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@BastamTajik: I don't know what “non-strongly causal” means, and no definition is provided, so it is hard to say more without some concrete details. I cited the example of strongly causal spacetimes since my guess is that non-strongly causal spacetimes should contain strongly causal spacetimes as a special case.
@BastamTajik: Any homotopy type is weakly equivalent to the nerve of a poset and therefore to the nerve of a thin category. However, for a Lorentzian manifold, under mild restrictions (e.g., stably causal), any two points x, y can be connected by future-directed timelike curves to another point z, which means the relevant poset is filtered and the nerve of the corresponding thin category is weakly contractible.
In many cases, the category is filtered and the nerve of a filtered category is weakly contractible. For example, this is the case for a Lorentzian manifold with its causality structure.
How exactly does the fact that κ-Lindelöf spaces are not closed under products in topological spaces imply that the category of κ-Lindelöf spaces is not locally presentable?
@SanaeKochiya: Every module of a finite von Neumann dimension is also a finitely generated projective module. However, there are finitely generated projective modules of an infinite dimension, e.g., M itself, if M is a type II_∞ factor. That's why the usual K-theory of a type II_∞ factor is trivial, but if you take the K-theory of modules of a finite von Neumann dimension, you get a nontrivial K-group, namely, R.
@SanaeKochiya: M-modules of a finite von Neumann dimension are precisely finite direct sums of M-modules of the form pM, where p is a finite projection in M.
As a remark, a far more natural choice for the K-theory of type II_∞ factors is to start with the category of modules (or W*-modules) of a finite von Neumann dimension. In this case, we do get a meaningful K-theory group, namely, R.
@PraphullaKoushik: They are not quite the same, as we have just seen: first-order differential operators form a strictly bigger class than derivative endomorphisms.