Definition 7.2.5.1. Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 7.2.1.22), and let $\operatorname{\mathcal{C}}$ be a category which is enriched over $\mathrm{h} \mathit{\operatorname{Kan}}$. We will say that $\operatorname{\mathcal{C}}$ is homotopy filtered if it is nonempty and satisfies the following condition for each $n \geq 1$:
- $(\ast _ n)$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ and for every morphism of simplicial sets $\sigma : \operatorname{\partial \Delta }^{n-1} \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$, there exists a morphism $v: Y \rightarrow Z$ for which the composite morphism
\[ \operatorname{\partial \Delta }^{n-1} \xrightarrow { \sigma } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y) \xrightarrow { v \circ } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \]is nullhomotopic.