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 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

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 15th 2010
   • (edited Dec 15th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSome of you may remember quite a long time ago discussion on the $n$Cat cafe on whether and how we have an internal hom on dg-algebras, such that the pull-push construction in [[AKSZ theory]] becomes literally the transgression of the symplectic form on the target $\infty$-Lie algebroid $X$ to the $\infty$-Lie algebroid of field configuraitons $[\Sigma,X]$ by pull-push through $X \leftarrow \Sigma \times [\Sigma,X] \to [\Sigma,X]$. Back then I got stuck. I was lacking the right $\infty$-topos context that makes all this obvious. (Todd back then kindly provided an internal hom construction on differential coalgebras, but I never got anywhere with that, for lack of trying hard enough). Now it is clear how things work in [[dg-geometry]]: we go to the $\infty$-topos over $(cdgAlg_k^-)^{op}$ and consider the Isbell-duality adjunction $$ cdgAlg_k^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} Sh_\infty((cdgAlg_k^-)^{op}) =: \mathbf{H} \,. $$ We know how the $\infty$-Lie algebroids appearing in AKSZ theory are objects in $\mathbf{H}$ of the form $X = Spec \mathcal{O}(X)$, so we can compute the internal $\infty$-topos hom $[Spec \mathcal{O}(\Sigma), Spec \mathcal{O} X]$.

   Some of you may remember quite a long time ago discussion on the Cat cafe on whether and how we have an internal hom on dg-algebras, such that the pull-push construction in AKSZ theory becomes literally the transgression of the symplectic form on the target -Lie algebroid to the -Lie algebroid of field configuraitons by pull-push through .

   Back then I got stuck. I was lacking the right -topos context that makes all this obvious. (Todd back then kindly provided an internal hom construction on differential coalgebras, but I never got anywhere with that, for lack of trying hard enough). Now it is clear how things work in dg-geometry:

   we go to the -topos over and consider the Isbell-duality adjunction

   We know how the -Lie algebroids appearing in AKSZ theory are objects in of the form , so we can compute the internal -topos hom .

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 15th 2010
   • (edited Dec 15th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis is given by $$ [Spec A, Spec B] : Spec C \mapsto ([n] \mapsto Hom_{cdgAlg_k}(Q B, A \otimes C \otimes \Omega^\bullet_{poly}(\Delta^n)) ) \,, $$ where $Q B$ is a cofibrant replacement and where $C \in cdgAlg_k^-$. Now we want to show that this is represented by the kind of construction that appears in Dmitry's review of AKSZ...

   This is given by

   where is a cofibrant replacement and where .

   Now we want to show that this is represented by the kind of construction that appears in Dmitry’s review of AKSZ…

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 15th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexSorry, I have been editing the above two messages. I'll better stop posting now and continue this tomorrow morning instead.

   Sorry, I have been editing the above two messages. I’ll better stop posting now and continue this tomorrow morning instead.