Quasi-unitial Inner Kan Spaces

TL;DR

This paper demonstrates that semi-simplicial spaces with homotopy inner horn fillers and weak units serve as effective models for ∞-categories, comparing different quasi-unitality conditions relevant to Floer homotopy theory.

Contribution

It introduces a framework for modeling ∞-categories using semi-simplicial spaces with homotopy inner horn fillers and compares various quasi-unitality conditions.

Findings

Semi-simplicial spaces with these properties model ∞-categories effectively.

Comparison of three quasi-unitality conditions in the literature.

Applications to Floer homotopy theory.

Abstract

We show that semi-simplicial spaces that i) admit inner horn fillers up to homotopy and ii) possess units in a weak sense provide a viable model for $\infty$-categories. The existence of units can be expressed through various quasi-unitality conditions, and we compare the natural generalization of three such conditions found in the literature. This work is motivated by applications in Floer homotopy theory.