  Let $\mathcal{A}_{g,d}$ be the (topological) cobordism category of orientable
surfaces whose connected components are homeomorphic to either $S^1 \times I$
with one incoming and one outgoing boundary component or the surface
$\Sigma_{g,d}$ of genus $g$ and $d$ boundary components that are all incoming.
In this paper, we study the homotopy type of the classifying space of the
cobordism categories $\mathcal{A}_{g,d}$ and the associated (ordinary)
cobordism categories of their connected components $\mathbb{A}_d$.
$\mathcal{A}_{0,2}$ is the cobordism category of complex annuli that was
considered by Costello and $\mathbb{A}_2$ is homotopy equivalent with the
positive boundary 1-dimensional embedded cobordism category of
Galatius-Madsen-Tillmann-Weiss. We identify their homotopy type with the
infinite loop spaces associated with certain Thom spectra.
