A Second Course in Topos Quantum Theory pp 247-289 | Cite as

# Extending the Topos Quantum Theory Approach

## Abstract

As it has been developed so far, the mathematical formalism of topos quantum theory only allows for taking into consideration one physical system at a time. This has clearly some limitations, in particular when trying to consider composite systems. Hence it comes natural to try and enlarge the mathematical formalism so that is is possible to take into consideration various physical systems at the same time. This implies considering a topos somewhat “larger” than the topos \(\textbf {Sets}^{\mathcal {V}(\mathcal {H})^{\mathrm {op}}}\). In particular what needs to be “enlarged” is the category \( \mathcal {V}(\mathcal {H})\). In fact this category only refers to the physical system with associated von Neumann algebra \(\mathcal {N}\), whose category of abliean subalgebras is given by \(\mathcal {V}(\mathcal {H})\). However we would like to consider all physical systems, each of which, has associated to it a different von Neumann algebra. To account for this, one possibility would be to construct a category in which each element is itself a topos which represents the mathematical formalism of a physical system. Then one would have to construct a mapping which associates to each physical system its associated topos. The aim would be to turn this map into a geometric morphism of some sort between topoi, such that it possesses nice properties which would help to better understand composite systems. In the following we will present all work done so far in this direction, which is an exposition of the results obtained in [30]. As it will be clear in due course, there are still many open problems to be addressed.

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