# 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.

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