ScalingStacks

Remark 3.14. We find it useful to think of a collar-gluing as the data of a manifold $M$ together with a codimension-1 properly embedded submanifold $M_{0}\subset M$ that splits the manifold $M$ into two disconnected parts, $M^{\prime}$ and $M^{\prime\prime}$. Such data is afforded by gluing two manifolds with boundary along a common boundary. The actual data of a collar-gluing specifies that named just above, in addition to a bi-collaring $M_{0}\times\mathbb{R}\hookrightarrow M$ of $M_{0}\subset M$.