ScalingStacks

Definition 2.1. $\mfld_{n}$ is the symmetric monoidal topological category for which an object is a topological $n$-manifold that admits a finite good cover, which is to say a finite open cover by Euclidean spaces with the property that each non-empty intersection of terms in the cover is itself homeomorphic to a Euclidean space. The morphism spaces are spaces of embeddings, endowed with the compact-open topology. The symmetric monoidal structure is disjoint union.³³ 3 Thus, any $M\in\mfld_{n}$ has finitely many connected components, each of which is the interior of a compact manifold with (possibly empty) boundary. This size restriction is not an essential requirement; since all noncompact manifolds are built as sequential colimits of such smaller manifolds, this smallness condition could be removed and one could instead add to Definition 3.15 the requirement that a homology theory preserves sequential colimits.