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 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 a canonical map see Kevin Buzzard's talk at the 2022 Grothendieck conference.[1]
If I is an ideal of a ringR, then there is a canonical surjective ring homomorphism from R onto the quotient ringR/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 functionalfv defined by fv(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 spectraf*: Spec(S) → Spec(R) is also called the structure map.
In topology, a canonical map is a function f mapping a 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 [x] modulo R.[2]
canonical, also, natural, transformation, related, concept, category, theory, canonical, algebraic, variety, into, projective, space, canonical, bundle, mathematics, canonical, also, called, natural, morphism, between, objects, that, arises, naturally, from, d. 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 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 a canonical map see Kevin Buzzard s talk at the 2022 Grothendieck conference 1 Examples EditIf 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 fv defined by fv l l 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 a 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 x modulo R 2 References Edit Buzzard Kevin Grothendieck Conference Talk Vialar Thierry 2016 12 07 Handbook of Mathematics BoD Books on Demand p 274 ISBN 9782955199008 This mathematics related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Canonical map amp oldid 1111119532, wikipedia, wiki, book, books, library,
