Timeline for What is the definition of "canonical"?
Current License: CC BY-SA 2.5
4 events
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| Jun 2, 2019 at 11:05 | comment | added | Pietro Majer | An example you mentioned: if $X$ and $Y$ are categories (say $X$ small) with a given continuous functor $g:Y\to X$, the rule to associate to any object $x\in X$ a certain limit in $Y$ produces a map $f:X\to Y$, and this is automatically a functor, according to the adjoint functor theorem of Freyd. We call this map canonical, although the whole construction may rely on some assumed arbitrary choice (possibly non canonical!) for the representative in the isomorphism class of each limit. | |
| Jun 2, 2019 at 11:05 | comment | added | Pietro Majer | And rule is from the latin regula, which is the exact correspondent of κανών (straight piece of wood/measuring rod/ruler). Thus "canonical" is (constructed) following a rule. Moreover, the uniqueness of the canonical choice produces further regularity between objects: E.g. if $X$ is a Hilbert space and $Y$ a closed subspace of its, and $f$ maps $x$ to its closest point in $Y$, this canonical choice makes the projector $f$ linear. And I agree with your point of view that the building rule maybe was given arbitrarily (as it is in the human world). | |
| Mar 29, 2010 at 17:18 | comment | added | abcdxyz | I think I share some of your concerns. | |
| Mar 29, 2010 at 13:07 | history | answered | Thomas Kragh | CC BY-SA 2.5 |