  This paper studies colimits of sequences of finite Chu spaces and their
ramifications. Besides generic Chu spaces, we consider extensional and
biextensional variants. In the corresponding categories we first characterize
the monics and then the existence (or the lack thereof) of the desired
colimits. In each case, we provide a characterization of the finite objects in
terms of monomorphisms/injections. Bifinite Chu spaces are then expressed with
respect to the monics of generic Chu spaces, and universal, homogeneous Chu
spaces are shown to exist in this category. Unanticipated results driving this
development include the fact that while for generic Chu spaces monics consist
of an injective first and a surjective second component, in the extensional and
biextensional cases the surjectivity requirement can be dropped. Furthermore,
the desired colimits are only guaranteed to exist in the extensional case.
Finally, not all finite Chu spaces (considered set-theoretically) are finite
objects in their categories. This study opens up opportunities for further
investigations into recursively defined Chu spaces, as well as constructive
models of linear logic.
