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I have fine-tuned the definition of manifolds in differential cohesion a bit more (here).
I think now a good axiomatization is like this:
Let $V$ be a differentially cohesive homotopy type equipped with a framing. Then a $V$-manifold is an object $X$ such that there exists a $V$-cover, namely a correspondence
$\array{ && U \\ & \swarrow && \searrow \\ V && && X }$such that both morphisms are formally étale morphisms and such that $U \to X$ is in addition an effective epimorphism.
This style of definition very naturally leads to a good concept of integrable G-structures (in differential cohesion).
What I find particularly charming is that if we take such a correspondence and “prequantize” it in the sense of prequantized Lagrangian correspondences, i.e. if we pick a differential coefficient object $\mathbf{B}\mathbb{G}_{conn}$ and complete to a correspondence in the slice
$\array{ && U \\ & \swarrow && \searrow \\ V && \swArrow_{\simeq} && X \\ & {}_{\mathllap{\mathbf{L}_{WZW}}}\searrow && \swarrow_{\mathrlap{\mathbf{L}^X_{WZW}}} \\ && \mathbf{B}\mathbb{G}_{conn} }$then this captures precisely the globalization problem of WZW terms that we have been discussing elsewhere: on the left we pick a WZW term on the model space, and completing the diagram to the right means finding a globalization of this term to $X$ that locally restricts to the canonical term, up to equivalence.
I think I have now full proof of one direction of the corresponding obstruction (details in this pdf):
Theorem Given $V$ a differentially cohesive $\infty$-group, $X$ a $V$-manifold, and $\mathbf{L}_{WZW}$ an equivariant WZW-term on $V$, then an obstruction to $\mathbf{L}_{WZW}^X$ to exist as above is the existence of an integrable $QuantMorph(\mathbf{L}_{WZW}^{\mathbb{D}^V})$-structure on $X$,
(i.e. a lift of the structure group of the frame bundle to the quantomorphism n-group of the restriction $\mathbf{L}_{WZW}^{\mathbb{D}^V_e}$ of the WZW term to the infinitesimal neighbourhood of the neutral element in $V$, such that this lift restricts over a $V$-cover $U$ to the canonical $QuantMorph(\mathbf{L}_{WZW}^{\mathbb{D}^V_e})$-structure on $V$).
I still need to prove that this is not just a necessary but also a sufficient condition. This is harder…
With the definition of $V$-manifolds and their frame bundles in differential cohesion here and here, it is immediate that forming frame bundles extends to an $\infty$-functor $\tau_{(-)}$ from the sub-$\infty$-category of the $\infty$-topos $\mathbf{H}$ on the $V$-manifold with local diffeos between them, to the slice $\infty$-topos $\mathbf{H}_{/\mathbf{B}GL(V)}$.
But I need this functor internally: for $X$ a $V$-manifold I need a morphism in $\mathbf{H}$ of the form
$\mathbf{Aut}(X) \longrightarrow \underset{\mathbf{B}GL(V)}{\prod} \mathbf{Aut}(\tau_X)$where on the right $\mathbf{Aut}(\tau_X) \in \mathbf{H}_{/\mathbf{B}GL(V)}$ denotes the internal automorphism $\infty$-group of $\tau_X \in \mathbf{H}_{/\mathbf{B}GL(V)}$.
First I thought it’s obvious, but now I seem to be stuck…
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