@John In fact, Fong and Spivak did mention the Yoneda lemma in passing (Oct 12 version, p.20), though restricted to the case of preorders and using the contravariant (upper set) version for some reason. :-? (I dug this out when one of my professors …
Yeah you are right. I guess in programming languages phonology does not have a role to play so syntax shoulders everything about expression formation. This is also true, in a broader sense, for artificial languages like Esperanto, where there is muc…
@John Cool, thanks for the clarification! :-)
By the way, is there any way to efficiently "@" members in this forum? Sometimes I miss replies targeted at me because I don't receive notifications and forget to follow up on all my comments... XD
Hi @Grant, glad to hear you are interested. Off the top of my head is Berwick and Chomsky's 2016 book Why Only Us which is a fairly recent (and not too technical) summary of the main results of the field in the past few decades (the first chapter in…
Fong and Spivak (p.32, Theorem 1.115) give an "adjoint functor theorem for preorders", which I guess is just the adjoint-computing method in this lecture. In addition, Fong and Spivak explicitly say that this method is only applicable when the two p…
By the way, I see the following from Ellerman (2016) "Brain Functors: A mathematical model of intentional perception and action":
When a morphism is between objects of the same category, it is called a homomorphism or hom, and when between objec…
I quite enjoyed the general style and the informativeness of the booklet, which seems closely related with what we've been learning here as well. One thing I was less sure about was the article's take on natural language syntax, in particular the di…
I have two small questions:
1 - In the appealing flight analogy:
We can think of the arrows in our Hasse diagrams as one-way streets in two cities, \(X\) and \(Y\). And we can think of the blue dashed arrows as one-way plane flights from cities…
Merry Christmas! Just a small question: is this bit from the lecture still true? -
Actually Fong and Spivak use the opposite convention, writing \(x \le y\) to mean you can get \(y\) if you have \(x\)...
I see the following in Fong & Spiva…
Now that I'm back to review the lessons, I have a tiny question re John's wording in 7:
... If a function \(f : A \to B\) is a monotone map between preorders and it has an inverse, its inverse may not be a monotone map.
Does "may not be" here …
@Keith Yeah I came across that paragraph too when reading Mac Lane's textbook. :-)
It's been so long since I last logged in (due to overwhelming workload...) but now that I have a bit more time I'm determined to catch up with everything that happen…
@John Yes, now that I have understood this, I totally see why modern theorists want to define only one type of functor. I just got confused by the mentioning of "contravariant" here and there (especially when we were trying to understand the op-tric…
An additional note on the op-trick for other beginners:
The situation in my previous comment #20 requires that the arrow direction in the first component of the product hom-functor be eventually reversed. There are two ways to do this:
(i) via a n…
Keith's string diagram #18 is epiphanic! And huge thanks to @John (#15 #16) and @Chritopher (#10) for the methodological clarifications! :-bd Now that I (think I) have a better understanding, I'll write down some tips in case other beginners might f…
Another layman question from me: in this diagram
[
\begin{matrix}
& & h & & \\
& c & \rightarrow & c' &\\
f & \downarrow & & \downarrow & g\\
& d & \rightarrow & d' &\\
& & ? …
@Keith Yeah I noticed the terminological overlapping between CT and philosophy/linguistics too - I had initially thought we borrowed from CT but it turned out to be the other way around. :P Do you perhaps know where Lawvere made the philosophy–math …
Thanks very much @John! Yes I did make notes of the bits you quoted but failed to realize that commutation (is this the correct noun?) did not come for free in the natural transformation square either. Now everything is clear! :)
I have been confused by the 'op' thing for quite a while. On reading Anindya's answer:
If you think about how this might works on pairs of morphisms \((f, g)\) you'll notice it has to be contravariant in the first variable and covariant in the s…
Thanks a lot for the explanation, @Christopher and @Jesus! I realize I had misunderstood 'commutative' as 'sharing source and target in the diagram', the latter being trivially true for all squares obtained from the natural transformation definition…
This might be a super layman question, but I was wondering what a 'non-commutative' square (i.e. a 'failed' attempt to get a natural transformation) would look like. I am curious because from the definition and examples given both in the textbook an…
By the way, I found the following remark on Milewski's blog:
[T]he functor L is called the left adjoint to the functor R, while the functor R is the right adjoint to L. (Of course, left and right make sense only if you draw your diagrams one par…
Fredrick said
The error is in treating \( L \) and \( R \) as if they were left and right ajoints of each other, they are not.
I feel this is spot-on but see two similar formulations in Fong & Spivak's book:
"We say that \(f\) is the lef…
Thanks for the very informative comment, @Matthew! Indeed, very often disciplinary concepts could have been given less overloaded terminology (e.g. I can totally feel your "calculus" point!). Apart from this perhaps socio-historical matter, though, …