Math3ma

I'm starting to dive into the Path Integral formulation and this is taking me to turn things upside-down from what I thought of the order of what relies on what. I've started to get Baez's view [1] and given that, the question in [2] becames more and more important. But it's never too late to learn. An important element is to search the simplest cases posible, discrete better than continuous, finite beter than countable. If you think of a matrix as a number-labeled bipartite graph, and you glue the graphs of a sequence of matrices the only sensible way, then the product of the sequence is a "path sumatorial", look at here [3]. You sum productorials one for each path, the sum extending to all paths. That is readable in the original Feynman paper. That is for finite state spaces. For a continous state space (e. g. some real variables), the integral became a path integral, and you are integrating a functional over paths, so you need a Lebesgue measure on a space of paths to do a Lebesgue Integral. Baez has some old lectures that develop further.

[1] https://www.classe.cornell.edu/spr/1999-03/msg0015539.html

[2] https://hsm.stackexchange.com/questions/553/why-isnt-feynmans-path-integral-taught-more-widely-and-earlier-in-todays-acad

[3] https://www.math3ma.com/blog/matrices-probability-graphs

Great textbook, I learned all the topology I know from it. Previously, Category Theory was taught as a field that connects branches of math, and thus in terms of other concepts. But recently there's a movement to view Category Theory as the definitive underlying field of math (instead of set theory), and teach different fields of math in terms of Category Theory rather than vice versa (a new-new math in a sense). I learned Category Theory well before learning abstract algebra and topology, and the embedding of Topology in Category Theory was seamless and intuitive; I feel as though this book proves that this new CT-centric view of math education has merit.

One of the authors, Tai-Danae Bradley, also runs math3ma [1] and is a prominent figure in Applied Category Theory. I had the pleasure of hearing her talk, and her way of explaining abstractions is very easy to understand despite Category Theory being fairly obtuse at times (looking at you, Mac Lane!)

Also, an obligatory shilling of the Topos Institute [2]. They're a research institution based in Berkeley, and they have weekly talks on Category Theory that they release on youtube. If you're interested in the categorification of mathematics, you need to check them out.

[1] https://www.math3ma.com/

[2] https://topos.site/

As a Mathematician who worked in Theoretical Physics for a bit, I'm happy to hear that there's people out there interested in the area.

Maths often get a bad rep, but at their core they are probably the most interesting thing to learn, on par with music in my opinion.

I encourage anyone interested in Mathematics (regardless of your level) to check out Roger Penrose's The Road to Reality. I know it's technically a physics book, even though Penrose was originally a Mathematician, but trust me, he explains things in a way that open up your mind, even for seasoned veterans. He begins with basic calculus, but ends up covering Groups, Topology, Complex Analysis, Differential Geometry, etc etc etc.

And, on parallel, check out John Baez's guide on how to learn Maths and Physics: https://math.ucr.edu/home/baez/books.html

I love all of Baez's pages and work, he does some really interesting mathematics: https://math.ucr.edu/home/baez/

I'm also a big fan of Tai-Danae Bradley's work: https://www.math3ma.com/

Anyway, I hope people find these recommendations interesting or useful...

I'm not a natural mathematician (emphatically not!) and pretty much every attempt I've made to wrap my head around Category Theory has foundered on an onslaught of formalism by paragraph 2.

To be clear, I'm not knoocking that: for those who can and do grok things from that sort of treatment, all power to your elbow (and no small amount of admiration).

But it's not how my brain works. I need an intuitive understanding first: exemplars, informal pictures, whatever. I can only start unscrambling the formalism once I have the intuition.

The author's introductory blog post is a great example [0]. It's without doubt the best "bootstrap" I've come across - where "best" means "most understanable to me". I really admire people who can take abstract/complex subject matters and lay them out in a clear and accessible way for the layperson. Tai-Danae Bradley definitely fits in that Category (dubious pun intended!).

[0]: https://www.math3ma.com/blog/what-is-category-theory-anyway

EDIT: fixed typo

Plug for a friend's book that is forthcoming later this summer, "Topology: A Categorical Approach". [1]

I can't speak to its contents per se, because there isn't a preview yet, but I can speak to the quality of exposition in the lead author's math blog. [2]

I haven't ever dug too much into category theory for its own sake (usually just one-off chapters or appendices that get included in books on other topics), but my understanding is that it unites a lot of mathematical topics. As such, this book might be of more interest to you than, say, a classical point-set topology text, given your desire to uncover connections. That being said, there may be other category-theory-flavored books on other more strictly algebraic topics that would suit your fancy more.

[1] https://mitpress.mit.edu/books/topology

[2] https://www.math3ma.com/

As a high schooler interested in things that are probably out of my league, I love reading this kind of stuff. The author mentions their unconventional style at the beginning: hyperlinks, color, italics, exclamations, parentheses, side notes, etc. They address your thoughts and craft their explanations around them instead of being terse and optimizing for length.

The author's blog [1] has more of this style and has really helped me get deeper into category theory. Without it, I wouldn't have even begun to try and understand things like the Yoneda lemma (which still remains a mystery to me).

Does anyone know any other papers or resources with this style? Maybe something like "explorable explanations" [2]?

[1] https://www.math3ma.com/

[2] https://explorabl.es/