## Coarse Graining Presheaf on category of Boolean subalgebras

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## Coarse Graining Presheaf on category of Boolean subalgebras

One can alternatively define a coarse graining presheaf directly on the category $\mathcal{W}$ of Boolean subalgebras (see Category of Boolean subalgebras ).

Definition 2.5   A coarse graining presheaf on $\mathcal{W}$ is a contravariant functor 3.2 $\Theta:\mathcal{W}\rightarrow Set$ such that:

• each object $W\in\mathcal{W}$ gets mapped to itself i.e. $\Theta(W):=W$

• given a $\mathcal{W}$-morphism $i_{W_BW_A}:W_B\rightarrow W_A$ such that $W_B\subseteq W_A$ the corresponding $\Theta$-morphism is $\Theta(i_{W_BW_A}):=\theta_{W_AW_B}:W_A\rightarrow W_B$

The prove that $\Theta$, as defined above, is indeed a presheaf, is very similar to the prove given for G, therefore, for sake of brevity, I will not repropose it here.

Cecilia Flori 2007-03-17