In mathematics, a canonical map, also called a natural map, is a

map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...

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 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 i ...

(i.e.,

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

). 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 In algebraic geometry, a prestack ''F'' over a category ''C'' equipped with some Grothendieck topology is a category together with a functor ''p'': ''F'' → ''C'' satisfying a certain lifting condition and such that (when the fibers are groupoids ...

. 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 In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...

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 considere ...

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 In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...

from ''R'' onto the

quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...

''R/I'', that sends an element ''r'' to its coset ''I+r''. *If ''V'' is a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

, 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 ...

''f''_''v'' defined by ''f''_''v''(λ) = λ(''v''). *If is a homomorphism between commutative rings, 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 ...

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 In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...

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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

, 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