canonical map
{{Short description|Mathematical mapping between objects arising from their definitions}}
{{dablink|For the canonical map of an algebraic variety into projective space, see {{section link|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.{{Cite web|url=https://www.youtube.com/watch?v=-OjCMsqZ9ww&list=PLXlinOq24a9Q8GPa5_mQfLUEn8ZCg8pg-&index=24|title=Grothendieck Conference Talk|last=Buzzard|first=Kevin|website=YouTube |date=21 June 2022 }}
Examples
- If {{mvar|N}} is a normal subgroup of a group {{mvar|G}}, then there is a canonical surjective group homomorphism from {{mvar|G}} to the quotient group {{math|G / N}}, that sends an element {{mvar|g}} to the coset determined by {{mvar|g}}.
- If {{mvar|I}} is an ideal of a ring {{mvar|R}}, then there is a canonical surjective ring homomorphism from {{mvar|R}} onto the quotient ring {{math|R / I}}, that sends an element {{mvar|r}} to its coset {{math|I + r}}.
- If {{mvar|V}} is a vector space, then there is a canonical map from {{mvar|V}} to the second dual space of {{mvar|V}}, that sends a vector {{mvar|v}} to the linear functional {{mvar|fv}} defined by {{math|1=fv(λ) = λ(v)}}.
- If {{math|f: R → S}} is a homomorphism between commutative rings, then {{mvar|S}} can be viewed as an algebra over {{mvar|R}}. The ring homomorphism {{mvar|f}} is then called the structure map (for the algebra structure). The corresponding map on the prime spectra {{math|f *: Spec(S) → Spec(R)}} is also called the structure map.
- If {{mvar|E}} is a vector bundle over a topological space {{mvar|X}}, then the projection map from {{mvar|E}} to {{mvar|X}} is the structure map.
- In topology, a canonical map is a function {{mvar|f}} mapping a set {{math|X → X / R}} ({{math|X mod R}}), where {{mvar|R}} is an equivalence relation on {{mvar|X}}, that takes each {{mvar|x}} in {{mvar|X}} to the equivalence class {{math|[x] mod R}}.{{Cite book|url=https://books.google.com/books?id=Eq4UDgAAQBAJ&pg=PA274|title=Handbook of Mathematics|last=Vialar|first=Thierry|date=2016-12-07|publisher=BoD - Books on Demand|isbn=9782955199008|language=en|pages=274}}