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Dmitri Pavlov's user avatar
Dmitri Pavlov's user avatar
Dmitri Pavlov's user avatar
Dmitri Pavlov
  • Member for 15 years, 2 months
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What is the nerve of this category?
@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.
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What is the nerve of this category?
@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.
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What is the nerve of this category?
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.
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Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?
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?
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$K_0$ group of an infinite factor
@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.
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$K_0$ group of an infinite factor
@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.
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$K_0$ group of an infinite factor
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.
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Regarding first order differential operator and derivative endomorphism
@PraphullaKoushik: They are not quite the same, as we have just seen: first-order differential operators form a strictly bigger class than derivative endomorphisms.
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