Thanks for the plugin! Just got it up and running on WordPress 2.6 here.

I’ll try to keep the instructions updated if I discover additional necessary tweaks, but overall 2.6 required minimal patching.

a sheaf with respect to target space and a co-sheaf with respect to base space.

Sorry, “base space” should read “parameter space” (= worldvolume):

a sheaf with respect to target space and a co-sheaf with respect to parameter space.

True. On the other hand, I suppose one could easily fix the failure of higher dimensional causal diamonds to be closed under intersection by passing to something like “convex open subsets the boundary of whose closure is piecewise lightlike”. The main point still being, I think, that all these sets are diffeomorphic to open balls.

But generally, there has been surprisingly little (as far as I can see: none(?)) discussion in the AQFT literature on cosheaf properties. Even the plain fact that a “net” is a special kind of co-presheaf (or pre-cosheaf if you like) is usually not mentioned. The only place that I know of is in writing of Bert Schroer, for instance his Lectures on AQFT and operator algebras #, where the term itself is mentioned when nets are introduced, though the sheaf-theoretic implications are not further explored.

(One also finds on the net the curious text Non-abelian quantum algebraic topology # which mentions AQFT co-presheaves and also the van Kampen theorem (which is secretly # about codescent, i.e. about cosheaves) but does not seem to connect them.)

In summary: it is not clear to me if the answer to “Should Haag-Kastler nets be taken to satisfy the co-sheaf condition?” is really “No.”

On a more speculative note: as we know, the chiral deRham complex # is a sheaf of vertex operator algebras on target space. On the other hand, vertex operator algebras are expected (people are telling me) to give rise to local nets (on parameter space, i.e on the worldvolume) by smearing vertex operators with test functions supported in causal subsets. It seems not unnatural therefore to speculate that the chiral deRham complex may be regarded as something which is a sheaf with respect to target space and a co-sheaf with respect to base space. Would seem to make sense. But I don’t know.

one has to be a bit careful that the Haag-Kastler nets are not defined on all open subsets, but just on the “causal subsets”: those which are intersections of the future of one point with the past of another.

That’s fine in 1+1 dimensions, where the intersection of two causal sets (often called “causal diamonds” in that context) is a causal set.

That property no longer holds in higher dimensions, which means that one has to extend the definition a bit. The subject of how, exactly, to extend the definition was the subject of debate between me and my anonymous interlocutor.

I’m not sure about this “split property for wedges” business. But I have a suspicion that it, too, may be special to 1+1 dimensions.

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Regarding the question of whether a given net (= co-flabby co-presheaf) is actually a cosheaf:

one has to be a bit careful that the Haag-Kastler nets are not defined on all open subsets, but just on the “causal subsets”: those which are intersections of the future of one point with the past of another.

This means first of all that neigther the torus interior nor the union of two disjoint causal subsets — the two proposed counterexamples in the comment by the commenter who signed as “anonymous hostile coward” # qualify.

More generally it means that we need to check the co-sheaf property on covers which exist as such in the category of causal subsets: we can cover one causal subset $O$ by a bunch $O_i\subset O$ of smaller ones all sitting inside, such that ${\cup }_i{O}_i=O$.

Whether or not a local net satisfies codescent (i.e. the co-sheaf property) with respect to such a cover is a little subtle, I suppose, if we are talking about von Neumann algebra valued nets, as we should.

I am not really sure yet about the details, but I notice that sometimes a certain extra property is found on the local net $A$, namely that it has the split property for wedges (see reference below). This property implies that:

for $O_1$ and $O_2$ causal subsets touching “in one point”, and for $O$ the causal hull of that (smallest causal subset containing both), we have $$A(O_1)\vee A(O_2)=A(O),$$ where $C\vee D$ is the von Neumann algebra generated by vN algebras $C$ and $D$.

The split property also implies the time slice axiom on $A$. In total, this seems to imply that $A$ is a cosheaf, but I’d need to check that.

This stuff about split property I am taking from Lechner: On the construction of quantum field theories with factorizing S-matrix around p. 25,, which in turn builds on Müger’s arXiv:hep-th/9705019.

But what an extraordinary night! The world is a lot better off now, even though there remain many, many challenges.

Nice speech by McCain. A truly, classy and humble speech by Obama—-contrast with the “i have the mandate” speech by Bush on 2004 after winning the election, barely.

Makes me less depressed than the re-election of the Conservatives (fortunately, a minority) here in Canada few weeks ago.