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 (e.g., a sign convention).
A closely related notion is that of 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 (that is, 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.
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 multiplicative set in commutative ring , then there is a canonical ring homomorphism from to the localization that sends element to in the localization. If is the localization of commutative ring by , and is the localization of by , where is a prime ideal such that , there is a canonical ring homomorphism mapping as an element of to as an element of . If φ is the canonical map and ψ is the canonical map , then α is the unique map such that . 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 (for the algebra structure). The corresponding map on the prime spectra is also called the structure map. More generally, a scheme X over a scheme S is one equipped with a structure morphism ; for a scheme over a field k (e.g., a variety), this is a morphism . 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 .
See also Natural transformation
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