Y]$ is $P$. Examples include locally discrete 2-categories, and locally graded and locally indexed categories.

Categories are called locally presentable if they are accessible and cocomplete. The “locally” in the name is intended to disambiguate between such categories and the finitely presentable objects in the 2-category Cat.

Unfortunately, these meanings can overlap. For instance, the term “locally cartesian closed 2-category” could in theory refer either to a 2-category whose slice 2-categories are locally cartesian closed, or a 2-category each of whose hom-categories are locally cartesian closed. For this reason, it is often worth considering whether a different term than “locally” may be used when naming such a concept.

(Note that for the bicategory of spans in some finitely complete category $E$ the hom-category $Span(E)(A, B)$ is the slice category $E/(A \times B)$. Thus, hom-wise properties of $Span(E)$ coincide with slice-wise properties of $E$.)

Last revised on September 26, 2022 at 05:47:49. See the history of this page for a list of all contributions to it.