ScalingStacks

Proof. Via Example 3.10, each augmented associative algebra $A\to\uno$ in $\mathcal{V}$ determines a symmetric monoidal functor $A\colon\disk^{\partial,\sf or}_{1}\to\mathcal{V}$. Applying $\otimes$-excision in this simplest case of the collar-gluing $[-1,1)\underset{(-1,1)}{\bigcup}(-1,1]\cong[-1,1]$, we have that the bar construction ${\sf Bar}(A)$ is identifiable as the factorization homology over the closed 1-disk:

$${\sf Bar}(A)~\simeq~\int_{\mathbb{D}^{1}}A~.$$

Proposition 5.5 gives the first and last of the following identifications

	$\displaystyle\underset{\mathbb{D}^{1}\times\mathbb{R}^{n-1}}{\int}\free_{n}(V)$	$\displaystyle\simeq$	$\displaystyle\coprod_{i\geq 0}\conf_{i}(\mathbb{D}^{1}\times\mathbb{R}^{n-1},\partial\mathbb{D}^{1}\times\mathbb{R}^{n-1})\underset{\Sigma_{i}}{\otimes}V^{\otimes i}$
		$\displaystyle\simeq$	$\displaystyle\coprod_{i\geq 0}\Sigma^{i}\conf_{i}(\mathbb{R}^{n-1})\underset{\Sigma_{i}}{\otimes}V^{\otimes i}$
		$\displaystyle\simeq$	$\displaystyle\coprod_{i\geq 0}\conf_{i}(\mathbb{R}^{n-1})\underset{\Sigma_{i}}{\otimes}(\Sigma V)^{\otimes i}$
		$\displaystyle\simeq$	$\displaystyle\free_{n-1}(\Sigma V)~.$

The second identification follows from the $\Sigma_{i}$-equivariant equivalence of spaces

$$\conf_{i}(\mathbb{D}^{1}\times M,\partial\mathbb{D}^{1}\times M)\simeq\mathbb{D}^{i}\times\conf_{i}(M)\big/\partial\mathbb{D}^{i}\times\conf_{i}(M)\simeq\Sigma^{i}\conf_{i}(M)$$

in the case $M=\mathbb{R}^{n-1}$. The third equivalence is a coproduct of a composite of two equivalences: $\Sigma^{i}X\otimes V^{\otimes i}\simeq X\otimes\Sigma^{i}(V^{\otimes i})\simeq X\otimes(\Sigma V)^{\otimes i}$. The first of these equivalences uses that tensoring with spaces preserves colimits among spaces – an assertion which is direct from definitions. The second of these equivalences directly uses the assumption that the symmetric monoidal structure of $\mathcal{V}$ distributes over colimits.

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