The article at hand provides a new tool for studying the classifying spaces of categories, in particular those that behave like cobordism categories. While the classifying space of the cobordism category of smooth manifolds \(\mathrm{Cob}_d\) was computed by [S. Galatius et al., Acta Math. 202, No. 2, 195–239 (2009; Zbl 1221.57039)] using a parametrized Pontrjagin-Thom construction, many other cobordism-like categories cannot be studied in such a local manner. The author lists several examples of interest: the h-cobordism category, the category of cobordisms equipped with positive scalar curvature, the cobordism category associated to a Waldhausen category (similar to Quillen’s \(Q(\mathcal{C})\)), and the cobordism categories of Poincaré objects that arise in recent work on Hermitian K-theory [B. Calmès et al., “Hermitian K-theory for stable \(\infty\)-categories. I: Foundations”, Preprint, arXiv:2009.07223].
The main theorem of this article is the eponymous additivity theorem for cobordism categories, which allows to construct homotopy pullback squares of classifying spaces, even when the spaces themselves cannot be computed directly. In its simplest form, when applied to ordinary categories, it says: Consider a functor \(P:\mathcal{C} \to \mathcal{D}\) that is both a cartesian and a cocartesian fibration (in the sense of Grothendieck). Then for any functor \(F: \mathcal{D}' \to \mathcal{D}\), the classifying space of the pullback of categories \(\mathcal{D}' \times_{\mathcal{D}} \mathcal{C}\) is homotopy equivalent to the homotopy pullback of classifying spaces \(B(\mathcal{D}') \times_{B(\mathcal{D})}^h B(\mathcal{C})\). To suit a wide range of applications this is proven in three flavours: for quasicategories, for weakly unital semi-Segal spaces, and for weakly unital topological categories.
As an application the author reproves the Genauer’s fiber-sequence for cobordism categories with horizontal boundaries and the Bökstedt-Madsen delooping of the cobordism category.