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Hey all. Here's the thing. , so I have a measure space. According to Johnstone's 'Topos theory' (page 213), let $(X,\Sigma,\mu)$ be a measure space, we can define a Grothendieck pretopology on it (and hence a Grothendieck topology) if we take the poset $\Sigma$ of measurable sets on $X$ as a category and take the covering families to be the countable families of inclusions { $U_{i}\rightarrow U\mid i=1,2,3,...$ }, such that $\mu(U)-\mu(\overset{\infty}{\underset{i=1}{\bigcup}}U_{i})=0$. This site can be used to define sheaves of measurable functions on measure spaces (according to Dana Scott). What I wanted to do here was to define a fine sheaf that's acyclic on the resulting site and an injective resolution.

The measure space I'm working on is a Borel space whose underlying topological space is a discrete topological space, so therefore it's paracompact and Hausdorff. I wanted to know how the concepts of paracompact and Hausdorff would translate to the resulting Grothendieck topology, since I know that on a paracompact Hausdorff space a fine sheaf is acyclic. What I did was I've been looking at locales as the natural setting to deal with things like paracompactness and Hausdorffness on sites.

So my questions are:

1 - What the definition of a fine sheaf would be on a locale.

2 - I want to know if the definitions of paracompact and Hausdorff are in some way equivalent when going from topological spaces to locales, I want to know if I can ultimately define a fine sheaf that's acyclic on the resulting paracompact Hausdorff locale (in case it's in fact paracompact and Hausdorff too).

Anyone have a good reference? A good book on this?

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# Fine and acyclic sheaves on locales

Hey all. Here's the thing. so I have a measure space. According to Johnstone's 'Topos theory' (page 213), let $(X,\Sigma,\mu)$ be a measure space, we can define a Grothendieck pretopology on it (and hence a Grothendieck topology) if we take the poset $\Sigma$ of measurable sets on $X$ as a category and take the covering families to be the countable families of inclusions { $U_{i}\rightarrow U\mid i=1,2,3,...$ } such that $\mu(U)-\mu(\overset{\infty}{\underset{i=1}{\bigcup}}U_{i})=0$. This site can be used to define sheaves of measurable functions on measure spaces (according to Dana Scott). What I wanted to do here was to define a fine sheaf that's acyclic on the resulting site and an injective resolution.

The measure space I'm working on is a Borel space whose underlying topological space is a discrete topological space, so therefore it's paracompact and Hausdorff. I wanted to know how the concepts of paracompact and Hausdorff would translate to the resulting Grothendieck topology, since I know that on a paracompact Hausdorff space a fine sheaf is acyclic. What I did was I've been looking at locales as the natural setting to deal with things like paracompactness and Hausdorffness on sites.

So my questions are:

1 - What the definition of a fine sheaf would be on a locale.

2 - I want to know if the definitions of paracompact and Hausdorff are in some way equivalent when going from topological spaces to locales, I want to know if I can ultimately define a fine sheaf that's acyclic on the resulting paracompact Hausdorff locale (in case it's in fact paracompact and Hausdorff too).

Anyone have a good reference? A good book on this?