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nLab > Latest Changes: geometry of physics – basic notions of category theory

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2018

    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.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 23rd 2018

    Not sure which page this on after the split, but something needs fixing in Example 1.24. Is it just equation (6)?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2018

    Thanks for catching this! Yes, the order if i *i^\ast and r *r^\ast in equation (6) was wrong. Should be fixed now.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2021

    There was still a left/right mixup in statement/proof of this Prop.. Hope to have fixed it now.

    diff, v40, current

    • CommentRowNumber5.
    • CommentAuthorGuest
    • CommentTimeMar 3rd 2023

    Hello,

    This page says, in example 1.36

    These hence form an adjoint triple

    Disc⊣U⊣coDisc. Disc \;\dashv\; U \;\dashv\; coDisc \,.

    Hence the adjunction unit of Disc⊣UDisc \dashv U and the adjunction counit of U⊣coDiscU \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))⟶εX⟶ηcoDisc(U(X)). 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⊣UDisc \dashv U and the adjunction unit of U⊣coDiscU \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

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2023

    Yes, thanks for catching this! Fixed now (here).

    diff, v43, current

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeMar 3rd 2023

    Another thing. Definition 1.41 contains the diagram

    f=η c∘R(f˜)AAAA c η c↙ ↘ f R(L(c)) ⟶R(f˜) R(d) L(c) ⟶f˜ d 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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2023

    Yes, thanks for catching this! Fixed now (here).

    diff, v45, current

    1. fixed lemma 1.68 and proposition 1.69 where the induced adjoint modality is in the wrong order
    2. fixed the last diagram in the proof of proposition 1.77 where the leftmost functor is incorrectly labelled id instead of L
    3. fixed the first diagram in the proof of lemma 1.68 where the morphism LCRCLX -> RCLX should be \epsilon^\bigcirc, instead of \eta^\bigcirc

    CanaanZhou

    diff, v46, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2023

    Thanks for chasing typos! I appreciate it.

    • CommentRowNumber11.
    • CommentAuthorsebasgar
    • CommentTimeJun 11th 2024

    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:

    1. fixed the diagram in remark 1.28 where the labelling for the rightmost morphism was incorrect.
    2. fixed the first diagram of definition 1.33, where the categories had been incorrectly labelled given the morphisms used later on.
    3. fixed some misc typos and corrected weird wording

    diff, v47, current

    • CommentRowNumber12.
    • CommentAuthorsebasgar
    • CommentTimeJun 11th 2024

    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⊣RL\;\dashv\;R should read L⊣CL\;\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?

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2024

    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.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2024

    I have edited Remark 1.34 now (here), thanks for pointing this out.

    I did mean to use the symbols as I did (so that “LL” is always the Leftmost, “RR” the Rightmost), but I can see how it may be confusing. So I have now adjusted the wording around it.

    diff, v48, current

    • CommentRowNumber15.
    • CommentAuthorsebasgar
    • CommentTimeJun 11th 2024
    • (edited Jun 11th 2024)

    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⊣GF\;\dashv\; G are a pair of functors between categories with GG mapping in the opposite direction as FF, such that FF and GG 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⊣C⊣RL\;\dashv\; C\; \dashv \; R, hence this would mean that L,R:𝒞 1→𝒞 2L, R: \mathcal{C}_1 \rightarrow \mathcal{C}_2. Therefore does it make sense to talk about the adjunction “L⊣RL\;\dashv\; R”? If so then the content might not be reflecting this to an unfamiliar audience.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2024

    No, that’s what I tried to clarify (but clearly I failed):

    The “LL” and “RR” in the sentence on adjoint triples are not the same “LL” and “RR” as in the previous sentence on adjoint pairs.

    Let me try to reword further…

    • CommentRowNumber17.
    • CommentAuthorsebasgar
    • CommentTimeJun 11th 2024
    • (edited Jun 11th 2024)

    I completely agree that L,RL,R are not the same between the sections on pairs and triples.

    I am strictly not understanding the triples section:

    L⊣C⊣R(12)L \;\dashv\; C \;\dashv\; R \,\quad\quad(12)

    …

    Notice that in the case of an adjoint triple (12), the adjunction unit of C⊣RC\;\dashv\;R and the adjunction counit of L⊣RL\;\dashv\;R …

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2024
    • (edited Jun 11th 2024)

    Oh now I see, sorry for being slow, thanks for insisting.

    I have fixed that typo now

    (that sentence is just narrating the little diagram that follows, here)

    diff, v48, current

1 to 18 of 18

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