Stability of homogeneous functors

An \(n\)-homogeneous functor is a \(K^{n+1}\)-local object whose map to the terminal object belongs to \(K^n\). Hence these functors form the layer \((K^n/K^{n+1})_*\). Since \(C\) is pointed, an \(n\)-reduced functor has a canonical point: the map \(P_0F\to F\) identifies with \(*\to F\). Thus the category of \(n\)-homogeneous functors identifies with the category of pointed objects in this layer. It is stable by Proposition 5.87; compare [Anel et al. 2025, Corollary 4.3.5].