Abstract
When analysing the origin of the twisted presheaves, it is clear that the reason we do get a twist is because the group moves the abelian algebras around, i.e. the group action is defined on the base category itself. Thus a possible way of avoiding the occurrence of twists is by imposing that the group does not act on the base category. The category of abelian von-Neumann sub-algebras with no group acting on it will be denoted \(\mathcal{V}_{f}(\mathcal{H})\), where we have added the subscript f (for fixed) to distinguish this situation from the case in which the group does act. Obviously, if one just defined sheaves over \(\mathcal{V}_{f}(\mathcal{H})\), then there would be no group action at all, therefore something extra is needed. As we will see this ‘extra’ will be the introduction of an intermediate category which will be used as an intermediate base category. On such an intermediate category the group is allowed to act, thus the sheaves defined over it will admit a group action. Once this is done, everything is “pushed down” to the fixed category \(\mathcal{V}_{f}(\mathcal{H})\). As we will see, the sheaves defined in this way will admit a group action which now takes place at an intermediate stage, but will not produce any twists since the final base category stays fixed.
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- 1.
Note that given \(w^{g_{1}}_{V}\) and \(w^{g_{2}}_{V}\) such that g 1≠g 2 then it is not possible to define an ordering between \(w^{g_{1}}_{V}\) and \(w^{g_{2}}_{V}\).
- 2.
Recall that \(p_{J}:(\varLambda J=\varLambda (\underline{Hom}(\mathcal{V}_{f}(\mathcal{H}),\mathcal{V}(\mathcal{H}))))\rightarrow \mathcal{V}_{f}(\mathcal{H})\).
- 3.
Note that here an element \(\mathcal{D}^{J}\) is called a diagram of type J since, intuitively, it ‘representes’ the category J onto the category \(\mathcal{D}\).
- 4.
Recall that \(A:J\rightarrow \mathbf {Sets}^{\mathcal{V}(\mathcal{H})}\) is such that A V (j)=A(j)(V), therefore (I(A(j))) ϕ :=A(j) ϕ(V)=A ϕ(V)(j).
- 5.
Note that g:x′→x and A(g):A(x)→A(x′), therefore ag:=A(g)(a) for a∈A(x), while gh:=f(g)∘h for h:y→f(x′).
- 6.
Note that it is possible to define a topology on \(\breve{\underline{\mathbb {R}}}\). To this end let us consider the set
$$ \mathcal{R}=\coprod_{V\in \mathcal{V}_f(\mathcal{H})}\breve{\underline{\mathbb {R}}}^{\leftrightarrow}_V =\bigcup_{V\in \mathcal{V}_f(\mathcal{H})}\{V\}\times\breve{\underline{\mathbb {R}}}^{\leftrightarrow}_V $$(16.69)where each \(\breve{\underline{\mathbb {R}}}^{\leftrightarrow}_{V}:=\coprod_{\phi_{i}\in \mathit{Hom}(\downarrow \!{V}, \mathcal{V}(\mathcal{H}))}\underline{\mathbb {R}}^{\leftrightarrow}_{\phi_{i}(V)}\).
This set represents a bundle over \(\mathcal{V}_{f}(\mathcal{H})\) with bundle map \(p_{\mathcal{R}}:\mathcal{R}\rightarrow \mathcal{V}_{f}(\mathcal{H})\); \(p_{\mathcal{R}}(\mu, \nu)=V=p_{J}(\phi_{i})\), where V is the context such that \((\mu, \nu)\in\underline{\mathbb {R}}^{\leftrightarrow}_{\phi_{i}(V)}\). In this setting \(p^{-1}_{\mathcal{R}}(V)=\breve{\underline{\mathbb {R}}}^{\leftrightarrow}_{V}\) are the fibres of the map \(p_{\mathcal{R}}\). We would like to define a topology on \(\mathcal{R}\) with the minimal requirement that the map \(p_{\mathcal{R}}\) is continuous. We know that the category \(\mathcal{V}_{f}(\mathcal{H})\) has the Alexandroff topology whose basis open sets are of the form ↓V for some \(V\in \mathcal{V}_{f}(\mathcal{H})\). Thus we are looking for a topology such that the pullback \(p_{\underline{\mathbb {R}}}^{-1}(\downarrow\!\! V):=\coprod_{V^{\prime}\in\downarrow\!\! V}\underline{\breve{\mathbb {R}}}_{V^{\prime}}\) is open in \(\mathcal{R}\).
Given the correspondence between sheaves and étale bundles, we know that each \(\underline{\mathbb {R}}^{\leftrightarrow}\) is equipped with the discrete topology in which all sub-objects are open (in particular each \(\underline{\mathbb {R}}^{\leftrightarrow}_{V}\) has the discrete topology). Since the F functor preserves monics, if \(\underline{Q}\subseteq\underline{\mathbb {R}}^{\leftrightarrow}\) is open, then \(F(\underline{Q})\subseteq F(\underline{\mathbb {R}}^{\leftrightarrow})\) is open, where \(F(\underline{Q}):=\coprod_{\phi_{i}\in \mathit{Hom}(\downarrow \!{V}, \mathcal{V}(\mathcal{H}))}\underline{Q}_{\phi_{i}(V)}\).
Therefore we define a sub-sheaf, \(\underline{\breve{Q}}\), of \(\breve{\underline{\mathbb {R}}}^{\leftrightarrow}\) to be open if for each \(V\in \mathcal{V}_{f}(\mathcal{H})\) the set \(\underline{\breve{Q}}_{V}\subseteq \breve{\underline{\mathbb {R}}}_{V}\) is open, i.e., each \(\underline{Q}_{\phi_{i}(V)}\subseteq \underline{\mathbb {R}}^{\leftrightarrow}_{\phi_{i}(V)}\) is open in the discrete topology on \(\underline{\mathbb {R}}^{\leftrightarrow}_{\phi_{i}(V)}\). It follows that the sheaf \(\breve{\underline{\mathbb {R}}}^{\leftrightarrow}\) gets induced the discrete topology in which all sub-objects are open. In this setting the ‘horizontal’ topology on the base category \(\mathcal{V}_{f}(\mathcal{H})\) would be accounted for by the sheave maps.
For each ↓V we then obtain the open set \(p_{\underline{\mathbb {R}}}^{-1}(\downarrow\!\! V)\) which has value \(\underline{\breve{\mathbb {R}}}_{V^{\prime}}\) at contexts V′∈↓V and ∅ everywhere else.
References
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A. Doering, C.J. Isham, Classical and Quantum Probabilities as Truth Values (2011). arXiv:1102.2213v1
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P.T. Johnstone, Sketches of an Elephant A Topos Theory Compendium, vols. 1, 2 (Oxford University Press, London, 2002)
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Flori, C. (2013). Group Action in Topos Quantum Theory. In: A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol 868. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35713-8_16
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DOI: https://doi.org/10.1007/978-3-642-35713-8_16
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