ScalingStacks

Proposition 5.5. Let $M$ be an $n$-manifold, possibly with boundary. Let $V\in\mathcal{V}$ be an object of a symmetric monoidal $\oo$-category which is $\otimes$-presentable. There is an equivalence

$$\int_{M}\free_{n}(V)~\simeq~\coprod_{i\geq 0}\conf_{i}(M,\partial M)\underset{\Sigma_{i}}{\otimes}V^{\otimes i}$$

between the factorization homology of an $n$-manifold $M$, possibly with boundary, with coefficients in $\free_{n}(V)$ and the coproduct of the configuration spaces of $M$ labeled by $V$ quotient the subspace where at least one point lies in the boundary of $M$.