Proposition 3.9 (0N3W) · ScalingStacks

0N3W

Proposition 3.9. Given a map $\varphi:B\rightarrow B^{\prime}$ of spaces over $\BTop(n)$ and $M$ a $B$ -framed $n$ -manifold and $A$ a $B^{\prime}$ -framed $n$ -disk algebra, composition with the map $\varphi$ defines a $B^{\prime}$ -framed $n$ -manifold $\varphi M$ , and restriction along $\varphi$ defines a $B$ -framed $n$ -disk algebra $\varphi A$ . There is a natural equivalence

\int_{\varphi M}A\simeq\int_{M}\varphi A

between the $B$ -framed and $B^{\prime}$ -framed factorization homologies.

0N3X

Proof. It suffices to show that the forgetful functor $\disk_{n/M}^{B}\rightarrow\disk_{n/M}$ is an equivalence. By definition, this functor is the projection from the double overcategory:

\disk_{n/M}^{B}~:=~(\disk_{n/B})_{/M}\longrightarrow\disk_{n/M}.

This functor is a pullback of the likewise functor $\bigl((\spaces_{/\BTop(n)})_{/B}\bigr)_{/M}\to(\spaces_{/\BTop(n)})_{/M}$ , which is an equivalence by Lemma 2.5.

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