Kerodon

Remark 3.1.7.4. In the situation of Corollary 3.1.7.2, the Kan complex $Q_{}$ (and the anodyne morphism $f$) can be chosen to depend functorially on $X_{}$. This follows from the proof of Proposition 3.1.7.1 given below, but there are other (arguably more elegant) ways to achieve the same result. For example, we can take $Q$ to be the simplicial set $\operatorname{Ex}^{\infty }(X)$ of Construction 3.3.6.1 (see Propositions 3.3.6.9 and 3.3.6.7), or the singular simplicial set $\operatorname{Sing}_{\bullet }(|X|)$ (see Proposition 1.2.5.8 and Theorem 3.6.4.1). These constructions also have non-aesthetic advantages: for example, the functors $X \mapsto \operatorname{Ex}^{\infty }(X)$ and $X \mapsto \operatorname{Sing}_{\bullet }(|X|)$ both preserve finite limits.