Tricki
a repository of mathematical know-how

"Canonical"

"Canonical"

A word that is really bugging me. It seem to mean so many different things!

I will try to write something about this.

Related to this, another word that comes up from time to time with different and sometimes "intuitive" meanings is "natural", mostly in the contexts of Category Theory and Universal Algebra (for example, as in "natural isomorphism").

Maybe this issue could be treated in the same article as "canonical", or perhaps in a different one.

I do plan to do this sometime soon (if not immediately), and I will indeed address naturality and canonicality in the same article. (The article I have in mind is something like a cross between ``Examples of canonical and natural structures" and ``How to use canonical/natural structures". If anyone else reading this wants to write them before I get to it, please feel free. Let me remark that there are already a few articles, such as How to compute the (co)homology of a space, and How to use tensor products and evaluation maps in representation theory, which have examples of natural constructions in them, and show some ways to exploit them. (And will eventually show many more.) So I think it will make sense to have links between the ``canonicality'' page and these pages. (And as the homological/categorical type pages build up, this should give rise to many more examples that can be linked to.)

A third word to mention is "universal".

Sorry, I just saw there is a thread about "universal".