Dwyer-Kan equivalences of Segal animae

Definition 24.1.4 (Dwyer-Kan equivalences). A morphism \(f\colon X \to Y\) of Segal animae is called a Dwyer-Kan equivalence if the following two conditions are satisfied:

(1)

(Essential surjectivity): For every object \(y \in Y_0\) there exists an object \(x \in X_0\) satisfying \(f(x) \cong y\) in \(Y_0\).

(2)

(Fully faithfulness): For objects \(x,x' \in X_0\), the induced map \(f\colon \Hom _X(x,x') \to \Hom _Y(fx, fx')\) is an equivalence of animae.