This question is formulated jointly with Neil Barton.
In several mathematical fields, the term "canonical" appears with
with regard to objects, maps, structures and presentations. It is not
clear if there is something unmistakable signified by this term through
math, or if people just mean different things in different
contexts by the term. Some examples:
In category theory, if we have universal property, the
the single card is canonical. It seems here that the fact is that the card
is only determined by certain data in the category. In addition, this type of schema can be used to select objects with certain properties which are canonical in the sense that they are unique up to isomorphism.
In set theory, L is a canonical model. Here it is unique and definable. In addition, its construction depends only on the ordinals – two models of ZF with the same ordinals build the same version of L.
In set theory, other models are called "canonical" but this is not
clearly how it can be so, since they are not unique in some
manners. For example, there is no analog of the above fact for L with respect to CFA models with an unlimited number of measurable cardinals. No matter how we extend the ZFC + theory "There is an appropriate class of measurables", there will be no single model of this theory until the specification of the ordinals plus a defined size parameter. See here.
Presentations of objects can be canonical: being the simplest
that of fractions, the presentation of which is canonical in case the
the numerator and denominator have no common factors (for example,
presentation of 4/8 is 1/2). But this also applies to other areas; see here.
Sometimes the canonicity seems to be relative. Given a finite dimensional vector space, there is a canonical way to define an isomorphism between V and its double V * from a choice of a base for V. This determines a base for V *, and therefore the initial basic choice for V gives a canonical isomorphism from V to V **. But two stages can be more canonical than one: the resulting isomorphism between V and V ** does not vary with the choice of the base, and can indeed be defined without reference to any base. See here.
Our sweet questions:
(a) Does the term "canonical" appear in your field? If yes, what is the
meaning of the term? Is it relative or absolute?
(b) What role does canonicity play in your field? For example, does this help solve problems, set research goals or just make the results more interesting?