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Background. Contextuality is one of the key characteristic features of quantum mechanics. Recent advances suggest that it provides the “magic ” ingredient enabling quantum computation [5]. However, the study of quantum contextuality has largely been carried out in a concrete, example-driven fashion, which makes it appear highly specific to quantum mechanics. Recent work by the present authors [1, 2] and others [3] has exposed the general mathematical struc-ture of contextuality, enabling more general and systematic results. The key idea from [1] is to understand contextuality as arising where we have a family of data which is locally but not globally consistent. This can be understood and very effectively visualised (see fig. 1a) in topological terms. We have a base space of contexts (typically sets of variables which can be jointly measured or observed); and local (i.e. contextual) data or observations — typically valuations of the variables in the context — are fibred over the base space, forming the total space. In this setting, contextuality arises as the absence of global sections, or valuations on all the variables that reconcile the local data. In topological language, we can say that the bundle space of observations is “twisted”, resulting in a topological obstruction to forming a global section. This perspective provides a unifying description of a number of phenomena which at first sight seem very different, including quantum contextuality as well as phenomena in classical computa