Title: I Think Indexed Enriched Categories are Nice.
First proposed by Lawvere in 1973, indexed enriched categories are a very complicated mathematical structure studied in only the most privileged ivory towers. In this talk I intend to bring them down to earth by exploring their viability as a data structure. I will cheat by only considering enrichments where the axioms are satisfied trivially. I will explain how these "trivial" enriched categories are nice data structures for holding solutions to algebraic path problems. Then I will explain how making them indexed allows us to work with distributed, concurrent, and compositional algebraic path problems. This talk will be in line with the research programme I started to develop before I quit and left for industry. Namely, that indexed categories and Grothendieck constructions may be used to develop a general theory of compositional computation. This talk will feature Idris2 code and categorical terminology side-by-side so that those familiar with either language will be able to follow along.
@johncarlosbaez Hi! Yes I'm having fun and realising that enjoying my work is just another skill to develop. Maybe Christian should pitch CQL to my company. We could definitely use some more category magic.
@johncarlosbaez Thanks I fixed the $$'s. Re: semiring vs rig...I think I've notice that computer scientists often use semiring even if it's not technically correct.
Hey what's it called when you have a graph whose nodes are smaller graphs? Like a hierarchical graph? or a meta-graph? What word would be used by a graph theorist?
Though people say 'octave', there are only 7 different notes in the major scale. This is annoying if you're trying to play scales in melodies with, say, 8 beats per measure. The scale keeps drifting out of synch.
One solution is to add an extra note to your scale! In this video, jazz cat Adam Maness explains the virtues of the 'bebop major scale', where you add a minor 6th to the major scale:
1 2 3 4 5 ♭6 6 7
Just playing this scale up and down, 8 beats per measure, already suggests some melodies. Even more so if you play it with both hands in 'contrary motion' - up with one hand, down with the other. Listen to the video and you'll see what I mean!
Why do they call this the 'major sixth diminished scale'? Barry Harris introduced this term: he said this scale is derived from a major 6th chord (1 3 5 6) and a diminished 7th chord starting at the 2 (that is, 2 4 ♭6 7).
You can get other bebop scales by putting the extra note somewhere else.
@johncarlosbaez I often play the bebop scale with 4,flat 5 and flat 7,8 as 8th notes instead of quarter notes. It sounds more Jazzy that way, interesting how having one less note encourages a jazzier rhythm. I never thought of it that way