Brown's conditions for representability

Definition 3.4.1 (Brown’s conditions). Let \[ F\colon \Ho (\An _{*,\geq 1})\catop \longrightarrow \Set \] be a functor. We say that:

(1)

The functor \(F\) is representable if it is naturally isomorphic to \([-,Z]_*\) for some \(Z\in \An _{*,\geq 1}\);

(2)

The functor \(F\) satisfies the wedge axiom if, for every small collection \((X_i)_{i\in I}\) of connected pointed animae, the canonical map \[ F\left (\bigvee _{i\in I}X_i\right )\longrightarrow \prod _{i\in I}F(X_i) \] is a bijection;

(3)

The functor \(F\) satisfies the Mayer–Vietoris property if every pushout square

in \(\An _{*,\geq 1}\) induces a surjection \[ (i^*,j^*)\colon F(X)\longrightarrow F(A)\times _{F(C)}F(B). \]