ScalingStacks

Remark 3.17. One can complete the $\oo$-category $\mfld_{n}$ as follows: first, formally adjoin, for every collar-gluing $M^{\prime}\underset{M_{0}\times\mathbb{R}}{\bigcup}M^{\prime\prime}\cong M$, the colimit of the simplicial object ${\sf Bar}_{\bullet}(M^{\prime},M_{0}\times\mathbb{R},M^{\prime\prime})$; second, Dwyer-Kan localize by forcing the natural map from this new object $|{\sf Bar}_{\bullet}(M^{\prime},M_{0}\times\mathbb{R},M^{\prime\prime})|\rightarrow M$ to be an equivalence. Denote this completion of the $\oo$-category of manifolds as $\widehat{\mfld}_{n}$. The completion functor $\mfld_{n}\rightarrow\widehat{\mfld}_{n}$ is the universal homology theory: that is, we now have the suggestive equivalence

$$\int_{M}\mathbb{R}^{n}~\simeq~M$$

as objects of $\widehat{\mfld}_{n}$ (the lefthand side is not defined in $\mfld_{n}$). By this universal property, a $\otimes$-excisive functor $\mfld_{n}\rightarrow\mathcal{V}$ is equivalent to a symmetric monoidal functor $\widehat{\mfld}_{n}\rightarrow\mathcal{V}$ that preserves geometric realizations of simplicial objects.