B \to \cV$ be a symmetric monoidal functor.
The counit of this adjunction evaluates on $\cF$ as a morphism $\int A \rightarrow \cF$, where $A=\cF_{|\RR^n}$ is the $\disk^B_n$-algebra defined by the values of $\cF$ on disjoint unions of $B$-framed Euclidean $n$-spaces. 
It remains to verify that this counit is an equivalence.
To that end, consider the full $\infty$-subcategory $\mathcal M}\def\cN{\mathcal N}\def\cO{\mathcal O}\def\cP{\mathcal P\subset \mfld_n^B$ consisting of those objects $M$ for which this counit $\int_MA \xrightarrow{\simeq} \cF(M)$ is an equivalence.  
We wish to show the inclusion $\mathcal M}\def\cN{\mathcal N}\def\cO{\mathcal O}\def\cP{\mathcal P \hookrightarrow \mfld_n^B$ is an equivalence.  

 
Both $\int A$ and $\cF$ are symmetric monoidal functors.
So the full $\infty$-subcategory $\mathcal M}\def\cN{\mathcal N}\def\cO{\mathcal O}\def\cP{\mathcal P\subset \mfld_n^B$ is symmetric monoidal.
By definition, $\mathcal M}\def\cN{\mathcal N}\def\cO{\mathcal O}\def\cP{\mathcal P$ contains $B$-framed Euclidean spaces. 
We conclude the containment $\disk_n^B\subset \mathcal M}\def\cN{\mathcal N}\def\cO{\mathcal O}\def\cP{\mathcal P$.

Now, for each $0\leq k\leq n$, consider the base change $B_k$ of $B\to \BO(n)$ along the map $\BO(k)\times \BO(n-k) \xrightarrow{\oplus} \BO(n)$ then projected $B_k \to \BO(k)\times \BO(n-k)\xrightarrow{\sf proj}\BO(k)$.  
We will show, by induction on $k$, that $\mathcal M}\def\cN{\mathcal N}\def\cO{\mathcal O}\def\cP{\mathcal P$ contains the image of the evident functor $\mfld_k^{B_k}\to \mfld_n^B$.  
The previous paragraph gives that this is the case for $k=0$.  


By assumption, $\cF$ satisfies $\otimes$-excision, while Lemma~\ref{excision} states that $\int A$ satisfies $\otimes$-excision as well.  
We conclude that $\mathcal M}\def\cN{\mathcal N}\def\cO{\mathcal O}\def\cP{\mathcal P$ is closed under collar-gluings. 
Therefore $\mathcal M}\def\cN{\mathcal N}\def\cO{\mathcal O}\def\cP{\mathcal P$ contains the image of any $B_k$-framed $k$-manifold that can be witnessed by a finite iteration of collar-gluings from disjoint unions of $B_{k-1}$-framed $(k-1)$-manifolds.  
By induction on $k$, we are therefore reduced to showing each $B_k$-framed $k$-manifold can be witnessed as a finite iteration of collar-gluings involving $B_k$-framed $k$-manifolds in the image of $\mfld_{k-1}^{B_{k-1}} \to \mfld_k^{B_k} \to \mfld_n^B$.  

Well, the reigning assumption that manifolds be finitary guarantees, for each $B_k$-framed $k$-manifold $M$, the existence of a compactification $\overline{M}$ of the underlying manifold of $M$ as a smooth $k$-manifold with boundary.
Choose a proper Morse function $f\colon \overline{M} \to [-R,R]$ for which $f^{-1}(\{\pm R\}) = \partial \overline{M}$.  
Such a Morse function witnesses $M$ as a finite collar-gluing involving $B_k$-framed $k$-manifolds of whose underlying manifold is diffeomorphic to $M_0\times \RR$ for some $(k-1)$-manifold $M_0$.  
Such a $(k-1)$-manifold is therefore isomorphic to one that is equipped with a $B_{k-1}$-framing.




\end{proof}



\begin{remark} 
In~\cite{aft2} a version of Theorem~\ref{homology} is established for structured singular manifolds equipped with a stratified tangential structure -- 
this will be surveyed in~\S\ref{sec.singular}.  
The result, as it appears there, is slightly more general than Theorem~\ref{homology} above: the assumption that $\cV$ be $\otimes$-presentable can be weakened to the assumption that the underlying $\infty$-category of $\cV$ is presentable yet the symmetric monoidal structure distributes only over geometric realizations and filtered colimits (i.e., over sifted colimits).  

\end{remark}



\begin{remark}
In~\cite{oldfact} a version of Theorem~\ref{homology} is established for topological manifolds in place of smooth manifolds.  
In essence, the only feature of the smooth category used in the proof of Theorem~\ref{homology} was the existence of open handlebody decompositions, granted by Morse functions.
But such open handlebody decompositions are guaranteed in the topological setting, through the results of~\cite{moise} for $n=3$, of~\cite{quinn} as it concerns the complement of a point for $n=4$ together with smoothing theory~\cite{kirbysieb}, of~\cite{quinn} for $n=5$, and of~\cite{kirbysieb} for $n>5$.

\end{remark}

















\medskip 





For much of this article, we will primarily concern ourselves with homology theories in the sense of Definition~\ref{exc}. 
We point out, however that there are very interesting functors symmetric monoidal functors $\mfld_n\to \cV$ that are not $\otimes$-excisive. 
For instance, as detailed in~\cite{qcloops}, for $X$ a derived (commutative) stack over a fixed field $\Bbbk$, cotensoring to $X$ followed by taking ring of functions defines a symmetric monoidal functor:
\[
\mfld_n \xrightarrow{~M\mapsto X^M~} {\sf Stacks}(\Bbbk)\xrightarrow{~\cO~} \m_{\Bbbk}~
\]
As the case $n=1$, the value of this functor on the circle is the Hochschild homology of $X$: $\cO(X^{S^1})\simeq \hh_\bullet(X)$ (compare with Corollary~\ref{HH}).
If $X$ is not affine, the above functor will generically fail to be $\otimes$-excisive. 
Now, the cotensor $X^M$ only depends on the underlying homotopy type of $M$, as we shall see in Proposition \ref{tensor}.
This is a feature of $X$ being a derived \emph{commutative} stack.
Should $X$ be a derived stack over $\disk_n$-algebras as in~\cite{thez}, there is a refined version of the cotensor $X^M$ that is sensitive to the manifold structure of $M$.   
Derived stacks over $\disk_n$-algebras arise as the outcome of deformation quantization of a shifted symplectic derived (commutative) stacks, such as in Rozansky--Witten theory.  



































\section{Nonabelian Poincar\'e duality}\label{sec.nonabel}

Applying Theorem \ref{homology}, we offer a slightly different perspective, and proof, of the nonabelian Poincar\'e duality of Salvatore \cite{salvatore}, Segal~\cite{segallocal}, and Lurie \cite{dag}, which calculates factorization homology of iterated loop spaces as compactly-supported mapping spaces.




\begin{definition} For a space $B$, $\Space_B:=(\spaces_{/B})^{B/}$ is the $\infty$-category of {\it retractive} spaces over $B$, i.e., spaces over $B$ equipped with a section. The $\oo$-category $\Space^{\geq n}_B$ is the full $\oo$-subcategory of $\Space_B$ consisting of those $X\rightleftarrows B$ for which the retraction is $n$-connective, that is, the homomorphism $\pi_q X \rightarrow \pi_q B$ is an isomorphism for $q< n$ with any choice of base-point of $B$.\end{definition}


\begin{remark}
Equivalently, an object $X\in \Space_B^{\geq n}$ may be thought of as a fibration $X\rightarrow B$ equipped with a section for which, for each $b\in B$, the fiber $X_b$ is