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 being mapped against each other. In general it is the map which preserves the widest amount of structure, and it tends to be unique. In the rare cases where latitude in choice remains, the map is either conventionally agreed upon to be the most useful for further analysis, or sometimes simply the most elegant or beautiful known.

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.

A canonical isomorphism is a canonical map that is also an isomorphism (i.e., invertible).

In some contexts, it is necessary to address an issue of choices of canonical maps or canonical isomorphisms; see prestack for a typical example.

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.
• In topology, a canonical map is a function f mapping a set X → X (X modulo R), where R is an equivalence relation in X, that takes each x in X to the equivalence class [[x]] modulo R.^[1]

References