Abstract
In this chapter we will describe a different way in which topos theory was utilised to describe quantum theory. This approach is called covariant topos quantum theory and it was first put forward in [42]. The aim of this approach is to combine, on the one hand, algebraic quantum theory by describing a system via a C ∗-algebra \(\mathcal {A}\) and, on the other, Bohr’s idea of classical snapshots which enables one to talk about physical quantities, only with respect to a suitable context of compatible physical quantities.
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Notes
- 1.
Note that we have denoted covariant functors by an overline \(\overline {X}\) to distinguish them from contravariant functors which we denote by an underline \( \underline {X}\).
- 2.
We recall that the downwards Alexandroff topology on \(\mathcal {V}(\mathcal { A})\) is the topology for which a subset \(U\subseteq \mathcal {V}(\mathcal {A})\) is open if it is a downwards closed set, i.e. \(U=\{C'\subseteq C|C\in \mathcal {V}(\mathcal {A})\}\).
- 3.
Here \(\overline {\mathbb {R}}\) represents the internal locale of Dedekind reals.
- 4.
\(\overline {\mathcal {\mathbb {I}\mathbb {R}}}\) represents the internal Scotts interval domain (see Appendix A.2).
- 5.
Here \(P(\overline {Y})\) denotes the power object of \(\overline {Y}\).
- 6.
In the following the set \(\mathbb {R}_u\) denotes the set of upper reals. The topology on \(\mathbb {R}_u\) is generated by the lower half open intervals [−∞, y), \(y\in \mathbb {R}\).
- 7.
In the following the set \(\mathbb {R}_l\) denotes the set of lower reals. The topology on \(\mathbb {R}_l\) is generated by the upper half intervals (y, +∞], \(y\in \mathbb {R}\).
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Flori, C. (2018). Brief Introduction to Covariant Topos Quantum Theory. In: A Second Course in Topos Quantum Theory. Lecture Notes in Physics, vol 944. Springer, Cham. https://doi.org/10.1007/978-3-319-71108-9_11
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