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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 17th 2013
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexAdded some examples to [[comonadic functor]]. Prompted by this [MO discussion](http://mathoverflow.net/questions/136851/iterating-monad-comonads-structures/136937#136937) (does anyone know how such monadic-comonadic iterations are referred to in the literature?).

   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?).

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 16th 2015
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 16th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks David. I have added to the [section](http://ncatlab.org/nlab/show/comonadic+functor#Necessity) 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 16th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexand then I have edited the text a little, trying to polish a bit more, please check if you agree.

   and then I have edited the text a little, trying to polish a bit more, please check if you agree.

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 16th 2015
   • (edited Sep 16th 2015)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexLooking 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeSep 16th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, 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]]).

   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).

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 27th 2018
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAccording to Todd [here](https://mathoverflow.net/a/275588/447), 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?

   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?

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 27th 2018
   • (edited Sep 27th 2018)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThis 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.

   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.