Simplicial Sets and No-Go Theorems

Simplicial sets are a combinatorial way of describing topological
spaces. We describe a simplicial set of the space of measurements and a
simplicial set of the space of possible outcomes of the measurements. An
experiment is then a map of simplicial sets from the space of outcomes to
the space of measurements. Basically, an outcome goes to the measurement
that corresponds to it. Global sections of such a map are actual results of
the experiment. By looking at different types of such maps, we formulate
different types of quantum systems and no-go theorems. One of the benefits
of this point of view is that the notion of compatibility (which restricts
the space of measurements) and entanglement (which restricts the space of
outcomes) are seen as intimately related phenomena.