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OK, moderately well-read in software engineering but not in math; experienced programmer; start reading the intro, sounds promising; but early on I find some presentation problems which I think are going to impede you reaching the audience you describe in the first few paragraphs. Just a bit of a commentary here; if I persist I may send a more detailed review -- although I note that there is no invitation to do so, or address for comments, in the Introduction?

Top of p.iv, "almost no introductory material available on monoidal categories." Wait, it was all about categories, now it is suddenly monoidal ones -- what is monoidal, is that a subset of the book subject, or a more precise term? Badly need a definition when first introducing a new term.

(Actually, that just highlights the point that the term "category" has been assumed and not defined to this point -- despite the audience not being expected to know what it is.)

Lemon pie diagram: of _course_ I can grasp the basic idea, but in what sense does the diagram illustrate category theory? Where are is the category (categories?), is it the pie, the lines, the nodes? It's probably blindingly obvious _after you understand the topic_ but at the opening of the book, the point needs to be explained even if the explanation seems redundant to the author.

Top paragraph on p.v flirts with a definition but doesn't actually define anything. Is a category "structures and coherence"? One "involves" (vague) "a collection of objects, a collection of morphisms relating objects, and a formula for combining any chain of morphisms into a morphism" and OK, here is a hint of familiar ground: this sounds a lot like category theory might be a generalization of classes, instances and methods, and functional composition, of software engineering. But the sentence about chains of chains ending in a happy "That's it!" doesn't really seem to follow. Onward.

Bottom of page 1, we finally get to some substance, although I really don't like the "Y is a meter" explanation because that whole paragraph kind of bodges the distinction between italic-capital-letter X and Y, which I am kinda sure you intend to represent classes or domains, with the word "object" which my prior training really wants to confine to singular instances of a domain. So what comes out of the map isn't a "meter Y" but "the reading displayed by the meter Y" but that doesn't work either.

Anyway, very shortly you drop in another new notation, "functions f from big-R to big-R" What does the big-R mean here? I'm casting about -- real-world-objects? set of Real numbers? Probably real numbers because the following features are about arithmetic comparisons and operations. But say it, dammit.

Order-preserving, ok, but -- why "metric-preserving"? How is absolute-difference a "metric"? But... this is an exercise and a challenging one, and I'm intrigued. Later.




There is a link for posting comments/feedback on the main arxiv page: https://docs.google.com/document/d/160G9OFcP5DWT8Stn7TxdVx83... There are ~381 comments there already, please post yours: it looks like the authors are genuinely interested in listening to feedback and addressing them perhaps in future revisions of the book.


I agree that to an inexperienced reader, most of your points would be valid. The downside is, it would be too verbose (imagine defining what big R is, for example). It being too verbose would stray away a different subset of readers, which arguably is more invested in learning (the number of readers that know what R and a monoid are but not category theory).

I think that the best way forward in your case is simply googling terms that the writers took for granted and which you don't understand (a monoid, for example), because the book material is one of the best I've seen on category theory.


While a monoid in abstract algebra is easy enough to understand from Wikipedia. Googling for monoids in the context of category theory could easily land you at nlab I.e https://ncatlab.org/nlab/show/monoid+in+a+monoidal+category I’m not sure that’s helpful to new a learner of the subject.

I recall that my personal experience of learning about CT was unnecessarily difficult due to being thrown into a bunch of definitions way to soon. (Which is understandable since there isn’t much else to it.) but I needed to un-learn my preconceptions about what information the theory encodes first, which was rally hard. Understanding that objects and morphisms are really _only_ about composability and _nothing else_ is challenging when your mind desperatly wants to see concrete thing like sets or numbers or what have you in those objects. Definition don’t help much until you can shed that urge. (To new learners I suggest to think of “objects” as the number of dots on domino tiles, or the poles of magnets, they describe where things may, or may not, compose, nothing more. category theory is just about naming patterns and rules of composition)

In particular even knowing the existence of monoidal categories is a distraction when getting to grips with how catgories as such generalize and abstracts monoids.


> CT was unnecessarily difficult due to being thrown into a bunch of definitions way to soon

I had a similar experience. Luckily, due to reading many definitions of the basic things (such as monoids), I reached a more complete? intuition of the concept if you will. I think that given enough definitions, people will converge to the same intuition of abstract concepts.

> To new learners I suggest to think of “objects” as the number of dots on domino tiles, or the poles of magnets, they describe where things may, or may not, compose, nothing more

Wouldn't a simpler intuition of circles and arrows be easier?


I meant that as examples of concrete simple things (that aren’t functions, types, sets or any other not fully understood concept) with composition rules that could be modeled in a category.

Categories might be drawn using arrows and circles. But that’s just notation.


A glossary as an annex sounds like a way to address this... Big-R and monoid would definitely warrant an entry.


Yep, that seems like an elegant solution.




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