Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab nonassociative noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)