Sunday, July 13, 2014

Irrationally contented in this vale of tears

I enjoyed one of my not-too-significant, probably-unimportant, nonetheless-delightful little mathy revelations a couple days ago, which followed on “remembering”, as Plato/Socrates would call it, that natural constructions tend to be functorial; and the result was
\[ \Sigma \varphi = \Sigma \vartheta \vee \vartheta\star\Omega\Sigma\vartheta \]... and now I must apologize, for mathematician is usually working at the top of a large wobbly stack of definitions and usually can't even see the one two or three layers down...
  • A Cateogory, as the mathematician intends it, has a collection of objects, and possibly a collection of relations between pairs of objects, and an operation of composing adjacent relations between three objects, and... stuff. For instance, you might have the family of human languages for objects, with translating dictionaries as relations between them. If you have a french-english dictionary and an english-italian dictionary, you might attempt to compose them into an experimental french-italian dictionary, and this might have suprising consequences!
  • Functors are the natural relations between categories that give you a “category of categories”; A functor connects objects of one category to objects of the other, as well as connecting relations between objects to relations between respective connected objects --- but because of the echoing clearly heard in “category of categories”, there are furthermore relations between functors with the same origin and same landing ...
  • A construction “being functorial” is an informal way of saying: we first thought of it in terms of the objects of some category, and then realized it related to the relations between the objects as well; more echoing... we like echoes.
And that's what happened through Friday; a construction I usually think of only in terms of objects (homotopical figures), I recognized anew was also realized on relations between them (continuous maps).

Anyways, these weird socratic-recollections congealed into something mathematically-writable after I joined an impromptu schola for to sing a Requiem Mass for Fr. Kenneth Walker — it seems he once attended school with some of my neighbors, before I or they moved in to town. It was a beautiful sorrow, and a beautiful evening, and a remarkably uncongested ride home with the choirmaster's wife as the full moon was rising.

All you out there, keep well; I hope to be back again next Sunday, too.

All honour to Mother Mary, and all Glory and Praise to God the Holy Trinity be; animae omnium fidelium defunctorum, misericordiae Domini, requiescant in pace.


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