In this paper we introduce an open-closed cobordism category with maps to a background space. We identify the classifying space of this category for certain classes of background space. The key ingredient is the homology stability of mapping class groups with twisted coefficients.
"An open-closed cobordism category with background space." Algebr. Geom. Topol. 9 (2) 833 - 863, 2009. https://doi.org/10.2140/agt.2009.9.833