... 1
... complete2
A set $ C=\{\alpha,
 \beta,\cdots, \gamma\}$ is said to be complete if all history are pair-wise disjoint and their logical `or' forms the unit history
... formalism3
The acronym `HPO' stands for `history projection operator' and was the name given by Isham to his own (non-topos based) approach to consistent-history quantum theory. This approach is distinguished by the fact that any history proposition is represented by a projection operator in a new Hilbert space that is the tensor product of the Hilbert spaces at the constituent times. In the older approaches, a history proposition is represented by a sum of products of projection operators, and this is almost always not itself a projection operator. Thus the HPO formalism is a natural framework with which to realise `temporal quantum logic'.
... history4
The unit history is the history which is always true
... theorem5
Kochen-Specker Theorem: if the dimension of $ \Hi$ is greater than 2, then there does not exist any valuation function $ V_{\vec{\Psi}}:\mathcal{O}\rightarrow\Rl$ from the set $ \mathcal{O}$ of all bounded self-adjoint operators $ \hat{A}$ of $ \Hi$ to the reals $ \Rl$ such that for all $ \hat{A}\in\mathcal{O}$ and all $ f:\Rl\rightarrow\Rl$ , the following holds $ V_{\vec{\Psi}}(f(\hat{A}))=f(V_{\vec{\Psi}}(\hat{A}))$ .
... realist6
By a `realist' theory we mean one in which the following conditions are satisfied: (i) propositions form a Boolean algebra; and (ii) propositions can always be assessed to be either true or false. As will be delineated in the following, in the topos approach to quantum theory both of these conditions are relaxed, leading to what Isham and Döring called a neo-realist theory.
... transform7
Given a commutative von Neumann algebra V, the Gel'fand transform is a map

  $\displaystyle V\rightarrow C(\Sig_V)$ (25)
  $\displaystyle \hat{A}\mapsto \bar{A}:\Sig_V\rightarrow\Cl$ (26)

where $ \Sig_V$ is the Gel'fand spectrum; $ \bar{A}$ is such that $ \forall\lambda\in\Sig_V$ $ \bar{A}(\lambda):=\lambda(\hat{A})$ .
... spectrum8
Given an algebra V, the Gel'fand spectrum, $ \Sig_V$ , is the set of all multiplicative, linear functionals, $ \lambda:V\rightarrow \Cl$ , of norm 1.
... points9
In a topos $ \tau$ , a `point' (or `global element'; or just `element') of an object $ O$ is defined to be a morphism from the terminal object, $ 1_\tau$ , to $ O$ .
... let10
We will denote the set of open subsets of a topological space, $ X$ , by $ \Sub_\op(X)$ .

... subsets11
Arguably, it is more appropriate to represent propositions in classical physics with Borel subsets, not just open ones. However, will not go into this subtlety here.
... topoi12
Of course, in the case of temporal logic, the Hilbert spaces $ \Hi_1$ and $ \Hi_2$ are isomorphic, and hence so are the associated topoi. However, their structural roles in the temporal logic are clearly different. In fact, in the closely related situation of composite systems it will generally be the case that $ \Hi_1$ and $ \Hi_2$ are not isomorphic. Therefore, in the following, we will not exploit this particular isomorphism.
... topoi13
We are here exploiting the trivial fact that, for any pair of categories $ {\cal C}_1,{\cal C}_2$ , we have $ ({\cal
 C}_1\times{\cal C}_2)^\op\simeq{{\cal C}_1}^\op\times{{\cal
 C}_2}^\op$ .
... that14
Since there is no state-vector reduction the existence of an operation $ \sqcap$ between truth values , that satisfies equation (100) is plausible. In fact, unlike the normal logical connective `$ \land$ ', the meaning of the temporal connective `$ \sqcap$ ' implies that the propositions it connects do not `interfere' with each other since they are asserted at different times: it is thus a sensible first guess to assume that their truth values are independent.

The distinction between the temporal connective `$ \sqcap$ ' and the logical connective `$ \wedge$ ' is discussed in details in various papers by Stachow and Mittelstaedt [33] ,[32], [29], [30]. In these papers they analyse quantum logic using the ideas of game theory. In particular they define logical connectives in terms of sequences of subsequent moves of possible attacks and defenses. They also introduce the concept of `commensurability property' which essentially defines the possibility of quantities being measured at the same time or not. The definition of logical connectives involves both possible attacks and defenses as well as the satisfaction of the commensurability property since logical connective relate propositions which refer to the same time. On the other hand, the definition of sequential connectives does not need the introduction of the commensurability properties since sequential connectives refer to propositions defined at different times, and thus can always be evaluated together. The commensurability property introduced by Stachow and Mittelstaedt can be seen as the game theory analogue of the commutation relation between operators in quantum theory. We note that the same type of analysis can be applied as a justification of Isham's choice of the tensor product as temporal connective in the HPO theory.

.... 15
This is correct since the projectors which appear on the right hand side of the equation are pair-wise orthogonal, thus the `or', $ \vee$ , can be replaced by the summation operation $ +$ of projector operators.