Canonical map

See also Natural transformation, a related concept in category theory.

For the canonical map of an algebraic variety into projective space, see Canonical bundle#Canonical maps.

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.

A closely related notion is a structure map or structure morphism; the map that comes with the given structure on the object. They are also sometimes called canonical maps.

Examples:

• If N is a normal subgroup of a group G, then there is a canonical map from G to the quotient group G/N that sends an element g to the coset that g belongs to.
• If V is a vector space, then there is a canonical map from V to the second dual space of V that sends a vector v to the linear functional f_v defined by f_v(λ) = λ(v).
• If f is a ring homomorphism from a commutative ring R to commutative ring S, then S can be viewed as an algebra over R. The ring homomorphism f is then called the structure map (for the algebra structure). The corresponding map on the prime spectra: Spec(S) →Spec(R) is also called the structure map.
• If E is a vector bundle over a topological space X, then the projection map from E to X is the structure map.