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