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Category theory using string diagrams (2014)
- birttAdenors 2y agoMight also be useful (one of the authors is the same). https://assets.cambridge.org/97810093/17863/frontmatter/9781009317863_frontmatter.pdf https://assets.cambridge.org/97810093/17863/frontmatter/9781...
- abdullahkhalids 2y agoThis is extremely powerful approach, because instead of manipulating algebraic expression, you just manipulate string diagrams according to well defined and rather simple rules. For example, large parts of quantum theory, quantum computing and quantum information have been reduced to this diagramatic approach to such a degree that novel research insights are emerging from these techniques. A good introduction is Picturing Quantum Processes [1]. [1] https://www.cambridge.org/gb/universitypress/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/picturing-quantum-processes-first-course-quantum-theory-and-diagrammatic-reasoning?format=HB https://www.cambridge.org/gb/universitypress/subjects/physic...
- elbear 2y agoThe book you linked sounds awesome. Too bad about the price.
- eigenket 2y agoYou can read the paper here which is a decent introduction https://arxiv.org/abs/1510.05468 https://arxiv.org/abs/1510.05468 The (much older) paper Kindergarten Quantum Mechanics is probably the classic in this field https://arxiv.org/abs/quant-ph/0510032 https://arxiv.org/abs/quant-ph/0510032
- elbear 2y agoThank you!
- deleted 2y ago[deleted]
- whooie 2y agoSee also the ZX-calculus, a graphical calculus for reasoning about linear maps applied to qubits (read: quantum circuits) [1]. The ZX-calculus derives its structure from categorical quantum mechanics and has found major use in several areas of quantum information science. [1] https://arxiv.org/abs/2012.13966 https://arxiv.org/abs/2012.13966
- carapace 2y agoOn page four it seems the only difference between the second and third string diagrams is that the 'h' node is lower on the XY string, topologically the diagrams are identical, is that a typo?
- Skeime 2y agoNo, not a typo. If you translated the two diagrams into classical notation, you’d get the two terms at the top of that page. They’re equal by naturality of alpha, but you need to concentrate on the formula a bit to see it. The great thing about string diagrams is that some of the inconsequential differences “vanish” into topologically identical diagrams.
- carapace 2y agoThank you. > If you translated the two diagrams into classical notation, you’d get the two terms at the top of that page. So the vertical height is meaningful? Not just the arrangement of nodes on the string? It seems like an ambiguous grammar (the two topologically equivalent diagrams) gives rise to two different parse trees (the two terms at the top of the page) which nevertheless have the same meaning? Sorry if I'm being dense. I spent a couple of hours last night trying to understand the paper and it was pretty frustrating.
- tel 2y agoIt's been a while since I read these, but I believe it's because h is below alpha. They're exploiting the "sliding equality" referenced at the bottom of page 7.
- carapace 2y agoAh. > They're exploiting the "sliding equality" referenced at the bottom of page 7. That was confusing too, and for the same reason: those are three topologically equivalent diagrams. So vertical height between nodes on different strings is meaningful? Is there an introductory paper or blog post that explains the structure of the string diagrams?
- zzhelezc 2y agoThese might be useful - Programming With Categories [1] and Category Theory for Programmers [2] (also on YT [3]). [1] http://brendanfong.com/programmingcats.html http://brendanfong.com/programmingcats.html [2] https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/ https://bartoszmilewski.com/2014/10/28/category-theory-for-p... [3] https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_ https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI...
- chithanh 2y agoVery nice. Also recommended reading is Eugenie Cheng, The Joy of Abstraction[1] which emphasizes diagram chasing to visualize abstract relationships. [1] https://doi.org/10.1017/9781108769389 https://doi.org/10.1017/9781108769389
- xanderlewis 2y agoEugenia Cheng is great, but the completely unwarranted political nonsense that fills this particular book is quite disappointing (I don’t seem to be alone; many reviewers concur). Mathematics is much more fun without all that.
- ykonstant 2y agoThis seems quite useful for complicated diagram chasing; has anyone here used this approach for work in homotopy theory or arithmetic geometry? I wonder if it is useful for simplifying long diagrammatic calculations and how robust it is regarding the usual errors in diagrammatic reasoning.
- mqefjh 2y ago[dead]
- vletal 2y agoI was into this stuff when Scala was at its peak. I still think that it made me a better developer and it's just fun trying to understand all these concepts, but after several years I came to the conclusion that category theory will never hit mainstream. Sure, the best ideas are already lurking into mainstream languages, but noone is building Monads and Functors. Implementations used in practice are not pure (flatMap on List accepts Sets/Options...). These days I would probably jump on the Rust hype train and learned more about memory management and safety instead.