mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a canonical map, also called a natural map, is a map or

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

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 ''N'' is a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...

of a group ''G'', then there is a canonical

surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...

group homomorphism from ''G'' to the

quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...

''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 In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...

of ''V,'' that sends a vector ''v'' to the linear functional ''f''_''v'' defined by ''f''_''v''(λ) = λ(''v''). *If is a homomorphism between

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

s, 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 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 In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...

'x''modulo ''R''.

References

Mathematical terminology {{math-stub