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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 17th 2013

    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?).

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 16th 2015

    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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2015

    Thanks David. I have added to the section statements that are at least sufficient to conclude that EM( W)H /*EM(\Box_W) \simeq \mathbf{H}_{/\ast}, namely that H\mathbf{H} is a topos and WW is inhabited.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2015

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

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 16th 2015
    • (edited Sep 16th 2015)

    Looking good.

    So the PDE case should go through in a similar way? Perhaps in explicit terms of SDG infinitesimals. Instead of the

    W(Q)=Γ W(Q) \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 (X)\Im(X)? That seems to be multiplying points by infinitesimal neighborhoods.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2015

    Yes, dependent product along a general morphism XYX \longrightarrow Y produces “spaces-of-sections-pointwise-over-YY”. And, yes, since the de Rham stack projectiuon ΣΣ\Sigma \to \Im \Sigma is a 1-epi, jet coalgebras over Σ\Sigma are equivalently objects in the slice over Σ\Im \Sigma. (In algebraic geometry these are the D-modules).

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 27th 2018

    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?

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 27th 2018
    • (edited Sep 27th 2018)

    This is quite a nice way to picture things: A possibility algebra/necessity coalgebra, AH/WA \in \mathbf{H}/W, requires a map, WA WA\sum_W A \to \prod_W A. Given a point in the total space, we need a section through that point.

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