Proposition 5.1 (0N53) · ScalingStacks

Proposition 5.1. The following diagram among $\infty$ -categories commutes:

where $U$ is the underlying space functor and the right downward arrow is the standard forgetful functor. In particular, there is a natural equivalence

\int_{M}A~\simeq~M\otimes A

between the factorization homology of $M$ with coefficients in $A$ and the tensor of the commutative algebra $A$ with the underlying space of $M$ .

0N54

Proof. The functor $-\otimes A\colon\mfld_{n}^{B}\to\mathcal{V}$ carries each contractible manifold to the underlying object of the commutative algebra $A$ . For $(A_{i})_{i\in I}$ a finite sequence of commutative algebras in $\mathcal{V}$ , the $I$ -fold coproduct in $\Alg_{\com}(\mathcal{V})$ is the pointwise tensor product $\underset{i\in I}{\bigotimes}A_{i}$ (see Proposition 3.2.4.7 of [Lu2]). It follows that this functor $-\otimes A$ is symmetric monoidal. From the defining expression of factorization homology as a colimit, there results a natural transformation

\int_{-}A\longrightarrow-\otimes A

between symmetric monoidal functors $\mfld_{n}^{B}\to\mathcal{V}$ , which evaluates as an equivalence on objects of $\disk_{n}^{B}$ . Lemma 3.18 grants that the domain of this natural transformation satisfies $\otimes$ -excision. Because a collar-gluing $M^{\prime}\underset{M_{0}\times\mathbb{R}}{\bigcup}M^{\prime\prime}\cong M$ determines a pushout of underlying spaces $M^{\prime}\underset{M_{0}}{\coprod}M^{\prime\prime}\simeq M$ , the codomain of this natural transformation too satisfies $\otimes$ -excision. That the natural transformation evaluates on each $B$ -framed $n$ -manifold $M$ as an equivalence then follows by induction on a handle decomposition on $M$ .

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