On an adjoint functor to the Thom functor (Q2781312)

Given a (locally trivial) bundle \(\xi\) with fiber \(\mathbb{R}^n\) and structure group \(O_n\), the Thom space of \(\xi\) is the quotient \(D(\xi)/S (\xi)\) of the total space of the unit disc bundle by the total space of its unit sphere bundle. Since it is useful to know when a space is the Thom space of a certain bundle, the given construction of a right adjoint to the Thom functor presents the ``de-Thomification'' problem like a dual of the ``de-looping'' one. With the suspension, left adjoint of the loop functor, every loop space is a space over the monad \(\Omega S\). Every Thom space is a space over the here constructed comonad.