A very good introductory reference for locale theory is Peter Johnstone's account:
- Johnstone, P.T. Stone Spaces. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press, 1982.
This book is remarkable as it exposes the story of the subject while at the same time containing all the required proofs in detail. As for topos theory there is.
- Johnstone, P.T. Sketches of an Elephant: a Topos Theory Compendium. Oxford Univ. Press 2002,
provides an excellent and rigorous introduction to the subject.
The standard reference for category theory,
- MacLane, S. Categories for the Working Mathematician. Texts in Mathematics 5. Springer-Verlag, 1971,
which you may see referred to as CWM, is still probably the best introductory account for category theory. This subject is now quite standard and so a number of texts are available. A good check as to whether a particular text really 'goes the distance' is to see whether an adjoint functor theorem (not just definition) is included.
References: Key Papers
A number of papers stand out in the literature as being of particular importance to our area of research.
- Joyal, A. and Tierney, M. An Extension of the Galois Theory of Grothendieck, Memoirs of the American Mathematical Society 309, 1984.
This paper contains an important classifying result for toposes but, for our purposes, it is useful as it explains how locales behave when moving from one topos to another.
- Vermeulen, J.J.C. Proper Maps of Locales. Journal of Pure and Applied Algerba 92, pp79-107, 1994.
In this paper Vermeulen replicates a number of results from the first paper but for a different class of maps (the proper ones).
- Priestley, H.A. Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, pp186-90, 1970.
In this paper Priestley examines Stone duality by representing distributive lattices as ordered Stone spaces. This technique is really the key that allows a lot of topological semilattice results to work. It introduces the patch topology. Stone's original duality result is in
- Stone, M.H. The theory of representation of Boolean algebras. Trans. Amer. Math Soc. 40, pp37-111. 1936.
- Johnstone, P.T. Tychonoff's theorem without the axiom of choice. Fund. Math. 113, pp21-35. 1981,
there is the localic form of the Tychonoff theorem providing a lot of the logical motivation for the work. Around that time there is also
- Hyland, J.M.E. Function-spaces in the category of locales. In Continuous Lattices, Springer LNM 871, pp264-81. 1981,
which describes exactly when exponentials exist within the category of locales. Finally there is
- Abramsky, S. and Vickers, S. Quantales, observational logic and process semantics. Mathematical Structures in Computer Science. 3 pp161-227. 1993. This paper outlines how the relevant free algebras in locale theory can always be re-expressed as free algebras of a more general variety. This technical insight has many applications to the subject.