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 comma 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 finite 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 k-theory lie-theory 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 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
    • CommentTimeMar 16th 2012

    created internal sheaf

    Mainly it was bugging me that I didn’t find a piece of literature that said it quite explicitly the way I do there, so I wanted to have that written down. To be expanded, eventually.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2012
    • (edited Mar 16th 2012)

    I have further expanded it (added two basic Propositions, more details in the definitions) and tried to prettify a bit more.

  1. I added the pedestrian definitions of presheaf and sheaf (as internal diagrams).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2012
    • (edited Aug 29th 2012)

    Thanks!

    I moved your new paragraph to become a subsection of the Definition-section, made my previous material there also a subsection, and added at the beginning of the Definition-section a little lead-in on how we will present two versions of the definition, one abstract, one more explicit.

    Of course much of the explicitness of the second definition (the one you added) is out-sourced to the entry internal diagram- But I guess that’s okay.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2012

    In this spirit I have also edited the very very last sentence of the entry (at then end of the References) making it now point to the two Definition-sections where previously it just vaguely referred to the explicit defintition. Good.

  2. Thanks, much better! I agree it’s not a problem that the actual explicitness is one further click away.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeSep 7th 2012

    Do you see a direct way to expand the concept to an internal generalization of non-set valued sheaves ? In my understanding this entry generalizes the set-valued sheaves only.

  3. You can certainly have presheaves valued in locally internal categories over the base topos, see the last section of internal diagram (where this is detailed in the language of indexed categories). Using the internal language, these look exactly like ordinary non-Set-valued presheaves; so I’d guess that to obtain a sensible notion of a sheaf, one could simply formulate the usual sheaf condition in the internal language.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2012

    The formulation In terms of external 2-sheaves works by passing to the 2-category of internal categories. In there you can do all of category theory and topos theory as you did externally.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2023
    • (edited Nov 8th 2023)