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Jade Master (Jan 28 2021 at 18:55):Anyone want to read (and discuss) this paper with me? http://www.tac.mta.ca/tac/reprints/articles/8/tr8abs.html
Chetan Vuppulury (Jan 28 2021 at 18:56):I have also been meaning to read this for a while. I would definitely be interested
Joe Moeller (Jan 28 2021 at 18:56):Me!
Jade Master (Jan 28 2021 at 18:57):Okay. So it shall be so. I was just thinking of posting questions here as I have them.
Eric Forgy (Jan 28 2021 at 18:57):I'd love to follow along, but not sure how much I could contribute.
Fawzi Hreiki (Jan 28 2021 at 19:00):I just reread that paper recently so sure
Fawzi Hreiki (Jan 28 2021 at 19:01):But maybe it’d be best to read his Metric Spaces paper first
Jade Master (Jan 28 2021 at 19:03):Fawzi Hreiki said:
But maybe it’d be best to read his Metric Spaces paper first
Oh that sounds good, which paper is that?
Fawzi Hreiki (Jan 28 2021 at 19:03):http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html
Eric Forgy (Jan 28 2021 at 19:08):I nodded through Section 1, but Section 2 
Kan extension is still over my head.
John Baez (Jan 28 2021 at 19:09):If a Lawvere paper isn't hard to read it must be some other guy named Lawvere.
Joe Moeller (Jan 28 2021 at 19:10):Riehl's chapter on Kan extensions is great imo.
https://math.jhu.edu/~eriehl/context.pdf
John Baez (Jan 28 2021 at 19:20):I can explain Kan extensions in ten minutes to anyone who understands adjoint functors. So can Joe!
John Baez (Jan 28 2021 at 19:20):So can Jade.
John Baez (Jan 28 2021 at 19:21):Most books make them seem complicated....
Eric Forgy (Jan 28 2021 at 19:22):That's because you guys rock :blush: :raised_hands:
John Baez (Jan 28 2021 at 19:22):Of course there's a lot of facts to learn about them, but the basic idea and the "why should I care" part is a ten-minute thing.
Fawzi Hreiki (Jan 28 2021 at 19:45):I think the problem is that most category theory books skim over 1-categorical extensions and lifts in the first chapter without giving examples (even though they're everywhere in super basic mathematics)
Fawzi Hreiki (Jan 28 2021 at 19:45):So by the time you get to 2-categorical (i.e. Kan) extensions and lifts, it's kind of out of nowhere
Eric Forgy (Jan 28 2021 at 19:46):Although the paper is over my head, I feel like the motivation resonates with why I am interested in CT. I'm making slow, but sure progress, but the road ahead seems long.
I recently wrote up this summary:
It is interesting for me to see convolution algebra popping up (bottom of page 10) since that is also important in discrete differential geometry.
John Baez (Jan 28 2021 at 19:51):It's worth remembering: every Lawvere paper is over everyone's head until they spend time reading the paper. This is the most famous thing about Lawvere.
John Baez (Jan 28 2021 at 19:52):If anyone finds any paper by Lawvere easy to understand, it's because they've spent a lot of time reading that paper.
Fawzi Hreiki (Jan 28 2021 at 19:52):Or they're Lawvere
Fawzi Hreiki (Jan 28 2021 at 19:52):Or I guess maybe Urs Schreiber
John Baez (Jan 28 2021 at 19:53):No, Urs had no magic understanding of Lawvere until he spent a lot of time studying Lawvere.
John Baez (Jan 28 2021 at 19:53):He spent a lot of time at it.
John Baez (Jan 28 2021 at 19:54):And then he wrote a lot about it....
Fawzi Hreiki (Jan 28 2021 at 19:54):I'm sure. I've been very slowly reading through his Hegel stuff on the nLab.
Fawzi Hreiki (Jan 28 2021 at 19:55):Also, just as a point of info for anyone, Urs' Geometry of Physics notes are fantastic.
Eric Forgy (Jan 28 2021 at 19:55):Urs definitely is gifted with magic of some kind. He works hard like everyone, but he has a magical ability to absorb things more quickly than most mortals (certainly me :sweat_smile: ).
Eric Forgy (Jan 28 2021 at 19:57):I remember when Urs was still in high school.
John Baez (Jan 28 2021 at 20:04):By "not magical" I meant that Urs did not just breeze through Lawvere.
John Baez (Jan 28 2021 at 20:08):It would be good for someone to explain a lot of Lawvere's ideas in a more pedagogical way: Lawvere often has no concern for whether he's being easy to follow (though in his two introductory books on category theory he really tries).
John Baez (Jan 28 2021 at 20:08):But I think most category theorists get around to understanding a lot of what Lawvere said.
Fawzi Hreiki (Jan 28 2021 at 20:09):I think though that despite this, the essential ideas themselves are really very simple. It just takes some wading
John Baez (Jan 28 2021 at 20:22):Yes, the most important ideas of his are simple; once you get them you can barely imagine not having them.
John Baez (Jan 28 2021 at 20:33):Here's my attempt to explain a few of Lawvere's ideas:
Start here:
Lawvere started out as a student of Clifford Truesdell, working on "continuum mechanics", which is the very practical branch of field theory that deals with fluids, elastic bodies and the like. In the process, Lawvere got very interested in the foundations of physics, particularly the notions of "continuum" and "physical theory". Somehow he decided that only category theory could give him the tools to really make progress in understanding these notions. After all, this was the 1960s, and revolution was in the air. So, he somehow got himself sent to Columbia University to learn category theory from Sam Eilenberg, one of the two founders of the subject. He later wrote:
In my own education I was fortunate to have two teachers who used the term "foundations" in a common-sense way (rather than in the speculative way of the Bolzano-Frege-Peano-Russell tradition). This way is exemplified by their work in Foundations of Algebraic Topology, published in 1952 by Eilenberg (with Steenrod), and The Mechanical Foundations of Elasticity and Fluid Mechanics, published in the same year by Truesdell. The orientation of these works seemed to be "concentrate the essence of practice and in turn use the result to guide practice".
It may seem like a big jump from the down-to-earth world of continuum mechanics to category theory, but to Lawvere the connection made perfect sense - and while I've always found his writings inpenetrable, after hearing him give four long lectures in Florence I think it makes sense to me too! Let's see if I can explain it.
Lawvere first observes that in the traditional approach to physical theories, there are two key players. First, there are "concrete particulars" - like specific ways for a violin string to oscillate, or specific ways for the planets to move around the sun. Second, there are "abstract generals": the physical laws that govern the motion of the violin string or the planets.
In traditional logic, an abstract general is called a "theory", while a concrete particular is called a "model" of this theory. A theory is usually presented by giving some mathematical language, some rules of deduction, and then some axioms. A model is typically some sort of map that sends everything in the theory to something in the world of sets and truth values, in such a way that all the axioms get mapped to "true".
Since theories involve playing around with symbols according to fixed rules, the study of theories is often called "syntax". Since the meaning of a theory is revealed when you look at its models, the study of models is called "semantics". The details vary a lot depending on what you want to do, and physicists rarely bother to formulate their theories axiomatically, but this general setup has been regarded as the ideal of rigor ever since the work of Bolzano, Frege, Peano and Russell around the turn of the 20th century.
And this is what Lawvere wanted to overthrow!
Actually, I'm sort of kidding. He didn't really want to "overthrow" this setup: he wanted to radically build on it. First, he wanted to free the notion of "model" from the chains of set theory. In other words, he wanted to consider models not just in the category of sets, but in other categories as well. And to do this, he wanted a new way of describing theories, which is less tied up in the nitty-gritty details of syntax.
To see what Lawvere did, we need to look at an example. But there are so many examples that first I should give you a vague sense of the range of examples.
John Baez (Jan 28 2021 at 20:33):... and then I go on and get into some details.
Eduardo Ochs (Jan 28 2021 at 23:41):John Baez said:
I can explain Kan extensions in ten minutes to anyone who understands adjoint functors. So can Joe!
I would love to see that!
I had to hammer Kan extensions a lot to make them fit into my tiny brain.
I only really understood them when I got the diagrams in section 7.9-7.11 here, but now it's time to learn how to translate that to how other people think about them...
Chetan Vuppulury (Jan 29 2021 at 05:47):@John Baez I never really got around to understanding Kan extensions. The 'why would I care about them' would be something I would love to know so that I can finally get into reading about them.
John Baez (Jan 29 2021 at 05:53):I suggest looking at my written "lectures" on category theory here:
Starting with lecture 49 I explain Kan extensions, but I explain them using the example of databases, so you might want to start further back, like lecture 34 or lecture 36, to see how databases are functors.
Chetan Vuppulury (Jan 29 2021 at 06:44):Thanks!
Paolo Perrone (Jan 29 2021 at 09:01):If you people make a "Kan extension seminar"-style blog summary of the article, I suggest the following title:
"Taking "Taking categories seriously" seriously".
Fawzi Hreiki (Jan 29 2021 at 10:33):Back to the original point, I think we should do it
Fawzi Hreiki (Jan 29 2021 at 10:36):So are we gonna start with the Metric Spaces paper?
Jade Master (Jan 29 2021 at 17:05):Sure sounds good
Chetan Vuppulury (Jan 29 2021 at 22:07):@Fawzi Hreiki Just to confirm, you are talking about 'Metric spaces, generalized logic and closed categories' right?
Fawzi Hreiki (Jan 29 2021 at 22:28):Yes
Fawzi Hreiki (Jan 29 2021 at 22:28):Then ‘Taking Categories Seriously’
Eduardo Ochs (Jan 30 2021 at 04:07):Paolo Perrone said:
If you people make a "Kan extension seminar"-style blog summary of the article, I suggest the following title:
"Taking "Taking categories seriously" seriously".
Or: ""Taking categories seriously" is a Kan extension" =)
Jacques Carette (Jan 30 2021 at 21:17):Wow, a link between Truesdell and Lawvere - I had not idea. His "An Essay Toward a Unified Theory of Special Functions" published in 1949 is really quite something. His F-Equation, which is a simultaneous first-order ODE and first-order recurrence, really does capture a huge number of special functions. Including some of the more unusual ones, like polylogarithms.
Christian Williams (Jan 31 2021 at 03:32):what would help in setting up these reading groups? a tool for easily polling meeting times, maybe something pinned announcing the material and current progress?
Fawzi Hreiki (Jan 31 2021 at 09:51):I’m not really sure. I haven’t done one of these before
John Baez (Jan 31 2021 at 17:40):Someone should be energetic about making this reading group happen. (Not me, I'm just a bystander.)
Maybe @Christian Williams should do it. He's pretty good at organizing things.
Christian Williams (Jan 31 2021 at 17:47):I was just asking a meta-question; not sure I have time now but it's possible.
John Baez (Jan 31 2021 at 17:52):Oh, okay. Anyway, I hope the people who want to do this actually do it. Sometimes everyone waits for everyone else to take the lead.
Fawzi Hreiki (Jan 31 2021 at 20:38):I don’t mind coordinating but I didn’t wanna jump the gun since @Jade Master started the topic.
John Baez (Jan 31 2021 at 20:47):Jade said she was willing to read Metric spaces, generalized logic and closed categories with people, so the next step is for people to start naming possible schedules for reading it... and/or set up a Doodle poll.
John Baez (Jan 31 2021 at 20:47):If everyone politely waits for everyone else it won't happen. :upside_down:
Jade Master (Jan 31 2021 at 21:27):@Fawzi Hreiki sure go ahead. I've been busy lately so it would be a help :)
Fawzi Hreiki (Jan 31 2021 at 21:43):Ok cool. So I've had a glance at the paper. Its 37 pages long but a lot of that is (re)introducing enriched generalities. It's split into an introduction and five sections (although sections 4 and 5 are very short).
I'm not sure what people's workloads/schedules are like but I think, to keep it relaxed, we could do roughly a section per 3 days (joining together sections 4 and 5).
Fawzi Hreiki (Jan 31 2021 at 22:06):I wouldn’t mind starting right away but that of course depends on everyone else.
Christian Williams (Feb 01 2021 at 02:20):if y'all decide to meet sometime, I'd join the conversation. it's a really cool paper, though definitely went over my head the first time I tried a couple years ago. should go better this time.
Joe Moeller (Feb 01 2021 at 02:22):Yeah, I'm also abstaining from the scheduling part of this, but if it fits my schedule, I'll attend.
Eric Forgy (Feb 01 2021 at 06:37):Skimming the "Metric spaces" paper, it seems to hit on a lot of things I'm interested in, so I'd be interested in following along.
GhaS Shee (Feb 13 2021 at 08:15):Super nice paper !
I really be inspired and I feel like having a ticket to the Lawvere's world!
By the way, I lost my notes where I drew the adjunctions( kan extensions ) between forall and exists
I cannot remember what is x
where
exists p. -| x -| forall p.
Does anyone help me !?
Edit:
Oops, I found some link that mentions it.
x was the inverse image of first projection.
https://projecteuclid.org/download/pdf_1/euclid.bia/1403013959