Not signed in (Sign In)

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 book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex 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 foundation 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 homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory 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 stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft 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.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2021
    • (edited Oct 8th 2021)

    Am giving this its own page, for a coherent discussion, and to supercede the scattered remarks in various entries.

    Meaning to amplify that the cohesion is a direct consequence of the fact that the GG-orbit category is reflective in the G\prec\!\! G-slice of the (2,1)(2,1)-category of delooping groupoids.

    Not done yet, though, but need to save.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 8th 2021

    Does anything interesting emerge from the differential cohomology hexagon in these cases?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2021

    I am not sure what to make of the differential cohomology hexagon for the “singular” version of cohesion. This is related to the fact that here the conceptual meaning of the modalities is rather different from that in “smooth” cohesion. For instance the singular-cohesive analog of the shape modalty is (speaking in the slice over 1\prec \!\! 1) the operation that sends an orbispace to its naive quotient space (while the analog of the sharp modality sends a smooth homotopy quotient to its orbi-singularization). This means that in the corresponding hexagon, the vertices now have entirely different “meaning” from what they used to have relative to smooth cohesion. Somebody needs to figure out what this meaning of the singular-cohesive hexagon (which certainly exists) is.

    On the other hand, for H\mathbf{H} a cohesive \infty-topos, its globally equivariant version GloHGlo \mathbf{H} is still “smooth cohesive” over GloGrpd Glo Grpd_\infty, and the hexagon with respect to that dimension of cohesion expresses “naive” equivariant differential cohomology (this is not so naïve really, it’s Bredon-type equivariance after all and e.g. equivariant K-theory is in here, but, for better or worse, “naïve” has become a technical term here, much like “perverse” or “sober” elsewhere).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2021

    Now the entry has some readable shape.

    Still lacking at the end a comment on siftedness to prove that the far left adjoint preserves finite products. But need to get some lunch first now.

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2021
    • (edited Oct 10th 2021)

    To complete the proof, I have now added (here) pointer to the new section “Cohesion” (here) at adjoint quadruple.

    diff, v5, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 11th 2021

    Made the main proof (here) more explicit by including pointer to the new explicit proof at slice of presheaves is presheaves on slice.

    diff, v6, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 13th 2021
    • (edited Oct 13th 2021)

    added a Lemma (here) making explicit that the left adjoint morphism of (2,1)(2,1)-sites preserves finite products after extension to free coproduct completions.

    This is under the assumption of discrete equivariance groups, using that here the GG-orbit category is just the connected objects inside all GG-sets, so that the extension of the left adjoint to free coproduct completions is just the 0-truncation functor τ 0\tau_0 in the slice Grpd /BGGrpd_{/B G}, and hence preserves finite products since all nn-truncation operations on higher toposes do.

    With this it follows “formally” that the leftmost adjoint in the adjoint quadruple preserves finite products. Charles Rezk in his note works harder (in section 7.3, around p. 23) to deduce this for GG a compact Lie group.

    diff, v9, current