Canonical map

In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. In general, it is the map which preserves the widest amount of structure,[1] and it tends to be unique. In the rare cases where latitude in choices remains, the map is either conventionally agreed upon to be the most useful for further analysis, or sometimes the most elegant map known up to date.

A standard form of canonical map involves some function mapping a set ${\displaystyle X}$ to the set ${\displaystyle X/R}$ (${\displaystyle X}$ modulo ${\displaystyle R}$), where ${\displaystyle R}$ is an equivalence relation on ${\displaystyle X}$.[2] A closely related notion is a structure map or structure morphism; the map or morphism that comes with the given structure on the object. These are also sometimes called canonical maps.

A canonical isomorphism is a canonical map that is also an isomorphism (i.e., invertible). In some contexts, it might be necessary to address an issue of choices of canonical maps or canonical isomorphisms; for a typical example, see prestack.