ScalingStacks

Proof. Regard the associative algebra $A$ as a symmetric monoidal functor $A\colon\disk^{\sf or}_{1}\to\mathcal{V}$, as in Section 3.2. Consider the standard collar-gluing $\mathbb{R}\underset{\mathbb{R}\sqcup\mathbb{R}}{\bigcup}\mathbb{R}\cong S^{1}$ by hemispheres. Lemma 3.18, which states that factorization homology staisfies $\otimes$-excision, determines the first of the equivalences in the expression:

$$\int_{S^{1}}A~\simeq~\int_{\mathbb{R}}A\underset{{\displaystyle\int_{{S^{0}\times\mathbb{R}}}\!A}}{\bigotimes}\int_{\mathbb{R}}A~\simeq~A\underset{A\otimes A^{\op}}{\bigotimes}A~\simeq~\hh_{*}(A)~.$$

The second equivalence is by inspecting values, and the final equivalence is definitional.

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