33 lines
2.5 KiB
Markdown
33 lines
2.5 KiB
Markdown
---
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title: "Canonical map"
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chunk: 1/1
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source: "https://en.wikipedia.org/wiki/Canonical_map"
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category: "reference"
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tags: "science, encyclopedia"
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date_saved: "2026-05-05T12:04:28.941026+00:00"
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instance: "kb-cron"
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---
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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).
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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.
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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.
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== Examples ==
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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.
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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.
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If V is a finite-dimenstional 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).
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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. More generally, a scheme X over a scheme S is one equipped with a structure morphism X → S; for a scheme over a field k (e.g., a variety), this is a morphism X → Spec(k).
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If E is a vector bundle over a topological space X, then the projection map from E to X is the structure map.
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In topology, a canonical map is a function f mapping a set X → X / R (X mod R), where R is an equivalence relation on X, that takes each x in X to the equivalence class [x] mod R.
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== See also ==
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Natural transformation
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== References ==
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== External links ==
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Buzzard, Kevin (21 June 2022). "Grothendieck's approach to equality". YouTube, on the problem of defining a canonical map.{{cite web}}: CS1 maint: postscript (link) |