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There is a higher extension of the classifying space $B \mathrm{Diff}$ of the diffeomorphism group implicit in the (infinity,n)-category of cobordisms with (X,zeta)-structure $\mathrm{Bord}_n^{(X,\zeta)}$ (as in arXiv:0905.0465). I would like to speak of that space in more direct terms than via the full cobordism hypothesis. I believe I know how to do it, but since tracing back to the "definition" of $\mathrm{Bord}_n^{(X,\zeta)}$ has its subtleties, this here is to ask for a sanity check.

So for $\Sigma$ a closed manifold (cobordism between empty manifolds) equipped with $(X,\zeta)$-structure $\sigma$, there is the space

$$ B \left(\mathrm{Diff}_{(X,\zeta)}(\Sigma,\sigma)\right) := B \left(\Omega^{\mathrm{dim}(\Sigma)}_{(\Sigma,\sigma)} \mathrm{Bord}_n^{(X,\zeta)}\right) $$

delooping the $\infty$-group of automorphism of $(\Sigma,\sigma)$ regarded as a $k$-morphism in $\mathrm{Bord}_n^{(X,\zeta)}$. (Notice that there is no geometric realization on the right here, I hope the text makes clear what I mean).

This space is important in its own right, even aside from the full generality of the cobordism hypothesis, and what I am after here is a direct definition of this space, not going via the cobordism hypothesis.

In a previous question I was checking whether such a direct description might exists at a "purely combinatorial" level, without invoking geometry and stacks. But as the answer there shows, at least the obvious guess as to what that would be fails, and so I will have to use the word "stack" in the following. Sorry for that.

So I am thinking it goes like this (my question is for sanity check of what I am saying now):

Let $\mathrm{Diff}_{(X,\zeta)}(\Sigma,\sigma)$ be the smooth group $\infty$-stack which sits in the homotopy fiber product

$$ \array{ \mathrm{Diff}_{(X,\zeta)}(\Sigma,\sigma) &\longrightarrow& \mathbf{Aut}_{/\rho}(\sigma) \\ \downarrow && \downarrow \\ \mathrm{Diff}(\Sigma) &\longrightarrow& \mathbf{Aut}_{/BO(n)}(\iota\tau_\Sigma) } \,, $$

where

  • $\iota \tau_\Sigma : \Sigma \longrightarrow BO(n)$ is the classifying map of the $n$-stabilized tangent bundle;

  • $\rho: X \longrightarrow BO(n)$ is the classifying map of $\zeta$:

  • $\mathbf{Aut}_{/BO(n)}(\iota\tau_\Sigma)$ is the automorphism $\infty$-group of $\iota \tau_\Sigma$ in the slice over $BO(n)$;

  • $\mathbf{Aut}_{/\rho}(\sigma)$ is the automorphism $\infty$-group of $\sigma$ in the slice (of the slice over $B O(n)$) over $\rho$.

Then $B\left(\mathrm{Diff}_{(X,\zeta)}(\Sigma,\sigma)\right)$ is the classifying space of that smooth $\infty$-group.

This is really straightforward but may look a bit involved. I have spelled this out in more detail with some more illustration around def. 3.2.10 in my note Local prequantum field theory. Following that definition in that note are propositions and proofs (or what I presently believe are such) of the basic properties of this $\mathrm{Diff}_{(X,\zeta)}(\Sigma,\sigma)$, characterizing it as an extension of $\mathrm{Diff}(\Sigma)$, characterizing the induced extension of the mapping class group etc. These statements come out the way they should, it seems. In particular they reproduce Segal's integral extensions of the MCG via $p_1$-structures and the corresponding topological modular functor as a special case. That and its generalizations is what I am really after.

However, I am a bit vague on how to produce precise proof (if such exists) of the statement that the $B\left(\mathrm{Diff}_{(X,\zeta)}(\Sigma,\sigma)\right)$ as defined above by homotopy pullback is indeed an incarnation of $B \left(\Omega^{\mathrm{dim}(\Sigma)}_{(\Sigma,\sigma)} \mathrm{Bord}_n^{(X,\zeta)}\right)$. Is it?

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