Groupoid core axiom

Axiom E (Groupoid core axiom). For every \(\infty \)-category \(C\), there is an anima \(C^{\simeq }\) called the (groupoid) core of \(C\), which comes equipped with a functor \(\gamma _C \colon C^{\simeq } \to C\). Every functor \(F\colon X \to C\) from an anima \(X\) factors through \(\gamma _C\), and for functors \(G,H\colon X \to C^{\simeq }\) every natural isomorphism \(\gamma _C \circ G \cong \gamma _C \circ H\) may be lifted to a natural isomorphism \(G \cong H\).

Furthermore, the map \(\lra {0,1}\colon * \sqcup * \to [1]\) factors through an equivalence \(* \sqcup * \iso [1]^{\simeq }\).