# 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?

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Starting from a discrete space is not very interesting. As far as fine sheaf on a locale goes, have a look at the Voisin definition at ncatlab.org/nlab/show/fine+sheaf. The definition of a partition of unity on a locale is probably what you are after, however, so this advice might be circular. –  David Roberts Sep 23 '11 at 3:02
I don't think it addresses your question, but there is a thesis on measure theory and topoi by Matthew Jackson which you might find interesting: andrew.cmu.edu/user/awodey/students/jackson.pdf –  Peter Arndt Sep 23 '11 at 11:24
Thank you both, I actually downloaded Matthew Jackson's thesis a while back, I know the discrete topological structure is in itself not interesting, but that's just the space I'm working on, really have to look deeper into that but it seems right from the point of view of what I need this whole construction for –  Mario Carrasco Sep 23 '11 at 19:29
If you have a measurable space X and some subspace Y⊂X, then functions on X are precisely arbitrary pairs of functions on Y and X\Y. The same is true for morphisms of sheaves on X. Therefore any sheaf on the site that you described is fine, flabby, and soft for trivial reasons. –  Dmitri Pavlov Sep 23 '11 at 23:07
@Mario: I am not sure what do you mean by the extension property, but any measurable function defined on a measurable subset of a measurable space can be extended to a measurable function defined on the whole space by declaring it to be 0 outside of the original domain. Locally constant functions stay locally constant under this transformation. –  Dmitri Pavlov Oct 2 '11 at 22:55