The homotopy type of the Poincaré cobordism category for surfaces

We define a version of the surface cobordism category ${\mathrm{C}\mathrm{o}\mathrm{b}}_2^{\mathrm{S}\mathrm{G}}(X)$ over a base space $X$ where surfaces are considered up to self homotopy equivalences instead of diffeomorphisms. We prove the induced functor $B{\mathrm{C}\mathrm{o}\mathrm{b}}_2^{\mathrm{S}\mathrm{G}}(-):\mathcal{S}\to \mathcal{S}$ is not $1$-excisive. We show its first derivative ${\mathrm{\partial }}_{1}B{\mathrm{C}\mathrm{o}\mathrm{b}}_2^{\mathrm{S}\mathrm{G}}(-)$ in the Goodwillie sense is equivalent to a Thom spectrum over $B{\mathrm{h}\mathrm{a}\mathrm{u}\mathrm{t}}_{\ast }^{+}(S^2)$.