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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 $n$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 $[Spec \mathcal{O}(\Sigma), Spec \mathcal{O} X]$.

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

   $$[Spec A, Spec B] : Spec C \mapsto ([n] \mapsto Hom_{cdgAlg_k}(Q B, A \otimes C \otimes \Omega^\bullet_{poly}(\Delta^n)) ) \,,$$

   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…

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.