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Since the page geometry of physics – categories and toposes did not save anymore, due to rendering timeouts caused by its size, I have to decompose it, hereby, into sub-pages that are saved and then re-!included separately.
With our new announcement system this means, for better or worse, that I will now have to “announce” these subsections separately. Please bear with me.
Not sure which page this on after the split, but something needs fixing in Example 1.24. Is it just equation (6)?
Thanks for catching this! Yes, the order if $i^\ast$ and $r^\ast$ in equation (6) was wrong. Should be fixed now.
There was still a left/right mixup in statement/proof of this Prop.. Hope to have fixed it now.
Hello,
This page says, in example 1.36
These hence form an adjoint triple
$Disc \;\dashv\; U \;\dashv\; coDisc \,.$Hence the adjunction unit of $Disc \dashv U$ and the adjunction counit of $U \dashv coDisc$ exhibit every topology on a given set as “in between the opposite extremes” of the discrete and the co-discrete
$Disc(U(X)) \overset{\epsilon}{\longrightarrow} X \overset{\eta}{\longrightarrow} coDisc(U(X)) \,.$
But this seems to be backwards? I think it should instead say
Hence the adjunction counit of $Disc \dashv U$ and the adjunction unit of $U \dashv coDisc$ exhibit every topology on a given set as “in between the opposite extremes” of the discrete and the co-discrete
I find the concept of adjunctions confusing so I am not sure.
Adrian
Another thing. Definition 1.41 contains the diagram
$f \;=\; \eta_c \circ R(\widetilde f) \phantom{AAAA} \array{ && c \\ & {}^{\mathllap{\eta_c}}\swarrow && \searrow^{\mathrlap{f}} \\ R(L(c)) &&\underset{R (\widetilde f)}{\longrightarrow}&& R(d) \\ \\ L(c) &&\underset{ \widetilde f}{\longrightarrow}&& d }$
but clearly the equation on the left doesn’t match the diagram on the right; I think that the order of composition should be reversed.
CanaanZhou
Thanks for chasing typos! I appreciate it.
Just made my first batch of edits on the site!
I thought the edits were more or less trivial so I didn’t make comments then, but seeing the discussion I have annotated them below to match the previous records:
There was also one more correction, but as I wasn’t certain and hence decided against making it:
[v47] In Remark 1.34, between (12) and (13): I think $L\;\dashv\;R$ should read $L\;\dashv\; C$.
I am generally inexperienced with the content in the site (I come from a background in physics but left academia after my masters). I am using the Geometry of Physics series as my main source towards learning mathematical physics in my spare time. Therefore, will likely find more typos and mistakes along the way. I still don’t fully understand adjoints, hence my hesitation to make this particular change.
As a general rule would it be better for me to comment on the forums on what I think is wrong and let more experienced members change it? I originally thought my diff would be manually verified as I “hadn’t been vetted”, and am now generally afraid to make an incorrect correction in the future that just gets pushed globally. If I am still “encouraged” to make the changes myself, is there any preference towards accumulating errors over a larger period of time and making a single revision, or are more frequent smaller revisions ok?
Thanks for fixing typos etc.! I appreciate it.
As for accumulated or smaller revisions: I may find smaller revisions easier to follow and to react to, but whatever works for you.
The new wording is much better; unfortunately I still don’t quite follow the notation used in the paragraph following your revision.
As I understand it (in layman’s terms) adjoint functors $F\;\dashv\; G$ are a pair of functors between categories with $G$ mapping in the opposite direction as $F$, such that $F$ and $G$ have some special properties. As the page on adjoint triples shows, in a triple, the first and last functors map between the same categories. In our case we have that $L\;\dashv\; C\; \dashv \; R$, hence this would mean that $L, R: \mathcal{C}_1 \rightarrow \mathcal{C}_2$. Therefore does it make sense to talk about the adjunction “$L\;\dashv\; R$”? If so then the content might not be reflecting this to an unfamiliar audience.
No, that’s what I tried to clarify (but clearly I failed):
The “$L$” and “$R$” in the sentence on adjoint triples are not the same “$L$” and “$R$” as in the previous sentence on adjoint pairs.
Let me try to reword further…
I completely agree that $L,R$ are not the same between the sections on pairs and triples.
I am strictly not understanding the triples section:
$L \;\dashv\; C \;\dashv\; R \,\quad\quad(12)$
…
Notice that in the case of an adjoint triple (12), the adjunction unit of $C\;\dashv\;R$ and the adjunction counit of $L\;\dashv\;R$ …
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