Quasi-unital $\infty$-Categories

TL;DR

This paper demonstrates that the unital structure of an $mbda$-category can be uniquely reconstructed from its non-unital form given appropriate units, advancing the understanding of $mbda$-categories and their applications.

Contribution

It establishes an analogous result to Lurie's quasi-unital algebras for $mbda$-categories, showing how to recover units from non-unital structures.

Findings

Unital structures can be uniquely recovered from non-unital structures in $mbda$-categories.

Provides a foundation for proving the 1-dimensional cobordism hypothesis.

Extends Lurie's theory to a new class of higher categories.

Abstract

Inspired by Lurie's theory of quasi-unital algebras we prove an analogous result for $\infty$-categories. In particular, 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 for the 1-dimensional cobordism hypothesis, as described in a forthcoming paper.