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Some of you may remember quite a long time ago discussion on the nCat 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 X to the ∞-Lie algebroid of field configuraitons [Σ,X] by pull-push through X←Σ×[Σ,X]→[Σ,X].
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 (cdgAlg−k)op and consider the Isbell-duality adjunction
cdgAlgopk𝒪←→SpecSh∞((cdgAlg−k)op)=:H.We know how the ∞-Lie algebroids appearing in AKSZ theory are objects in H of the form X=Spec𝒪(X), so we can compute the internal ∞-topos hom [Spec𝒪(Σ),Spec𝒪X].
This is given by
[SpecA,SpecB]:SpecC↦([n]↦HomcdgAlgk(QB,A⊗C⊗Ω•poly(Δn))),where QB is a cofibrant replacement and where C∈cdgAlg−k.
Now we want to show that this is represented by the kind of construction that appears in Dmitry’s review of AKSZ…
Sorry, I have been editing the above two messages. I’ll better stop posting now and continue this tomorrow morning instead.
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