In 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 ...

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 of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

''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 In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...

determined by ''g''. *If ''I'' is an

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...

of a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

''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 In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...

''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 ...

s, then ''S'' can be viewed as an

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

''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''.

References

Mathematical terminology {{math-stub