nLab locally

This page is concerned with notions of locality in category theory. For notions in physics see instead e.g. causal locality or local field theory (which overlaps with a category-theoretic meaning via the notion of extended functorial field theories).

Meanings

In category theory, the term “locally” is used in several ways with different (and not always compatible) meanings.

Slice-wise

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

Examples include the notion of locally cartesian closed categories, i.e. those categories all whose slice categories are cartesian closed categories.

Hom-wise

A 2-category $K$ (or more generally an 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-small categories?, locally discrete 2-categories, and locally graded and locally indexed categories. Corresponding, we have terminology like local colimits.

Stalk-wise

In topos theory the term “local” typically refers to properties that hold after, possibly, passage to a cover or (if there are enough points) stalk-wise.

Compare the notion of local isomorphism or the term Local Homotopy Theory for $\infty$-sheaf-theory [Jardine 2015]

Via the case of 2-sheaves, i.e. category-valued stacks or internal categories in toposes, this notion of “local” also applies to categories.

Local presentability

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.

Ambiguity

Unfortunately, these meanings can overlap. For instance, the term “locally cartesian closed 2-category” could in principle 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 $E$.)

Related pages

• bi-

category: disambiguation