Yes, “co-observable” is something I just made up, in need of a term. But they do form a co-algebra, as any homology does!

A more fancy terminology for the spectrum of co-observables is: The *motive* of the phase space:

Let $Phase$ be a phase space, and $E$ a ring spectrum, in the given ambient cohesive $\infty$-topos. Then $E$-valued observables are $[\Sigma^\infty Phase, E]$, while $E$-value co-observables are $E \wedge \Sigma^\infty Phase$. But the geometric suspension spectrum $\Sigma^\infty Phase$, that’s the motive of phase space.

]]>Yeah, my first thought was whether this relates to holography as Koszul duality. But at this point, all we see is plain linear duality of Hopf algebras.

In this context, the bulk/boundary correspondence is very much brought about by the simple fact that round chord diagrams and Jacobi diagrams have a bulk and a boundary.

]]>Is term ’co-observable’ as used in physics new with you? It seems computer scientists use it in a coalgebraic setting for the supervisory control of discrete-event systems.

]]>This is a case of Koszul duality? I see at this MO answer, an instance is given as

- the relation between the homotopy groups of a topological space and its (co)homology groups

In your case, how does the duality between homology and cohomology produce that shift of dimension from bulk to boundary, if that makes sense?

]]>It finally dawned on us that the analysis here from Hypothesis H exhibits holographic gauge/gravity duality as concrete mathematical duality. Now on p. 4 of v2.

Comments are welcome.

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