Nerve of a category

Example 1.6.11 (Nerve of a 1-category). Let \(C\) be a (small, classical) 1-category. We define the nerve of \(C\) as the simplicial set \(N(C) \in \sSet \) given by \[ N(C)_n := \Hom _{\Cat ^{(1)}}([n],C), \] i.e. it is the composite \(\simp \catop \hookrightarrow (\Cat ^{(1)}_1)\catop \xrightarrow {\Hom _{\Cat ^{(1)}}(-,C)} \Set \). Here \(\Cat ^{(1)}_1\) denotes the 1-category of small 1-categories.¹ Again this construction is functorial in \(C\), resulting in a functor \[ N\colon \Cat ^{(1)}_1 \to \sSet . \]