Canonical map

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. In general, it is the map which preserves the widest amount of structure,[1] and it tends to be unique. In the rare cases where latitude in choices remains, the map is either conventionally agreed upon to be the most useful for further analysis, or sometimes the most elegant map known up to date.

A standard form of canonical map involves some function mapping a set ${\displaystyle X}$ to the set ${\displaystyle X/R}$ (${\displaystyle X}$ modulo ${\displaystyle R}$), where ${\displaystyle R}$ is an equivalence relation on ${\displaystyle X}$.[2] 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.

Examples

• 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.
• 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 f_v defined by f_v(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 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.[3]

References

1. "The Definitive Glossary of Higher Mathematical Jargon — Canonical". Math Vault. 2019-08-01. Retrieved 2019-11-20.
2. Weisstein, Eric W. "Canonical Map". mathworld.wolfram.com. Retrieved 2019-11-20.
3. Vialar, Thierry (2016-12-07). Handbook of Mathematics. BoD - Books on Demand. p. 274. ISBN 9782955199008.