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Added some examples to comonadic functor. Prompted by this MO discussion (does anyone know how such monadic-comonadic iterations are referred to in the literature?).
I added an example concerning modalities. We should also be able to tell a similar story for the jet comonad.
I should think the trio – monadic functor, comonadic functor, monadicity theorem – could be integrated better.
Thanks David. I have added to the section statements that are at least sufficient to conclude that $EM(\Box_W) \simeq \mathbf{H}_{/\ast}$, namely that $\mathbf{H}$ is a topos and $W$ is inhabited.
and then I have edited the text a little, trying to polish a bit more, please check if you agree.
Looking good.
So the PDE case should go through in a similar way? Perhaps in explicit terms of SDG infinitesimals. Instead of the
$\prod_W (Q) = \Gamma_W(Q)$we need something like sections of the infinitesimal neighborhood of a point. Then base change that back. I guess that requires the language of jets.
And then the coalgebras are the ones that come from base change of bundles on $\Im(X)$? That seems to be multiplying points by infinitesimal neighborhoods.
Yes, dependent product along a general morphism $X \longrightarrow Y$ produces “spaces-of-sections-pointwise-over-$Y$”. And, yes, since the de Rham stack projectiuon $\Sigma \to \Im \Sigma$ is a 1-epi, jet coalgebras over $\Sigma$ are equivalently objects in the slice over $\Im \Sigma$. (In algebraic geometry these are the D-modules).
According to Todd here, when a monad is left adjoint to a comonad, then the algebras of the former are equivalent to the coalgebras of the latter. So the jet coalgebras in #6 are the same as ’infinitesimal neighborhood’ algebras, necessity coalgebras (#3) as possibility algebras, etc., right?
This is quite a nice way to picture things: A possibility algebra/necessity coalgebra, $A \in \mathbf{H}/W$, requires a map, $\sum_W A \to \prod_W A$. Given a point in the total space, we need a section through that point.
I cannot figure out what happened to this article. It now apparently redirects to monadic functor, and some stuff that used to be at comonadic functor is now gone. For example, a result of Mesablishvili on when free functors out of $Set$ are comonadic – a result I’ve found useful over the years – has seemingly vanished.
FWIW, the last Internet Archive snapshot of comonadic functor (dated 2021-01-28 12:08:59Z) still contains this reference at the end of the first subsection of §4, and its math markup is mostly intact.
Thanks, Guest – I appreciate it. I am supposing the explanation has something to do with the migration effort.
I am supposing the explanation has something to do with the migration effort.
Several other pages that have been broken in this way on Dec 29, for instance as reported here.
Richard seems to be too sick to continue with the migration effort. I think it would be best if the installation were re-instantiated to its form on 28 Dec 2021.
Thanks, Urs. I am sorry to hear of Richard’s illness, and hope he gets better soon.
It’s a tough situation. Even if Richard were experiencing excellent health and had more time, it seems to be a huge job for one person.
I had initiated the donation mechanism to the $n$Lab in order to use the money to hire a professional IT service to take care of the $n$Lab, so that we don’t need to bother volunteers with the task. We seem to have accumulated sufficient donations to go that route, but Richard had since been opposing it. Now I feel I have run out of ideas, for the time being.
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