The effective epi–mono factorization system

Existence of factorizations. Consider an arbitrary morphism \(f\colon U \to X\). Define \(V := \colim_{[n] \in \simp\catop} \check{C}_n(f)\). The augmentation of the Čech nerve gives a factorization By Lemma 2.32, the groupoid object \(\check{C}_{\bullet}(f)\) is the Čech nerve of its realization map \(e\). Hence \(e\) is an effective epimorphism and \(\check{C}_{\bullet}(e)\iso\check{C}_{\bullet}(f)\).The second map is a monomorphism. It suffices to show that the first projection \(p_1\colon V \times_X V \to V\) is an isomorphism, since the diagonal is a section of \(p_1\). Pullback along the effective epimorphism \(e\colon U \to V\) is conservative by Lemma 2.29(1). We may therefore test \(p_1\) after this pullback. Universality of groupoid colimits gives The augmented simplicial object \([n] \mapsto \check{C}_{n+1}(f)\) over \(U\) admits extra degeneracies, starting with the diagonal \(U \to U \times_X U = \check{C}_1(f)\). Its realization is therefore \(U\), so \(V \times_X U \to U\) is an isomorphism. Conservativity now implies that \(p_1\) is an isomorphism and hence that \(m\) is a monomorphism.Orthogonality. Consider a solid commutative diagram

where \(f\) is an effective epimorphism and \(i\colon A \hookrightarrow B\) is a monomorphism. Because \(i\) is a monomorphism, the anima of fillers is \((-1)\)-truncated. It remains only to prove that it is nonempty.Work in the slice \(C_{/B}\). Composition with \(i\) defines a fully faithful functor \(C_{/A}\hookrightarrow C_{/B}\), and this functor preserves colimits. The given map \(U\to A\) shows that every term \(\check{C}_n(f)=U^{\times^{n+1}_X}\) of the Čech nerve belongs to its essential image. Since \(X\) is the colimit of this Čech nerve in \(C_{/B}\), the object \(X\to B\) also belongs to the essential image. Thus there exists a map \(X\to A\) giving a filler. The anima of fillers is consequently contractible, proving orthogonality.