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Stream: learning: reading & references

Topic: Feedback on Ralph Sarkis book

Eric M Downes (Jun 13 2024 at 13:13):

I wanted somewhere to provide feedback on @Ralph Sarkis (excellent) intro-level book and he suggested we do it here as github no like latex. Ralph doesn't have a lot of time right now so this might just be me talking to myself, and that's fine.

Eric M Downes (Jun 13 2024 at 13:22):

I found this statement on p. 20 (Categories 1.1) confusing
"In other words, composition of functions is not a
binary operation ◦ : Set1 × Set1 → Set1, it is of type Set2 → Set1."

this sounds like you are saying composition takes a 2-morphism and returns an arrow... which is not false (2-morph witnessing the composition of $f,g$ returns $f\circ g$) except you never defined Set2, and we are talking about categories not n-categories yet so it seems crazy to get into witnesses of composition instead of on-the-nose equality...

Perhaps you meant to write that dom() and cod() are of types Set1 -> Set0?

I know introducing a fibered product at this stage is not strictly self-contained, but fwiw that's what MacLane does... he just doesn't call it a fibered product. :) He manually defines $\times_{\sf Set_1}$ as a binary relation in Ch 1, a subset of $\sf Set_1\times Set_1$, and uses that as the domain for the composition operator, until he has built the proper equipment to express it more generally. That might prove useful?

Ralph Sarkis (Jun 13 2024 at 14:00):

I did define $\mathbf{Set}_2$ in the definition of paths, it is the collection of pairs of functions $(f_1,f_2)$ such that $s(f_1) = t(f_2)$. I think it is worth to remind what it is in a footnote. Also, the 2 in subscript is a link to that definition.

Eric M Downes (Jun 13 2024 at 15:34):

ahhh thanks! My cheapo ipad pdf reader sucks and wasn't following the link. Sorry. I'll use one that doesn't suck. :)

Eric M Downes (Jun 13 2024 at 18:06):

Minor suggestions / comments, no major issues, so ignore as you like!

p.21

It might be nice to ask readers in an end-of-chapter exercise to use induction to prove that $\circ:\mathsf{C}_k\to\mathsf{C_{k-1}}$ extends to $\circ:\mathsf{C}_k\to\mathsf{C}_1$.

It would require adding a page or two, but regarding (7) and parallel arrows not commuting... How do you feel about Eduardo Ochs' suggestions of adding a total order labelling with quantifiers for some beginning diagram constructions? I don't mean to suggest doing this in all diagrams, just soon after diagrams are first introduced show a a few careful examples (such as the square diagram), where you break down the steps to help anyone who does find the geometric reasoning non-intuitive, and reference section 4 of Ochs' paper. (I've realized that some people just don't get diagrams; I have a friend who works in combinatorics, and can rattle off normal subgroups or prime divisiors of large numbers as if from memory, but rarely understands my intention when I draw a diagram. He doesn't like others' diagrams either, so it could be a rare cognitive disability, I don't know... but I pointed him to Ochs' material and he was at least able to deduce what I was saying logically.)

Normally I wouldn't even suggest it, but I think this is worth doing in this book because you start using diagrams almost immediately, to define functors, which I love! So its worthwhile at least offering a little more formality to those who need it.

p. 25

I love that you are introducing the delooping $\mathbf{B}G$ this soon.

I am glad you put in footnote 75 regarding FinSet. It could also be an exercise if you broke it down a bit.

Ex 110 -- glad to see something hinting at issues that can arise in idempotent completions.
(Also, I had forgotten what SOL meant by the time I got here)

I like that you introduce the idea of concepts (like concrete categories) at a natural time, before defining them, and offer a lookahead to the definition. I wish more mathematics writing did this.

p. 27 -- really like "functors ASAP" approach

p.28 -- you leave the definition of $u_{\sf C}:C_0\to C_1; ~u_{\sf C}:x\mapsto id_x$ to a footnote... if that isn't intentional, you could mention it when you define a category. (I did look harder this time, and while it is possible I still missed it, I am more certain I wasn't just skimming over it.)

Ralph Sarkis (Jun 13 2024 at 20:02):

Eric M Downes said:

It might be nice to ask readers in an end-of-chapter exercise to use induction to prove that $\circ:\mathsf{C}_k\to\mathsf{C_{k-1}}$ extends to $\circ:\mathsf{C}_k\to\mathsf{C}_1$.

I think induction and associativity are part of the mathematical maturity of the reader going into this book, although we do not use induction that much in basic category theory. On the other hand, maybe I could make this an exercise in diagram paving.

Ralph Sarkis (Jun 13 2024 at 20:05):

Eric M Downes said:

How do you feel about Eduardo Ochs' suggestions of adding a total order labelling with quantifiers for some beginning diagram constructions?

I agree with the sentiment that diagrams are hard to get, and I don't think I have found a good way to introduce them in this book (yet?), but I personally did not feel Eduardo's diagrams were helpful. I wish I could remember when this stuff clicked.

Ralph Sarkis (Jun 13 2024 at 20:09):

Thanks for the other things to think about :smile:

Eric M Downes (Jun 13 2024 at 20:20):

p. 29 -- constant functor -- minor quibble; you elsewhere (such as in the solutions to Ch 1) use $\Delta:X\to XX;~\Delta:x\mapsto (x,x)$.. maybe use $!_x$ instead for the unqiue constant functor to object $x$?

Eric M Downes (Jun 13 2024 at 20:23):

Ralph Sarkis said:

I think induction and associativity are part of the mathematical maturity of the reader going into this book, although we do not use induction that much in basic category theory. On the other hand, maybe I could make this an exercise in diagram paving.

I think that would be a cool exercise!

And agreed on the maturity expected. Hence the importance of having a proof use induction; its trivial for people who have seen it, but if somebody hasn't, its a warning to turn around.

Ralph Sarkis (Jun 14 2024 at 06:45):

Eric M Downes said:

p. 29 -- constant functor -- minor quibble; you elsewhere (such as in the solutions to Ch 1) use $\Delta:X\to XX;~\Delta:x\mapsto (x,x)$.. maybe use $!_x$ instead for the unqiue constant functor to object $x$?

Both of them are instances of the generalized diagonal functor which is defined later with the notation $\Delta$ :smirk:

Ralph Sarkis (Jun 14 2024 at 08:16):

I feel like my responses could seem overly negative, and I hope it does not deter you from writing feedback. It is easier to say when I like something like it is rather than say how I would change it in response to your comments, so the positive answers will probably be less detailed.

Eric M Downes (Jun 14 2024 at 16:33):

no worries at all, man. I love the level of thought that has gone into your presentation! Most of my criticisms have been pretty half-baked tbh! :)

I think you might have converted me to the C0 C1 C2 ... way of defining categories using explicit functions and restrictions. I had been implicitly using that way of thinking for a while, but hadn't fully embraced the notation, and had stopped short of outright defining things that way.

It seems clear at least from where I am at present, that this is The Way, in light of having an idea of where this all leads: presheaves, enrichment, higher cats, etc. but also the simplicity and cohesiveness of presentation for undergraduates.

So far that is what really sets your book apart from others at this level. You demand a little more notational sophistication than Leinster, and a lot more math than Spivak, Chang, et al. but not so much as Reihl, and the clarity really pays off. I think you targeted the right level.

So... a question; is there a reason you are using $u:{\sf C}_0\to{\sf C}_1$ instead of $id:{\sf C}_0\to{\sf C}_1$ outright? You appear to prefer single-letter functions $s,t$ instead of $dom,cod$ etc. so maybe its just that.

John Baez (Jun 14 2024 at 16:39):

By the way, one reason not to say $id : \mathsf{C}_0 \to \mathsf{C}_1$ is that for a lot of people $id$ means "identity morphism". And I think anyone who introduces a map like $id : \mathsf{C}_0 \to \mathsf{C}_1$, even if they call it $u : \mathsf{C}_0 \to \mathsf{C}_1$, should say that this is a map that sends each object to its identity morphism, but this map is not an identity morphism. Of course for an ideal reader this remark is unnecessary, but I find students get confused at first about this point: a phrase like "a map that maps each object to its identity morphism" tends to short-circuit their neurons when they first hear it.

Eric M Downes (Jun 14 2024 at 16:40):

That's a good point! Certainly it is very tempting to just write $id:x\to x$ for object $x$ when its clear in context, instead of $id_x:x\to x$ and I often am guilty of this :)

Julius Hamilton (Jun 15 2024 at 13:01):

Is @Ralph Sarkis looking for feedback on his book? Because I can provide some, in case that would give perspective on the intelligibility of the book to the neophyte. Could be a win-win.

Ralph Sarkis (Jun 15 2024 at 13:36):

I would not say I am actively looking for feedback, simply because I don't think I am done writing it (especially the later chapters). But I believe (with some evidence) it is already a good resource for learning, so if you are already reading (parts of) it, and you want to make comments or ask questions here, it will be appreciated.