If A is an object in a category
𝒜, the properties of 𝒜∕A (the category of objects over A) may be considered as local
properties of 𝒜. Using ‘local’ in this sense, the notion of local universality is defined
and some of its basic properties developed. These ideas are then applied in a brief
discussion of local adjunction and local limits. Finally two local tripleability theorems
are given.