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 I is an
ideal of a
ringR, then there is a canonical surjective
ring homomorphism from R onto the
quotient ringR/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 functionalfv defined by fv(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 spectraf*: Spec(S) → Spec(R) is also called 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]