4 ms·
- If you are confused, it is probably because FP languages are really only half the story of CT - Adjoints are all about joining things hence the name; much li
by reinman 7y ago
- If you are confused, it is probably because FP languages are really only half the story of CT
- Adjoints are all about joining things hence the name; much like chasing dependencies (arrows) between cells in a spreadsheet and then going back to the relational database where the sheet was extracted from. You have seamlessly jumped from algebraic morphisms to geometry without even thinking about it
- Homotopy is about finding "paths" between things and therefore imply some relative naming (coordinate) scheme is possible. The segments of a path arise due to torsors. Hence homotopy is associated with paths. UNIX paths and URLs come to mind
- bollu 7y ago- Adjoints as far as I understand are not about joining things, adjoints are about a generalization of inverses. Can you formally (and I mean mathematically) define what this intuition you have about adjoints ~= "joining things"? Also, the word "adjoint" does not come from "join". It comes from "adjoint" in the theory of complex operators: https://en.wikipedia.org/wiki/Adjoint_functors https://en.wikipedia.org/wiki/Adjoint_functors - Homotopy is not about finding paths, it is an equvalence relation _between paths_. a path is _homotopic_ to another. Homotopy is also a _continuous object_, unlike torsors, which are discrete objects (a continuous torsor is an affine space). So, while UNIX paths and URLs are "paths" in the sense of torsors, they have _nothing_ to do with homotopy (as it is classically defined). Unless you are using some weakened notion of homotopy that I am unaware of, in which case I'd love links.