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Morgan Rogers (he/him) (Apr 06 2020 at 10:08):Is this a stream where people can show off their nice diagrams!? :tada:
Jules Hedges (Apr 06 2020 at 13:32):Yeah, what's this for? It looks like everybody was subscribed to it, and it's not obvious... Is it for announcing your latest papers etc?
Christian Williams (Apr 06 2020 at 14:55):Sure! Just make sure to "tell" as well.
If they're really nice -- we're also going to have a "math art" stream.
Jules Hedges (Apr 06 2020 at 14:57):Ooooo, very nice, I like that
Christian Williams (Apr 06 2020 at 14:57):For all the new streams, I added a description. I wish it automatically displayed it in stream events; later today I'll make the explanations more explicit.
Chad Nester (Apr 06 2020 at 15:14):
Chad Nester (Apr 06 2020 at 15:14):My alien language is coming along nicely
Morgan Rogers (he/him) (Apr 06 2020 at 15:15):I especially appreciate the use of inverted latin characters :stuck_out_tongue_wink:
Jules Hedges (Apr 06 2020 at 15:16):Pretty! Something involving double categories?
Chad Nester (Apr 06 2020 at 15:20):Yup! I'm not quite sure what I'm going to call the double category but I'm leaning towards "exchange circuits". I think it's probably your sort of thing.
Jules Hedges (Apr 06 2020 at 16:06):Alright, since I happen to be working on slides right now...
comb.jpg context.jpg
Jules Hedges (Apr 06 2020 at 16:08):I've also got this commutative diagram sitting around that I made a couple of weeks ago, that I'm pretty proud of
(https://categorytheory.zulipchat.com/user_uploads/21317/6ev85YD0rupsXoJzaE4rqAnB/ES_oFL6XsAEDMlH.jpg)
Jules Hedges (Apr 06 2020 at 16:10):(These are actually related. The bottom thing is most of the associativity proof for an operad I'm working on that describes the combinatorics of how the top diagrams can be pasted together)
Jules Hedges (Apr 06 2020 at 16:13):For context: that diagram lives in the category of posets, where means the ordinal with elements, is disjoint union of posets and is ordinal sum
Jules Hedges (Apr 07 2020 at 11:18):Pictures? Pictures.
(https://categorytheory.zulipchat.com/user_uploads/21317/f5JoMR5d3g_q4gdw-AY6tkw3/composition1.jpg) (https://categorytheory.zulipchat.com/user_uploads/21317/14NK_Fsv-02PIV4dBRObe4Yz/composition2.jpg) (https://categorytheory.zulipchat.com/user_uploads/21317/7sKbKffP6v2QmwktiTV_jqIh/composition3.jpg)
Fabrizio Genovese (Apr 07 2020 at 11:24):This comes from the cryptography stream. It took me an hour to draw on a blackboard and I don't dare to imagine how long will it take me to tikz it:
IMG_20200326_224318.jpg
Sam Tenka (Apr 07 2020 at 17:42):@Fabrizio Genovese Is manual tikz the preferred method to draw such diagrams? It would be nice to have an "OCR" for this --- I know one of my labmates has such a system (https://papers.nips.cc/paper/7845-learning-to-infer-graphics-programs-from-hand-drawn-images.pdf) but not specialized to category theory diagrams.
Fabrizio Genovese (Apr 07 2020 at 18:07):Normally I'd go for manual tikz, but this will really end up being time consuming. There are ways to convert SVG to tikz, so I could also do with a good open source SVG editor for linux
Thomas Read (Apr 07 2020 at 18:36):Inkscape is pretty good for SVGs
John Baez (Apr 07 2020 at 18:44):Here is a proof by @Paul-André Melliès that the conjunction of the double negations of two formulas A and B implies the double negation of the conjunction of the two formulas:
John Baez (Apr 07 2020 at 18:44):He has more nice diagrams here:
https://www.irif.fr/~mellies/gallery.html
and more in his papers and talks.
Jules Hedges (Apr 07 2020 at 18:45):I think the standard method to typeset them is to give up and work on something else instead
Jules Hedges (Apr 07 2020 at 18:46):I actually have no idea how Paul-André does his diagrams, but I suspect Inkscape
John Baez (Apr 07 2020 at 18:46):He is, nominally at least, here.
Lê Thành Dũng (Tito) Nguyễn (Apr 07 2020 at 19:03):I've heard good things about http://ipe.otfried.org/ (though I use manual TikZ myself)
Nathaniel Virgo (Apr 07 2020 at 19:13):If I was publishing diagrams that complex, I'd be quite tempted to just draw them really neatly with a pen, Roger Penrose style.
Eduardo Ochs (Apr 07 2020 at 19:14):sheaves-for-children.png
From <https://arxiv.org/abs/2001.08338>, but I don't want to advertise
that paper before I finish rewriting its last sections.
Paul-André Melliès (Apr 07 2020 at 20:59):Although I would prefer to use an open source software like Inkscape for that purpose, I got used to it now and draw my diagrams with Illustrator.
Paul-André Melliès (Apr 07 2020 at 21:30):As a matter of fact, this diagram you have picked John and all my work on connecting game semantics to string diagrams was heavily influenced by your talks at CIRM back in 2006. Do you remember that golden age when we could all happily meet in the same room, close to the Calanques? An opportunity for me to thank you here!
http://iml.univ-mrs.fr/~lafont/Geocal-week1.html
http://math.ucr.edu/home/baez/universal/universal_hyper.pdf
John Baez (Apr 07 2020 at 21:33):Thanks to you too! Yes, I remember you drawing diagrams like that for me and explaining why the left adjoint of a monoidal functor is always lax monoidal. That was a fun time!
Jules Hedges (Apr 07 2020 at 21:48):Ah yes, I remember the golden age when multiple people could be in the same room
Pierre Cagne (Apr 08 2020 at 07:41):Jules Hedges said:
I actually have no idea how Paul-André does his diagrams, but I suspect Inkscape
I remember implementing those diagrams in TikZ at some point during my master thesis. It was not easy but it has the distinctive advantage that you can script stuff easily afterwards.
Ralph Sarkis (Apr 22 2020 at 22:07):I have not taken the time to rigorously study string diagrams, so when proving the fact that the composite of two monads related with a monad distributive law is also a monad, I drew one of my best diagrams.
Given two monads and and a monad distributive law (I'll let you check here what diagrams have to commute), we want to prove that is a monad, where $\cdot$ and $\diamond$ denote the vertical and horizontal compositions of natural transformations respectively.
I was able to show the unit diagram (the triangle) commutes quite easily, but for the multiplication, I was stuck for quite a while before giving up and ending up on this post. There, everything is stated with string diagrams and it made something click; I forgot about the commutative square that makes the horizontal composition well-defined. In the language of string diagrams (and in this context), I believe this is the fact that you can slide operations up and down independently if they do not involve the same wires. After that, I was able to get this beast (the double arrows are in the original square we want to show commute and all the numbered shapes inside have a simple justification, i.e.: use the diagrams in the definition of monad distributive law or use the square used to define ).
On the post I mentioned, the equivalent is this.
Christian Williams (Apr 22 2020 at 22:30):Beautiful! Great example of the power of two-dimensional syntax.
Nathanael Arkor (Apr 22 2020 at 22:35):Is that commutative diagram a minimal proof of coherence? If it is, you may have sold string diagrams to me with one diagram.
Fabrizio Genovese (Apr 22 2020 at 22:44):Ralph Sarkis said:
I have not taken the time to rigorously study string diagrams, so when proving the fact that the composite of two monads related with a monad distributive law is also a monad, I drew one of my best diagrams.
Given two monads and and a monad distributive law (I'll let you check here what diagrams have to commute), we want to prove that is a monad, where $\cdot$ and $\diamond$ denote the vertical and horizontal compositions of natural transformations respectively.
I was able to show the unit diagram (the triangle) commutes quite easily, but for the multiplication, I was stuck for quite a while before giving up and ending up on this post. There, everything is stated with string diagrams and it made something click; I forgot about the commutative square that makes the horizontal composition well-defined. In the language of string diagrams (and in this context), I believe this is the fact that you can slide operations up and down independently if they do not involve the same wires. After that, I was able to get this beast (the double arrows are in the original square we want to show commute and all the numbered shapes inside have a simple justification, i.e.: use the diagrams in the definition of monad distributive law or use the square used to define ).
On the post I mentioned, the equivalent is this.
Did you see this? https://arxiv.org/pdf/1401.7220.pdf It's a very nice paper by @Dan Marsden that explains how to do diagrammatic category theory. An entire chapter is dedicated to monads and there are also diagrams to express distributive laws that are very similar to yours. I believe this is being turned into a book, which should be out at some point!
Nathanael Arkor (Apr 22 2020 at 22:47):This is a very nice paper, though I think it would benefit from some side-by-side comparisons with the same proofs as commutative diagrams, to really give a sense of how they compare.
Nathanael Arkor (Apr 22 2020 at 22:47):I'd love to be horrified by some of the complicated coherences that the string diagrams were expressing :)
Ralph Sarkis (Apr 22 2020 at 23:08):Fabrizio Genovese said:
Did you see this? https://arxiv.org/pdf/1401.7220.pdf It's a very nice paper by Dan Marsden that explains how to do diagrammatic category theory. An entire chapter is dedicated to monads and there are also diagrams to express distributive laws that are very similar to yours. I believe this is being turned into a book, which should be out at some point!
Yes, it was in my bookmarks. I'll wait for the book then :)
Nathanael Arkor said:
Is that commutative diagram a minimal proof of coherence? If it is, you may have sold string diagrams to me with one diagram.
I don't know but I believe it follows all the steps of what the complete proof would be like with string diagrams.
Ralph Sarkis (Apr 22 2020 at 23:11):I was also considering reading Picturing Quantum Processes to kill two birds with one stone and learn a bit of quantum theory along with string diagrams.
John Baez (Apr 22 2020 at 23:41):To enjoy Picturing Quantum Processes I think one needs to be a bit interested in quantum mechanics as well as string diagrams. The same is true for Heunen and Vicary's book Introduction to Categorical Quantum Mechanics, which is a bit more advanced but really nice if one wants all the details.
Jules Hedges (Apr 23 2020 at 09:34):The learning curve in PQP is extremely gentle, I read about 800 pages of it despite knowing very little physics. (It gets steeper in a few places. If you mostly care about learning string diagrams you probably don't care about the gory details of Stinesprung dilation, for example...)
Fabrizio Genovese (Apr 23 2020 at 10:16):John Baez said:
To enjoy Picturing Quantum Processes I think one needs to be a bit interested in quantum mechanics as well as string diagrams. The same is true for Heunen and Vicary's book Introduction to Categorical Quantum Mechanics, which is a bit more advanced but really nice if one wants all the details.
I am not sure you need to like QM to enjoy Picturing Quantum Process. I think t he leitmotiv of the book is not quantum mechanics, but the general sense of amazement one gets in thinking "whoa! So you can do this, and this, and this just with pictures?!", which goes on for roughly 800 pages (even if the middle section of the book is the hardest one, and one may get a bit lost there).
Fabrizio Genovese (Apr 23 2020 at 10:18):Like, reducing Picturing Quantum Process to just "a book about quantum mechanics" is a mistake in my opinion. I would never present/introduce it that way
Sam Tenka (Apr 30 2020 at 17:38):rep-strings-a.jpg rep-strings-b.jpg
I've been compressing the content of basic representation theory (finite-dimensional complex rep.s of finite groups) in diagrams! The notation is pretty different from Cvitanović's birdtracks, as far as I can tell. Also, it generalizes more to the case of compact groups (e.g. the p-adic integers) than the case of lie groups, since we stay in the world of group rings rather than going to lie algebras.
I waffled over whether or not to normalize loops by the size of G. Now I think they /should/ be normalized, to better correspond to haar measure on infinite compact groups. Also, I cheated in drawing the Mackey criterion by drawing a related concept instead of the standard formulation: it is hard to write sums, especially dependent sums, in my notation!
John Baez (Apr 30 2020 at 18:56):Fabrizio Genovese said:
Like, reducing Picturing Quantum Process to just "a book about quantum mechanics" is a mistake in my opinion. I would never present/introduce it that way.
Okay, good! I haven't actually read it.
Fabrizio Genovese (Apr 30 2020 at 19:19):You should skim through it. I think it's all stuff you already know perfectly, but it's still a very fun read imho
Cole Comfort (Apr 30 2020 at 19:21):Fabrizio Genovese said:
You should skim through it. I think it's all stuff you already know perfectly, but it's still a very fun read imho
I think that stuff like the CPM construction for example have very little motivation from a mathematical or traditional cs point of view, so it is its own kind of genre.
John van de Wetering (May 20 2020 at 18:20):This is a recent quite nice-looking diagram of mine:
image.png
It's a proof in the ZH-calculus of a hyper-local-complementation law using !-box notation.
Source: https://arxiv.org/pdf/2003.13564.pdf
Jules Hedges (May 20 2020 at 18:44):This is clearly a map of the power grid of a starship
Aleks Kissinger (May 27 2020 at 16:48):As long as the rebels don't find out that there's a weak point just above the .
Eduardo Ochs (Jun 29 2020 at 22:02):Hi all, I just uploaded this to my page and to arxiv... it may be interesting to people who teach CT to outsiders, and @Paolo Perrone is explicitly mentioned in the text.
On my favorite conventions for drawing the missing diagrams in Category Theory
I used to believe that my conventions for drawing diagrams for categorical statements could be written down in one page or less, and that the only tricky part was the technique for reconstructing objects "from their names"... but then I found out that this is not so.
This is an attempt to explain, with motivations and examples, all the conventions behind a certain diagram, called the "Basic Example" in the text. Once the conventions are understood that diagram becomes a "skeleton" for a certain lemma related to the Yoneda Lemma, in the sense that both the statement and the proof of that lemma can be reconstructed from the diagram. The last sections discuss some simple ways to extend the conventions; we see how to express in diagrams the ("real") Yoneda Lemma and a corollary of it, how to define comma categories, and how to formalize the diagram for "geometric morphism for children".
People in CT usually only share their ways of visualizing things when their diagrams cross some threshold of of mathematical relevance --- and this usually happens when they prove new theorems with their diagrams, or when they can show that their diagrams can translate calculations that used to be huge into things that are much easier to visualize. The diagrammatic language that I present here lies below that threshold --- and so it is a "private" diagrammatic language, that I am making public as an attempt to establish a dialogue with other people who have also created their own private diagrammatic languages.
Links:
http://angg.twu.net/math-b.html#favorite-conventions
http://angg.twu.net/LATEX/2020favorite-conventions.pdf
Zx.png
Tim Hosgood (Jul 01 2020 at 13:11):
Tim Hosgood (Jul 01 2020 at 13:12):for a while i was working on a “personal” diagrammatic language for talking about relative algebraic geometry. i never made it formal, but it’s something that i would like to think about at some point
Tim Hosgood (Jul 01 2020 at 13:13):
Eduardo Ochs (Jul 01 2020 at 17:43):Looks great!!! How would you spell out your diagrammatic statement of Lemma 2.20?
Tim Hosgood (Jul 01 2020 at 18:18):Eduardo Ochs said:
Looks great!!! How would you spell out your diagrammatic statement of Lemma 2.20?
Let be a morphism of sheaves with a scheme. If there exists a Zariski affine cover such that is a scheme for all , then $F_1$ is a scheme.
Chad Nester (Oct 08 2020 at 13:41):I drew this picture for a paper I'm writing and I'm extremely proud of myself :laughter_tears: slug.png
Chad Nester (Oct 08 2020 at 13:48):for the "telling" part: As they move through time, processes leave a trail of effects on the world -- just like slugs leave a trail of slime as they move through space!
Chad Nester (Oct 08 2020 at 14:09):Here's its companion: process-slime.png
Cole Comfort (Oct 08 2020 at 14:33):Chad Nester said:
I drew this picture for a paper I'm writing and I'm extremely proud of myself :laughter_tears: slug.png
So this causal slug is in the pantheon of cosmological creatures alongside the world turtle....
RXSeF7D.jpg
Jules Hedges (Dec 25 2020 at 19:22):So my brother gave me a mug with a string diagram pulled from one of my papers
20201225_153159.jpg 20201225_153211.jpg 20201225_153218.jpg
Morgan Rogers (he/him) (Dec 26 2020 at 13:09):There's nothing quite as satisfying as drinking out of one's own research
Eduardo Ochs (Jul 19 2021 at 19:28):This is a way to draw a Lawvere-Tierney topology on a topos (bottow right), its associated Grothendieck topology (bottom left), and its associated nucleus (top right):
2020clops-and-tops-4-2.png
Eduardo Ochs (Jul 19 2021 at 19:31):The paper is here:
http://angg.twu.net/LATEX/2020clops-and-tops.pdf#page=22
Keith Elliott Peterson (Aug 07 2021 at 01:07):I got sick of drawing the graphs of functions of sets by hand, so I decided to try my hand at doing it programmatically.
It has functions:
sample1.png
Compositions of functions:
sample2.png
Function sets:
sample3.png
Currying of function sets:
sample4.png
and uncurrying function sets:
sample5.png
Should I keep going?
Jules Hedges (Aug 07 2021 at 13:03):Cool! Is this done with LaTeX, or something else?
Keith Elliott Peterson (Aug 08 2021 at 02:00):No, this is using Racket picts.
Also, after wasting today trying to get it to draw cartesian products with projections, I think I might have to redo the function drawer.