In mathematics, canonical example is frequently used to mean "archetype, standard example". Canonical also means "distinguished representative of a class", particularly one that does not require making any choice; this is also known as "natural", as in natural transformation. In geometry "tautological" is sometimes used as an alternative, to avoid confusion with other uses of "canonical". Specific uses of canonical include:
Canonical coordinates, sets of coordinates which can be used to describe a physical system at any given point in time
Canonical form, a natural unique representation of an object, or a preferred notation for some object
Canonical homomorphism, canonical isomorphism: an homomorphism that is uniquely defined by its main property, for example, the canonical homomorphism of the integers into the rational numbers, the canonical homomorphism of a vector space onto the quotient by a subspace. More precisely, a canonical morphism or canonical homomorphism is a morphism whose existence and unicity are provided by a universal property.
Canonical polyhedron, in geometry a polyhedron whose edges are all tangent to a common sphere, whose center is the average of its vertices; see Midsphere
Canonical representative, in set theory a standard member of each element of a set partition