The main purpose of this work was to study equilibrium states in the topos approach to quantum theory. This is a relatively new approach to quantum theory then we also tried to present it in a self-contained way -at least from a mathematical point of view. Concerning equilibrium, we focused on two features: time translation invariance (stationary states) and the KMS condition. We characterize invariant states (in particular stationary states) by means of the so called "measures on the spectral presheaf". We couldn't do the same for the KMS condition and, as argued in this work, we think that there are good reasons to believe that it is not possible.Then, we studied the KMS condition using pseudo-states. For this purpose we introduced the idea of a "topos associated to a state" and proved that in the topos of any state there are at least two propositions (clopen subobjects of the spectral presheaf) which are true (in the sense of the pseudo-state) if, and only if, the state is KMS. One of these propositions is not of the kind "daseinization of a projector" and it can be seen as a way of giving physical meaning to a proposition which is not in the interpretation of the PL(S) language.

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A key ingredient of the Kochen-Specker theorem is the so-called functional composition principle, which asserts that hidden states must ascribe values to observables in a way that is consistent with all functional relations between them. This principle is motivated by the assumption that, like functions of observables in classical mechanics, a function g(A) of an observable A in quantum theory is simply a logically possible observable derived from A, and that measuring g(A) consists in measuring A and post-processing the resulting value via g. In this paper we show that, under suitable conditions, this reasonable assumption is in conflict with the collapse postulate. We then discuss possible solutions to this conflict, and we argue that the most justifiable and less radical one consists in adapting the collapse postulate by taking measurement contexts into account. This dependence on contexts arises for precisely the same reason why noncontextual hidden-variable models are ruled out by the Kochen-Specker theorem, namely that an observable A can typically be written as a function A = g(B) = h(C) of noncommuting observables B, C.