In this talk, we will discuss the notion of a complete Segal space – a model of an infinity category, and then study the infinity category of $n$-bordisms.

Higher category theory is the generalization of category theory to a context where there are not only morphisms between objects, but generally $k$-morphisms between $(k−1)$-morphisms, for all $k \in \N$. The theory of higher categories or $(\infty, 1)$-categories, as it is sometimes called, however, can be very intractable at times. That is why there are now several models which allow us to understand what a higher category should be. Among these models is the theory of quasi-categories, introduced by Bordman and Vogt, and much studied by Joyal and Lurie. There are also other very prominent models such as simplicial categories (Dwyer and Kan), relative categories (Dwyer and Kan), and Segal categories (Hirschowitz and Simpson). One of those models, complete Segal spaces, was introduced by Charles Rezk in his seminal paper “A model for the homotopy theory of homotopy theory”. Later they were shown to be a model for $(\infty, 1)$-categories.

One major application of higher category theory and one of the driving forces in developing it has been extended topological quantum field theory. This has recently led to what may become one of the central theorems of higher category theory, the proof of the cobordism hypothesis, conjectured by Baez and Dolan. Lurie suggested passing to $(\infty, n)$-categories for a proof of the Cobordism Hypothesis in arbitrary dimension $n$. However, finding an explicit model for such a higher category poses one of the difficulties in rigorously defining these $n$-dimensional TFTs, which are called “fully extended”. Our focus will be on the $(\infty, 1)$-category $\mathrm{Bord}_n^{(n -1)}$, a variant of the fully extended $\mathrm{Bord}_{n}$. Our goal is to sketch a detailed construction of the $(\infty, 1)$-category of $n$-bordisms as a complete Segal space.