The Kapteyn universe is an early (late 19th century/early 20th century) model of the universe, which consisted of what is now known as the Milky Way galaxy.
It was a model developed before it was established that the spiral nebulae seen throughout the sky were, in fact, extremely distant structures similar to the Milky Way itself. When this was realized, such nebulae were eventually termed galaxies, borrowing a term originally that meant the Milky Way itself. The great debate took place before this realization was well-established.
As it was known at the time, the Kapteyn universe was assumed to be larger than the Milky Way is now known to be, i.e., the stars spread further, the most distant stars assumed to be further than they are. This misconception occurred because reddening was not sufficiently taken into account. Reddening was known, but analysis at the time suggested its effect was relatively minor.