Equivalences in Segal animae

Definition 24.1.6. Let \(X \in \Seg (\An )\) be a Segal anima. A morphism \(f\colon x \to y\) in \(X\) is called an isomorphism if there exist \(2\)-simplices \(\sigma , \tau \in X_2\) satisfying the relations \(d_0(\sigma ) \simeq f\), \(d_1(\sigma ) \simeq \id _x\), \(d_2(\tau ) \simeq f\) and \(d_1(\tau ) \simeq \id _y\). In other words: \(f\) is an isomorphism if there exist morphisms \(g,h\colon y \to x\) together with homotopies \(gf \simeq \id _x\) and \(fg \simeq \id _y\).

We write \(X_1^{\sim } \subseteq X_1\) for the full subanima spanned by the isomorphisms.