I am looking for one (or more) reference about properties of the category of von Neumann algebra. 


More precisely, in an answer of a previous [question](https://mathoverflow.net/questions/23601/monoidal-structures-on-von-neumann-algebras), Dmitri Pavlov mentions
that the $W^\*$ category is complete and cocomplete. I would be happy to have a reference for these facts. 


On a related note, at the page 84 of the volume 3 of his treatise "Theory of 
operator algebras", Takesaki says:


*"Namely, 
if one insists to have the universal property for the inductive limit, then one has 
to treat a non-separable von Neumann algebra as the inductive limit of a sequence 
of separable von Neumann algebras. Therefore, we shall not consider the general 
theory of inductive limits of von Neumann algebras."* 


This got me curious. Where can I find a reference for an example of an inductive system 
of separable von Neumann algebras with non separable limit ? Is it possible for this example 
to be a countable inductive system ?


And this makes me wonder : why is nonseparability such a problem in the theory of von Neumann algebra, as Takesaki suggests ?



About the category of von neumann algebras
The standard reference for such matters is Guichardet's paper
Sur la catégorie des algèbres de von Neumann.
Bulletin des Sciences mathématiques 90 (1966), 41–64.
PDF file: <http://math.berkeley.edu/~pavlov/scans/guichardet.pdf>


I don't think separability (or the more general property of σ-finiteness) is important.
Requiring it is more of a tradition than a real necessity,
similar to requiring smooth manifolds to be second countable instead of just being paracompact


