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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.
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