ScalingStacks

Definition 3.15. A symmetric monoidal functor $\mathcal{F}:\mfld_{n}^{B}\rightarrow\mathcal{V}$ satisfies $\otimes$-excision if, for each collar-gluing $M^{\prime}\underset{M_{0}\times\mathbb{R}}{\bigcup}M^{\prime\prime}\cong M$ among $B$-framed $n$-manifolds, the canonical morphism (6)

$$\mathcal{F}(M^{\prime})\underset{\mathcal{F}(M_{0}\times\mathbb{R})}{\bigotimes}\mathcal{F}(M^{\prime\prime})\xrightarrow{~\simeq~}\mathcal{F}(M)$$

is an equivalence in $\mathcal{V}$. The $\oo$-category of homology theories for $B$-framed $n$-manifolds valued in $\mathcal{V}$ is the full $\oo$-subcategory

$$\mathbf{H}(\mfld_{n}^{B},\mathcal{V})~\subset~\Fun^{\otimes}(\mfld^{B}_{n},\mathcal{V})$$

consisting of those symmetric monoidal functors that satisfy $\otimes$-excision.