Segal animae

Definition 24.1.2 (Segal anima). A simplicial anima \(X\colon \simp \catop \to \An \) is called a Segal anima if the Segal condition is satisfied: For every \(n \geq 1\), the inclusion maps \(e_i\colon [1] \cong \{i- 1\leq i \} \hookrightarrow [n]\) induce an isomorphism of animae \[ (e_1^*, \dots , e_n^*)\colon X_n \iso X_1 \times _{X_0} X_1 \times _{X_0} \dots \times _{X_0} X_1. \] We write \(\Seg (\An ) \subseteq \sAn = \Fun (\simp \catop ,\An )\) for the full subcategory of Segal animae.