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. Often, it is a map which preserves the widest amount of structure. A choice of a canonical map sometimes depends on a convention.
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. 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.
For a discussion of the problem of defining a canonical map see Kevin Buzzard's talk at the 2022 Grothendieck conference.

Examples

If is a normal subgroup of a group, then there is a canonical surjective group homomorphism from to the quotient group, that sends an element to the coset determined by.
If is an ideal of a ring, then there is a canonical surjective ring homomorphism from onto the quotient ring, that sends an element to its coset.
If is a finite-dimenstional vector space, then there is a canonical map from to the second dual space of, that sends a vector to the linear functional defined by.
If is a homomorphism between commutative rings, then can be viewed as an algebra over. The ring homomorphism is then called the structure map. The corresponding map on the prime spectra is also called the structure map.
If is a vector bundle over a topological space, then the projection map from to is the structure map.
In topology, a canonical map is a function mapping a set , where is an equivalence relation on, that takes each in to the equivalence class.