Epimorphisms in animae

(1) Suppose first that \(f\) is an epimorphism. For any local system \(A\) on \(Y\), the Mayer–Vietoris sequence associated to the pushout square

identifies the restriction map \(H^*(Y,A)\to H^*(X,f^*A)\) with an isomorphism. Thus \(f\) is acyclic.Conversely, suppose that \(f\) is acyclic and set \(C:=Y\sqcup_XY\). Let \(i_1,i_2\colon Y\to C\) be the two coprojections. For a local system \(A\) on \(C\), the Mayer–Vietoris sequence contains The two restrictions of \(A\) to \(X\) are canonically identified, and both restriction maps from the two copies of \(H^q(Y)\) to \(H^q(X)\) are isomorphisms by acyclicity of \(f\). Exactness therefore identifies \(H^q(C,A)\) with the diagonal copy of \(H^q(Y,i_1^*A)\). In particular, \(i_1\) is itself acyclic.By Lemma 5.11, the map \(f\) induces an isomorphism on \(\pi_0\) and a surjection on every fundamental group. We may consequently apply the Seifert–van Kampen theorem componentwise to the pushout defining \(C\). It gives a pushout square

in groups. Since the two maps out of \(\pi_1(X,x)\) agree and are surjective, the coprojection \(i_1\) induces an isomorphism on fundamental groups. The same van Kampen calculation on components shows that \(i_1\) induces an isomorphism on \(\pi_0\). Hence Lemma 5.12 applies and shows that \(i_1\) is an isomorphism. Since \(\nabla_f i_1=\id_Y\), the codiagonal \(\nabla_f\colon C\to Y\) is its inverse. Thus \(f\) is an epimorphism.(2) See [Hoyois 2019, Corollary 10].(3) By definition, Quillen's \(+\)-construction is the reflection of \(X\) into the subcategory of animae with hypoabelian fundamental group. By (2), this subcategory precisely consists of the objects \(Y\) for which \(Y \to *\) is right orthogonal to the epimorphisms, and the claim follows.