Dan Piponi on Nostr: Something it took me many years to appreciate is this: a lot of constructions in ...
Something it took me many years to appreciate is this: a lot of constructions in algebra can be interpreted as figuring out what you can precompute when you're not sure what's coming next.
The first example I was taught may have been the free group on some generators. You're basically saying "well I don't know how to multiply these things so I'll just record whatever I need now so when I do learn how to multiply elements I've already done as much work as possible".
Sometimes there's even a kind of cost model. As I've mentioned before, a tensor product is basically what you need to store from x and y when you know that at some point in the future you're going to compute many bilinear functions f(x,y), but you don't know what f actually is, arbitrary multiplication is expensive, but memory is cheap along with addition and multiplication by a constant.
In a sense everything I said here is wrong. For example in mathematics you have considerable freedom to choose what sets actually represent a tensor product so it makes no sense to identify a construction with a particular "implementation". Nonetheless the definition of the tensor product corresponds well to the sort of interface you might attach to an obvious precomputation for the bilinear function scenario.
(I think this is basically summarising what a Left Kan extension is.)
Published at
2026-04-26 21:07:44 UTCEvent JSON
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