This question is jointly formulated with Neil Barton. We want to know about the significance of canonicity in mathematics broadly. That is, both what it means in some detail, and why it is important.

In several mathematical fields, the term 'canonical' pops up with respect to objects, maps, structures, and presentations. It's not clear if there's something univocal meant by this term across mathematics, or whether people just mean different things in different contexts by the term. Some examples:

In category theory, if we have a universal property, the relevant unique map is canonical. It seems here that the point is that the map is uniquely determined by some data within the category. Furthermore, this kind of scheme can be used to pick out objects with certain properties that 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. Furthermore, its construction depends only on the ordinals-- any two models of ZF with the same ordinals construct the same version of L.

In set theory, other models are termed 'canonical' but it's not clear how this can be so, given that they are non-unique in certain ways. For example, there is no analogue of the above fact for L with respect to models of ZFC with unboundedly many measurable cardinals. No matter how we extend the theory ZFC + "There is a proper class of measurables," there will not be a unique model of this theory up to the specification of the ordinals plus a set-sized parameter. See here.

Presentations of objects can be canonical: The most simple being that of fractions, whose presentation is canonical just in case the numerator and denominator have no common factors (e.g. the canonical presentation of 4/8 is 1/2). But this applies to other areas too; see here.

Sometimes canonicity seems to be relative. Given a finite-dimensional vector space, there is a canonical way of defining an isomorphism between V and its dual V* from a choice of a basis for V. This determines a basis for V*, and thus the initial choice of basis for V yields a canonical isomorphism from V to V**. But two steps can be more canonical than one: The resulting isomorphism between V and V** does not vary with the choice of basis, and indeed can be defined without reference to any basis. See here.

Other examples can be found here.

Our soft questions:

(a) Does the term 'canonical' appear in your field? If so what is the sense of the term? Is it relative or absolute?

(b) What role does canonicity play in your field? For instance, does it help to solve problems, help set research goals, or simply make results more interesting?

somemodel just don't satisfy the same goal). But it's very unclear what that is meant tomeangiven that such a beast wouldn't be unique. $\endgroup$ – Neil Barton Feb 19 at 15:27