Proof of Cobordism Hypothesis

5 Unfolding of Higher Categories

The next step is to reinterpret the data of the categories \(\Bord _n\) in terms of \((\infty ,1)\)-categories. All categories and functors here are symmetric monoidal unless otherwise specified. We have currently reduced the cobordism hypothesis to a statement about the sequence \(\Bord _1 \to \Bord _2 \to \Bord _3 \to \dots \). To see that the data might be describable in terms of \((\infty ,1)\)-categories, we can observe that the inclusion of \(\Bord _n\) into \(\Bord _{n+1}\) is that it is \(n\)-connected, which is a notion analogous to the corresponding notion for spaces. In this case it amounts to the observation that the \(k\)-morphisms for \(k\leq n\) are the same in each category. Thus there aren’t as many levels of noninvertible morphisms where they differ, so this data should be able to be captured using a lower level of category.

The sequence \(\Bord _n\) is an example of what is called a skeletal sequence of categories, which is a way of formalizing the fact that certain adjoints exist and that each inclusion is sufficiently connected. We can replace \(\Bord _n\) with a sequence of symmetric monoidal \((\infty ,1)\) categories \(\Cob _{\partial }^{un}(n)\) that form a categorical chain complex: In otherwords there are symmetric monoidal coCartesian fibrations \(d:\Cob _{\partial }^{un}(n) \to \Cob _{\partial }^{un}(n-1)\) and isomorphisms of \(d^2\) with the constant map to the unit in a coherent way. \(\Cob _{\partial }^{un}(n)\) is the category of manifolds with boundary of dimension \(n-1\) with boundary with maps given by the space of cobordisms (not necessarily trivial on the boundary). The map between them is given by taking the boundary.

Let’s see how this replacement works for \(n=1\). \(\Bord _1=\Cob _{\partial }^{un}(1)\). The map \(\Bord _1 \to \Bord _2\) is essentially surjective, and for any \(X,Y\) in \(\Bord _2\), the category \(\Map _{\Bord _2}(X,Y)\) is equivalent to \(\Map _{\Bord _2}(1,Y\otimes X^{\vee })\). This means that to recover the map to \(\Bord _2\) we just need to remember \(Y \mapsto \Map _{\Bord _2}(1,Y)\) as a lax monoidal functor \(\Bord _1 \to \Cat _{(\infty ,1)}\). But by a version of the Grothendieck construction, this is equivalent to a coCartesion fibration to \(\Bord _1\), and one can show this gives exactly something equivalent to \(\Cob _{\partial }^{un}(2)\). This example can be spiced up to give a general correspondence between skeletal sequences and categorical chain complexes (for which the definitions have only been sketched here).

A similar construction for \((\infty ,n)\)-categories gives the following:

Proposition 5.1. The following data are equivalent for a symmetric monoidal \((\infty ,n-1)\) category with duals:
- • \(n-2\)-connected symmetric monoidal functors \(B_{n-1} \to B_n\), \(B_n\) a symmetric monoidal \((\infty ,n)\) category
- • Lax symmetric monoidal functors \(\Omega ^{n-2}B_{n-1} \to Cat_{\infty ,1}\)
- • Symmetric monoidal coCartesian fibrations \(C \to \Omega ^{n-2} B_{n-1}\).