Canonical map explained
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
- If is a normal subgroup of a group, then there is a canonical surjective group homomorphism from to the quotient group, that sends an element to the coset determined by .
- If is an ideal of a ring, then there is a canonical surjective ring homomorphism from onto the quotient ring, that sends an element to its coset .
- If is a vector space, then there is a canonical map from to the second dual space of, that sends a vector to the linear functional defined by .
- If is a homomorphism between commutative rings, then can be viewed as an algebra over . The ring homomorphism is then called the structure map (for the algebra structure). The corresponding map on the prime spectra is also called the structure map.
- If is a vector bundle over a topological space, then the projection map from to is the structure map.
- In topology, a canonical map is a function mapping a set, where is an equivalence relation on, that takes each in to the equivalence class .[2]
See also
Notes and References
- Web site: Grothendieck Conference Talk. Buzzard. Kevin. . 21 June 2022 .
- Book: Vialar, Thierry. Handbook of Mathematics. 2016-12-07. BoD - Books on Demand. 9782955199008. en. 274.
