[HN Gopher] Category theory using string diagrams (2014)
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Category theory using string diagrams (2014)
Author : soloist11
Score : 62 points
Date : 2024-06-28 22:43 UTC (1 days ago)
(HTM) web link (arxiv.org)
(TXT) w3m dump (arxiv.org)
| birttAdenors wrote:
| Might also be useful (one of the authors is the same).
| https://assets.cambridge.org/97810093/17863/frontmatter/9781...
| abdullahkhalids wrote:
| This 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/physic...
| elbear wrote:
| The book you linked sounds awesome. Too bad about the price.
| eigenket wrote:
| You can read the paper here which is a decent introduction
|
| 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
| whooie wrote:
| See 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
| carapace wrote:
| On 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 wrote:
| No, 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 wrote:
| Thank 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 wrote:
| It'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 wrote:
| Ah.
|
| > 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?
| eigenket wrote:
| The vertical height isn't relevant, but the relative
| vertical height between two "nodes" on different wires
| could (a-priori) be. The fact that you can reorder them
| as you like so only the topology matters is exactly
| encoding the fact that they are natural transformations,
| if they were some other map it wouldn't work.
| zzhelezc wrote:
| These might be useful - Programming With Categories [1] and
| Category Theory for Programmers [2] (also on YT [3]).
|
| [1] http://brendanfong.com/programmingcats.html
|
| [2] https://bartoszmilewski.com/2014/10/28/category-theory-
| for-p...
|
| [3]
| https://www.youtube.com/playlist?list=PLbgaMIhjbmEnaH_LTkxLI...
| chithanh wrote:
| Very 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
| xanderlewis wrote:
| Eugenia 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 wrote:
| This 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 wrote:
| https://github.com/akissinger/chyp
| vletal wrote:
| I 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.
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