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.
- 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 fv defined by fv(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.
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