Homotopy coherent nerve

Construction 24.3.2 (Homotopy coherent nerve). Consider a topologically enriched category \(D\). We will define an \(\infty \)-category \(\Ntop (D)\), called the homotopy coherent nerve of \(D\).

For every \([n] \in \simp \catop \), consider the (classical) 1-category \(D_n\) defined as follows:

• The objects of \(D_n\) are the objects of \(D\);
• The morphisms \(X \to Y\) of \(D_n\) are the continuous maps \(\abs {\Delta ^n} \to \Homtop _D(X,Y)\);
• The identity in \(D_n\) is the constant map \(\abs {\Delta ^n} \to \Homtop _D(X,X)\) with value \(\id _X\);
• The composition in \(D_n\) of morphisms \(f\colon \abs {\Delta ^n} \to \Homtop _D(X,Y)\) and \(g\colon \abs {\Delta ^n} \to \Homtop _D(Y,Z)\) is the composite \[ \abs {\Delta ^n} \xrightarrow {\Delta } \abs {\Delta ^n} \times \abs {\Delta ^n} \xrightarrow {g \times f} \Homtop _D(Y,Z) \times \Homtop _D(X,Y) \xrightarrow {\circ } \Homtop _D(X,Z). \]

Since \(\abs {\Delta ^n}\) is functorial in \([n] \in \simp \) (see Example 1.6.8) we see that the assignment \([n] \mapsto D_n\) determines a functor \[ D_{\bullet }\colon \simp \catop \to \Cat _1 \hookrightarrow \Cat _{\infty }. \] We then define the \(\infty \)-category \(\Ntop (D)\) as the colimit of this functor: \[ \Ntop (D) \quad := \quad \colim (D_{\bullet }\colon \simp \catop \to \Cat _{\infty }). \]