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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.^{[1]}

- 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*.^{[2]}