LURIE STABLE ∞-CATEGORIES // EXCISIVE FUNCTORS // SPECTRUM SPACE FUNCTOR
A stable ∞-category is a pointed ∞-category C with finite limits and colimits where the suspension functor Σ: C → C is an equivalence. The homotopy category hC is automatically triangulated.
Sp(C) = lim(... → C → C → C) via Ω
A functor F: C → D between ∞-categories is n-excisive if it takes strongly cocartesian (n+1)-cubes to cartesian cubes. The Goodwillie tower P_n F approximates F by n-excisive functors.
F → ... P_2 F → P_1 F → P_0 F
The spectrum space functor Ω^∞: Sp → S sends a spectrum E to its infinite loop space Ω^∞E = colim Ω^n E_n. Its left adjoint Σ^∞_+: S → Sp is the suspension spectrum functor.