Sheaf over a Locale

Question

Define a frame to be a lattice with arbitrary joins and where finite meets distribute over joins. Morphisms of frames are lattice morphisms distributing over arbitrary joins. Observing that the poset of open sets on a topological space is a frame, one can see that there is a contravariant functor $\Phi: {\bf Top } \rightarrow {\bf Frame}$, so that the category of frames can be thought of as a generalization of the opposite category of topological spaces. Forming the category of locales as the opposite category of the category of frames, one comes upon the question of setting up concepts for locales analogous to those for topological spaces.

For instance, I have read that one can establish measure theory over locales in a way that avoids many of the pitfalls of ordinary measure theory. Measure theory over locales avoids the Banach Tarski Paradox and the need to distinguish only certain sets as measurable.

All this motivates my particular question:

1. Does there exist a notion of a Sheaf over a locale?

2. If so, does it have any desirable properties or avoid analogous complications in topos theory?

3. Where can I read about such objects?

Answer 1

The definition of a sheaf on a topological space is (or can be phrased) entirely in terms of its frame of opens; the exact same definition works to define sheaves over a locale.

So, not only can you define sheaves on locales, it is more natural to do so than to define sheaves on topological spaces.

The main complication is that there are locales without 'enough points', and the corresponding toposes are also without 'enough points'.

More generally, you can define sheaves on sites — categories with a Grothendieck topology.

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Any Grothendieck topos has a localic reflection: the lattice $\operatorname{Sub}(1)$. Furthermore, If $L$ is a locale and $\mathcal{E} = \mathbf{Sh}(L)$, then $\operatorname{Sub}_{\mathcal{E}}(1) \cong L$.

Thus, the localic toposes — the ones that are equivalent to sheaves on a locale — are special and can be distinguished from general toposes.

In fact, these constructions are adjoint, making Locale a reflective subcategory of Topos.

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It turns out that Grothendieck toposes vary in two orthogonal directions — the topological direction is about defining sheaves on locales, and the algebraic direction is about defining actions of a group. Ultimately, it turns out every Grothendieck topos is (equivalent to) the category of sheaves on a localic groupoid.

(although, groupoids can be interpreted as an algebraic description of space, so this can be viewed as a blending of the two notions of space)

I really liked Sheaves in Geometry and Logic, by MacLane and Moerdijk.