Quasi-unitial Inner Kan Spaces

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 $\mathrm{\infty }$-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.