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. 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.

Examples

• If Template:Mvar is a normal subgroup of a group Template:Mvar, then there is a canonical surjective group homomorphism from Template:Mvar to the quotient group G / N, that sends an element Template:Mvar to the coset determined by Template:Mvar.
• If Template:Mvar is an ideal of a ring Template:Mvar, then there is a canonical surjective ring homomorphism from Template:Mvar onto the quotient ring R / I, that sends an element Template:Mvar to its coset I + r.
• If Template:Mvar is a vector space, then there is a canonical map from Template:Mvar to the second dual space of Template:Mvar, that sends a vector Template:Mvar to the linear functional Template:Mvar defined by f_v(λ) = λ(v).
• If f: R → S is a homomorphism between commutative rings, then Template:Mvar can be viewed as an algebra over Template:Mvar. The ring homomorphism Template:Mvar 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 Template:Mvar is a vector bundle over a topological space Template:Mvar, then the projection map from Template:Mvar to Template:Mvar is the structure map.
• In topology, a canonical map is a function Template:Mvar mapping a set X → X / R (X mod R), where Template:Mvar is an equivalence relation on Template:Mvar, that takes each Template:Mvar in Template:Mvar to the equivalence class [x] mod R.

See also

• Natural transformation