Theorem 3.26 (0N4M) · ScalingStacks

0N4M

Theorem 3.26. For $\mathcal{V}$ a symmetric monoidal $\oo$ -category which is $\otimes$ -presentable, and for $B\to\BTop(n)$ a map of spaces, there is an equivalence between $\infty$ -categories

from $\disk^{\partial,B}_{n}$ -algebras in $\mathcal{V}$ and homology theories of $B$ -framed $n$ -manifolds with coefficients in $\mathcal{V}$ . This equivalence is implemented by the factorization homology functor $\int$ and evaluation on $B$ -framed Euclidean $n$ -spaces and half-spaces.

0N4N

Proof. After Proposition 3.9 it is enough to consider the case where $B$ is equivalent to $\BTop(n)$ , and so we omit it from the notation and discussion. Let $\mathcal{F}:\mfld_{n}^{\partial}\rightarrow\mathcal{V}$ be a symmetric monoidal functor satisfying the $\otimes$ -excision condition, and let $A$ be the restriction of $\mathcal{F}$ to $\disk_{n}^{\partial}$ . For a manifold with boundary $\overline{M}$ , we prove that the canonical morphism $\int_{\overline{M}}A\rightarrow\mathcal{F}(\overline{M})$ is an equivalence. By $\otimes$ -excision applied to the collar-gluing $\partial\overline{M}\times[0,1)\underset{\partial\overline{M}\times\mathbb{R}}{\bigcup}M\cong\overline{M}$ , where $M$ is the interior of $\overline{M}$ , we obtain a diagram in $\mathcal{V}$

in which the vertical morphisms are equivalences. It now suffices to show that the top horizontal morphism is an equivalence. The equivalences of $\int_{M}A\rightarrow\mathcal{F}(M)$ and $\int_{\partial\overline{M}\times\mathbb{R}}A\rightarrow\mathcal{F}(\partial\overline{M}\times\mathbb{R})$ are given by Theorem 3.24; the last equivalence $\int_{\partial\overline{M}\times[0,1)}A\rightarrow\mathcal{F}({\partial\overline{M}\times[0,1)})$ by follows by Theorem 3.24 and Proposition 2.16. ∎