Quasi-unital $\infty$–categories

Abstract

Inspired by Lurie’s theory of quasi-unital algebras we prove an analogous result for $\infty$–categories. By constructing a suitable model category of non-unital complete Segal spaces, we show that the unital structure of an $\infty$–category can be uniquely recovered from the underlying non-unital structure once suitable candidates for units have been identified. The main result of this paper can be used to produce a proof of the $1$–dimensional cobordism hypothesis, as described in a forthcoming paper of the author.