Additivity of Thom spectra

Remark 10.2.17. As \(J\) is a homomorphism of commutative groups in animae, the assignment \(V \mapsto \xi _V\) carries sums to pointwise tensor products and inverses to pointwise inverses: \[ \xi _{V+W} \cong \xi _V \otimes \xi _W \qquad \text {and}\qquad \xi _{-V} \cong \xi _V^{-1}. \] The Thom spectrum itself is of course not additive: \(\th (V+W)\) is a colimit over \(A\), not a tensor product of colimits.