Definition 3.5.6.1. Let $X$ be a Kan complex and let $n \geq 0$ be an integer. We say that a morphism of Kan complexes $f: X \rightarrow Y$ exhibits $Y$ as a fundamental $n$-groupoid of $X$ if the following conditions are satisfied:
- $(a)$
The simplicial set $Y$ is an $n$-groupoid (Definition 3.5.5.1).
- $(b)$
The morphism $f$ is bijective on $m$-simplices for $m < n$.
- $(c)$
The morphism $f$ is surjective on $n$-simplices.
- $(d)$
If $\sigma $ and $\sigma '$ are $n$-simplices of $X$ satisfying $f(\sigma ) = f(\sigma ')$, then $\sigma $ and $\sigma '$ are homotopic relative to $\operatorname{\partial \Delta }^{n}$ (see Definition 3.2.1.3).