Thom spectrum of a map to BOP

Definition 10.2.24 (The Thom spectrum \(MB\)). Let \(\phi \colon B \to \BOP \) be a map of animae. We define its associated Thom spectrum \(MB\) by \[ MB := \colim \big (B\xrightarrow {\phi }\BOP \xrightarrow {J}\Sp \big ). \] Equivalently, \(\phi \) is a virtual vector bundle on \(B\), and \(MB=\th (\phi )\) in the notation of Construction 10.2.16. The notation suppresses \(\phi \), which will always be clear from the context. We allow \(B\) to be an arbitrary anima. For the Pontryagin–Thom theorem of Section 10.3, we will specialize to maps which land in the virtual-rank-zero component \(\bB O\subseteq \BOP \).