ScalingStacks

Construction 2.21 offers, for each collar-gluing $M\xrightarrow{f}[-1,1]$ among $B$-framed $n$-manifolds, a monoidal functor

$$f^{-1}\colon\disk^{\partial,\sf or}_{1/[-1,1]}\longrightarrow\mfld^{B}_{n/M}~.$$

In particular, for each symmetric monoidal functor $\mfld^{B}_{n}\xrightarrow{\mathcal{F}}\mathcal{V}$ with $\otimes$-presentable codomain, there is a canonical morphism in $\mathcal{V}$:

(6)

$$\mathcal{F}(M^{\prime})\underset{\mathcal{F}(M_{0}\times\mathbb{R})}{\bigotimes}\mathcal{F}(M^{\prime\prime})~\underset{\rm Cor~\ref{interval}}{\simeq}~\int_{[-1,1]}\mathcal{F}\circ f^{-1}~\longrightarrow~\mathcal{F}(M)~.$$