The comonad of the stabilization adjunction
$$\Sigma^{\infty}: \mathrm{Top}_* \rightleftarrows \mathrm{Sp}: \Omega^{\infty}$$
is part of another adjunction
$$\Sigma^{\infty}\Omega^{\infty}: \mathrm{ConnSp} \rightleftarrows E_{\infty}\mathrm{Ring} :\mathfrak{gl}_1$$
whose right adjoint has infinite loop space $GL_1 = \Omega^{\infty}\mathfrak{gl}_1$ . Could someone explain the analogy to Lie algebras and Lie groups? How is the connective spectrum of units like a Lie algebra? How is the tangent category, given by the stabilization of the codomain fibration, like the tangent bundle of a manifold? More specifically, is the tangent category of a manifold considered as an $\infty$-groupoid its tangent bundle? If not, what additional structure do we need on the manifold or its interpretation as a category?

Pic unrelated, just some obligatory Kurisu posting.