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In mathematics, a canonical map, also called a natural map, is: a map/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 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 (i.e., 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.

For a discussion of the problem of defining canonical map see Kevin Buzzard's talk at the 2022 Grothendieck conference.

Examples※

• If N is a normal subgroup of a group G, then there is a canonical surjective group homomorphism from G——to the quotient group G/N, that sends an element g to the coset determined by, g.
• If I is an ideal of a ring R, then there is a canonical surjective ring homomorphism from R onto the quotient ring R/I, that sends an element r to its coset I+r.
• 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: R → S is a homomorphism between commutative rings, 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 f: 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 set X → X/R (X modulo R), where R is an equivalence relation on X, that takes each x in X to the equivalence class ※ modulo R.

See also※

• Natural transformation

References※

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