Thom spectrum of a virtual vector bundle

Construction 10.2.16 (Thom spectrum of a virtual vector bundle). Let \(V\) be a virtual vector bundle on an anima \(A\). Composing with the J-homomorphism gives a stable spherical fibration \[ \xi _V \quad := \quad J \circ V\colon A \longrightarrow \Pic (\S ). \] If \(V\) has rank \(d\), then so does \(\xi _V\), since \(J\) carries the class of \(\R \) to \(\S [1]\). We define the Thom spectrum of \(V\) as \[ \th (V) \quad := \quad \th (\xi _V) \quad = \quad \colim \big (J \circ V\colon A \to \Sp \big ). \]