ScalingStacks

Definition 2.6 (T.2.3.4.1). Let $\mathcal{C}$ be a simplicial set and let $d\geq-1$ be an integer. We will say that $\mathcal{C}$ is a $d$-category if it is an $\infty$-category and the following additional conditions are satisfied:

1. (1)

   Given a pair of maps $f,f^{\prime}\colon\Delta^{d}\to\mathcal{C}$, if $f$ and $f^{\prime}$ are homotopic relative to $\partial\Delta^{d}$, then $f=f^{\prime}$.

2. (2)

   Given $m>d$ and a pair of maps $f,f^{\prime}\colon\Delta^{m}\to\mathcal{C}$, if $f\mid\partial\Delta^{m}=f^{\prime}\mid\partial\Delta^{m}$, then $f=f^{\prime}$.