Here's the basic idea: The nicest situation we can have between two representations (x,y) is that they can encode each other, so transformations (f,g) "round-t

Here's the basic idea: The nicest situation we can have between two representations (x,y) is that they can encode each other, so transformations (f,g) "round-trip": f x == y x == g y x == g (f x) f (g y) == y That isn't always possible, though. (and some mathematicians find it so boring that they work "up to isomorphism", meaning they don't even bother keeping track of the exact round-trippable transformations, and just assume we use any suitable coercion, writing down only the non-round-trippable transformations, just as we don't bother keeping track of exactly which and how many zeroes there are in 1+2 [or is that 1+0+2? or 1+0+0+2?] or which and how many null strings there are in the middle of "hi") Much more commonly (this always happens, if we allow degenerate encodings) is a situation where each representation can encode only a subset of the other, so we only have: f x == f (g (f x)) g (f (g y)) == g y because attempting to round trip is lossy, but the loss is idempotent: unlike analogue copying, further generations (which are already in the mutually-expressible subset) aren't any more lossy. [aside: does the 3-in-1 trinitarian nature of these equalities suggest that one transformative application may exist in three coequal, coeternal, consubstantial hypostatic applications? end of aside] Finally, the situation I'd contrived between plaintext and pdf is where a more expressive representation can encode a less expressive, but not the other way around, so we have an intermediate situation, where Procrustes only affects the more expressive side: x == g (f x) g (f (g y)) == g y If you squint, you should be able to see the relation between this abstract algebraic notion of adjunction and the linear algebraic notion of "best possible approximation to an inverse". Galois connections are also closely related. I don't think I can do it justice in an HN post, but old János also did a lot with these concepts, both theoretically exploring when the idempotents mentioned above are "regular" (related to regular expressions and the Kleene star) and practically suggesting that we should make Y-combinator-style autoreplicating mining-factory-complexes on the moon (in the asteroid belt?); see also https://news.ycombinator.com/item?id=38893829 https://news.ycombinator.com/item?id=38893829 ...