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
    • CommentTimeJun 4th 2010

    added to connected topos in the Examples-section the statement that the sheaf topos on CartSp (that which contains the quasi-topos of diffeological spaces) is connected.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 4th 2010

    I guess the proof that Sh(CartSp) is locally connected is easy?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2010
    • (edited Jun 4th 2010)

    Yes: for every sheaf topos with the property that constant presheaves are already sheaves, the left adjoint of the constant presheaf functor is also the left adjoint of the constant sheaf functor

    Write LL for sheafification, then:

    Hom Sh(C)(X,LConstS)Hom PSh(C)(X,LConstS)Hom PSh(C)(X,ConstS)Hom Set(lim X,S) Hom_{Sh(C)}(X , L Const S) \simeq Hom_{PSh(C)}(X , L Const S) \simeq Hom_{PSh(C)}(X, Const S) \simeq Hom_{Set}(\lim_\to X, S)

    So Π 0:=lim \Pi_0 := \lim_\to is the required left adjoint, whenever constant preshaves are sheaves. And this is clearly the case on CartSpCartSp, because all Cartesian spaces are connected (or alternatively: because all discrete topological spaces are uniquely diffeological spaces).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2010

    And to see that the the left adjoint Π 0=lim :Sh(CartSp)Set\Pi_0 = \lim_\to: Sh(CartSp) \to Set to LConst:SetSh(CartSp)LConst : Set \to Sh(CartSp) indeed sends every diffeological space to its set of connected components:

    write the colim lim X\lim_\to X as the colimit of the diagram constant on the point of shape the comma catgegory y/Xy/X with yy the Yoneda embedding. So the category of elements of XX. This category has exactly one connected component per “plot-connected component” of XX, since

    y/X={y(U) y(V) X}. y/X = \left\{ \array{ y(U) &&\to&& y(V) \\ & \searrow && \swarrow \\ && X } \right\} \,.

    So Π 0X=lim X\Pi_0 X = \lim_\to X is the set of plot-connetced components of the diffeological space XX.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 4th 2010

    Cool, thanks.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2010

    Cool,

    I like it, too. Of course it may not look like too deep a statement, but it is the tip of an iceberg: indeed I ran into the analog first for the (oo,1)-topos on CartSp.

    I am currently typing out more expositional notes on this here.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2010

    Okay, I made this a section at diffeological space: Connectedness