Not signed in

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

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

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 definitions 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 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 nforum 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

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 4th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to [[connected topos]] in the Examples-section the statement that the sheaf topos on [[CartSp]] (that which contains the quasi-topos of [[diffeological space]]s) is connected.

   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.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 4th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI guess the proof that Sh(CartSp) is locally connected is easy?

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

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 4th 2010
   • (edited Jun 4th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes: 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 $L$ for sheafification, then: $$ Hom_{Sh(C)}(X , L Const S) \simeq Hom_{PSh(C)}(X , L Const S) \simeq Hom_{PSh(C)}(X, Const S) \simeq Hom_{Set}(\lim_\to X, S) $$ So $\Pi_0 := \lim_\to$ is the required left adjoint, whenever constant preshaves are sheaves. And this is clearly the case on $CartSp$, because all Cartesian spaces are connected (or alternatively: because all discrete topological spaces are uniquely diffeological spaces).

   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 $L$ for sheafification, then:

   $$Hom_{Sh(C)}(X , L Const S) \simeq Hom_{PSh(C)}(X , L Const S) \simeq Hom_{PSh(C)}(X, Const S) \simeq Hom_{Set}(\lim_\to X, S)$$

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 4th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexAnd to see that the the left adjoint $\Pi_0 = \lim_\to: Sh(CartSp) \to Set$ to $LConst : Set \to Sh(CartSp)$ indeed sends every diffeological space to its set of connected components: write the colim $\lim_\to X$ as the colimit of the diagram constant on the point of shape the comma catgegory $y/X$ with $y$ the Yoneda embedding. So the category of elements of $X$. This category has exactly one connected component per "plot-connected component" of $X$, since $$ y/X = \left\{ \array{ y(U) &&\to&& y(V) \\ & \searrow && \swarrow \\ && X } \right\} \,. $$ So $\Pi_0 X = \lim_\to X$ is the set of plot-connetced components of the diffeological space $X$.

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

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

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

   So $\Pi_0 X = \lim_\to X$ is the set of plot-connetced components of the diffeological space $X$.

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 4th 2010
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexCool, thanks.

   Cool, thanks.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 4th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> 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 <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos+--+survey#LocallyConnected">here</a>.

   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.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJun 4th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexOkay, I made this a section at [[diffeological space]]: <a href="http://ncatlab.org/nlab/show/diffeological+space#Connectedness">Connectedness</a>

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