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I created split coequalizer and absolute coequalizer, the latter including a characterization of all absolute coequalizers via an “$n$-ary splitting.” While I was doing this, I noticed that monadic adjunction included a statement of the monadicity theorem without a link to the corresponding page, so I added one. (The discussion at the bottom of monadic adjunction should probably be merged into the page somehow.) Then I noticed that while we had a page preserved limit, we didn’t have reflected limit or created limit, so I created them. They could use some examples, however.
I would also like to include an example of how to actually use the monadicity theorem to prove that a functor is monadic. Something simpler than the classic example in CWM about compact Hausdorff spaces; maybe monadicity of categories over quivers? Probably not something that you would need the monadicity theorem for in practice, so that it can be simple and easy to understand.
thanks for the limits, Mike, I added your new entries to the floating limit toc
One thing: at created limit and preserved limit is found lots of $U$s that I think were meant to be $F$s. I am actually pretty sure, but in case I am badly mixed up somehow, please have a look and check!
I don’t see the U’s, but if they’re there, they shouldn’t be; I meant to write only about one functor called F.
I don’t see the U’s
I should have said: I changed them to $F$s!
if they’re there, they shouldn’t be
Okay, good.
I didn’t even see them in the edit history. But as long as it’s correct now, that’s what matters.
I started adding some more variants to the monadicity theorem. I particularly like Duskin’s version.
I didn’t even see them in the edit history.
here for instance
There appear to be some typos in absolute coequalizer: First, in the last diagram, it seems that the two parallel arrows should be $f_0 \circ -$ and $f_1\circ -$ (instead of $f \circ -$ and $g\circ -$), then, in the following paragraph, it seems that in $(h_0,h_1)\colon M \to N\times N$, $M$ and $N$ should be replaced by $P$ and $Q$, and finally, there is something strange with $h_{1-\epsilon_i}(p_n)$ – shouldn’t this $i$ be $n$ or $n-1$? (I didn’t read it carefully yet.) Perhaps I’m wrong, but please have a look.
Also, regarding the monadicity theorem, is there a particular reason that the equivalent condition with absolute coequalizers (instead of split coequalizers) is missing? (This basically says that a monadic functor creates coequalizers for the parallel pairs that are sent by $U$ to a pair with an absolute coequalizer – is this considered useless?)
I think you’re right about the typos.
is there a particular reason that the equivalent condition with absolute coequalizers (instead of split coequalizers) is missing?
Probably just that no one thought to add it. Note that since every split coequalizer is absolute, but not conversely, there are fewer U-split pairs than U-absolute pairs, so “U creates coequalizers of U-split pairs” is a weaker condition than “U creates coequalizers of U-absolute pairs.” So the version given currently on the page appears to be slightly more general.
Thanks – corrected the typos.
Regarding monadicity: But the version with absolute coequalizers is stronger in the opposite direction (monadicity implies creation of absolute coequalizers). Is there no use for the opposite direction of the PTT?
I’ve never seen one, but it’s conceivable. However, if I did see one, I might be more inclined to regard it as a special case of the general theorem that “a monadic functor creates all colimits preserved by the monad” rather than as part of the monadicity theorem.
I see. Thanks!
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