Towards Algebraic n-Categories of Manifolds and Cobordisms

Advancements in the study of extended topological quantum field theories have recently relied upon constructing suitable symmetric monoidal $n$-categories whose objects are smooth $m$-dimensional manifolds and whose $k$-morphisms are $(m+k)$-dimensional cobordisms, with composition given by gluing along boundaries and monoidal structure by disjoint union. A number of such constructions currently prevail in the literature, in particular those defined as symmetric monoidal bicategories and those obtained as symmetric monoidal $(\infty, n)$-categories, usually using the model of complete $n$-fold Segal spaces. The former have proven suitable for explicit computations, often admitting finite presentations that quickly yield new topological quantum field theories. The latter instead revel in their generality; with them, it is possible to consider full extension to all $n > 0$ and to consider all degrees of extension, using $m$-dimensional manifolds for any $m \geq 0$ as the objects.
There is an unfortunate divide between these two approaches to TQFTs - until the work in this thesis was developed, there was no complete direct investigation into obtaining fully weak homotopy bicategories from $2$-fold Segal spaces, which has left researchers without the technology needed to translate results about $(\infty, 2)$-categorical TQFTs into their bicategorical counterparts. Moreover, there has been no means to develop TQFTs using algebraic models of $n$-category for general $n$, such as those of Trimble or Batanin and Leinster. Such models may be more amenable to presentations by generators and relations, in a similar manner to bicategories.
In this thesis, we take steps towards developing homotopy $n$-categories of $n$-fold Segal spaces with the goal of application to higher categories of manifolds and cobordisms. In particular, we functorially obtain the unbiased homotopy bicategory of a Reedy fibrant $2$-fold Segal space, constructed by choosing sections of the Segal maps and homotopies between sections. We compare this with a simpler approach obtained by formally inverting the homotopy $1$-functors of the Segal maps; our belief is the former method will more easily extend to homotopy $n$-categories for $n > 2$. We then concretely establish a general Reedy fibrant replacement functor, which we specialize to projective fibrant $2$-fold Segal spaces. We obtain and discuss the resulting homotopy bicategory of a projective fibrant $2$-fold Segal space by this method. We then apply our constructions to obtain fundamental bigroupoids of topological spaces, which serves to illustrate our notion of homotopy bicategory in action. Finally, we consider some resulting characterizations of completeness and equivalences between complete $2$-fold Segal spaces.