First of all, Kevin is being quite modest in his comment above: his paper
Buzzard, Kevin. Integral models of certain Shimura curves. Duke Math. J. 87 (1997), no. 3, 591--612.
contains many basic results on integral models of Shimura curves over totally real fields, and is widely cited by workers in the field: 22 citations on MathSciNet. The most recent is a paper of mine:
Clark, Pete L. On the Hasse principle for Shimura curves. Israel J. Math. 171 (2009), 349--365.
http://alpha.math.uga.edu/~pete/plclarkarxiv7.pdf
Section 3 of this paper spends 2-3 pages summarizing results on the structure of the canonical integral model of a Shimura curve over $\mathbb{Q}$ (with applications to the existence of local points). From the introduction to this paper:
"This result [something about local points] follows readily enough from a description of their [certain Shimura curves over Q] integral canonical models. Unfortunately I know of no unique, complete reference for this material. I have myself written first (my 2003 Harvard thesis) and second (notes from a 2005 ISM course in Montreal) approximations of such a work, and in so doing I have come to respect the difficulty of this expository problem."
I wrote that about three years ago, and I still feel that way today. Here are the documents:
- http://alpha.math.uga.edu/~pete/thesis.pdf
is my thesis. "Chapter 0" is an exposition on Shimura curves: it is about 50 pages long.
- For my (incomplete) lecture notes from 2005, go to
http://alpha.math.uga.edu/~pete/expositions2012.html
and scroll down to "Shimura Curves". There are 12 files there, totalling 106 pages [perhaps I should also compile them into a single file]. On the other hand, the title of the course was Shimura Varieties, and although I don't so much as attempt to give the definition of a general Shimura variety, some of the discussion includes other PEL-type Shimura varieties like Hilbert and Siegel moduli space. These notes do not entirely supercede my thesis: each contains some material that the other omits.
When I applied for an NSF grant 3 years ago, I mentioned that if I got the grant, as part of my larger impact I would write a book on Shimura curves. Three years later I have written up some new material (as yet unreleased) but am wishing that I had not said that so directly: I would need at least a full semester off to make real progress (partly, of course, to better understand much of the material).
Let me explain the scope of the problem as follows: there does not even exist a single, reasonably comprehensive reference on the arithmetic geometry of the classical modular curves (i.e., $X_0(N)$ and such). This would-be bible of modular curves ought to contain most of the material from Shimura's book (260 pages) and the book of Katz and Mazur Arithmetic Moduli of Elliptic Curves (514 pages). These two books don't mess around and have little overlap, so you get a lower bound of, say, 700 pages that way.
Conversely, I claim that there is some reasonable topology on the arithmetic geometry of modular curves whose compactification is the theory of Shimura curves. The reason is that in many cases there are several ways to establish a result about modular curves, and "the right one" generalizes to Shimura curves with little trouble. (For example, to define the rational canonical model for classical modular curves, one could use the theory of Fourier expansions at the cusps -- which won't generalize -- or the theory of moduli spaces -- which generalizes immediately. Better yet is to use Shimura's theory of special points, which nowadays you need to know anyway to study Heegner point constructions.) Most of the remainder concerns quaternion arithmetic, which, while technical, is nowadays well understood and worked out.
A good place to start is in low dimensions and there a good introduction can be found in
J. Kock, 2003, Frobenius Algebras and 2-D Topological Quantum Field Theories, number 59 in London Mathematical Society Student Texts, Cambridge U.P., Cambridge.
There are also old notes of Quinn that take a very neat homotopy theoretic approach to some of the problems:
F. Quinn, 1995, Lectures on axiomatic topological quantum field theory, in D. Freed and K. Uhlenbeck, eds., Geometry and Quantum Field Theory, volume 1 of IAS/Park City Math- ematics Series, AMS/IAS,.
These do not really get near the physics but you seem to indicate that that is not the direction you want to go in.
I like the links with higher category theory. I realise that this is not everyone's `cup of tea' but it does have some useful insights. I wrote a set of notes for a workshop in Lisbon in 2011 which contain a lot that might be useful (or might not!). They are available at
http://ncatlab.org/nlab/files/HQFT-XMenagerie.pdf
My advice would be to raid the net getting this sort of resource (storing it on your hard disc rather than printing it all out!), then as you start working your way through some of the stuff you have found there will be explanations available ready at hand. Start with the main ideas and `back fill', i.e. don't try to learn everything you might need before you start. If when reading some source material an idea that you are not happy with comes up, search it out then, just enough to make progress beyond that point easy. (Of course this is how one progresses through lots of areas of maths so ....)
(Those notes of mine exist in several different forms and lengths, so in a longer version some idea may be more developed.... so ask!)
.... and don't forget the summaries in the n-Lab can be a very useful place to start a search.
(I had at the back of my mind just now a reference to a seminar that I had some notes on .... but no idea of the source. A little search found me:
http://ncatlab.org/nlab/show/UC+Riverside+Seminar+on+Cobordism+and+Topological+Field+Theories
... One other thing, there are lectures by Lurie on this stuff on YouTube:
https://www.youtube.com/watch?v=Bo8GNfN-Xn4
that are well worth watching.)
Best Answer
I haven't read all of DCCT so take this with a grain of salt, but after having spent a lot of time with it, this is how I would recommend getting started on the abstract stuff.
First one must learn classical Grothendieck Topos Theory. Chapter 1.2 of DCCT gives a pretty good motivation and some nice examples of sheaves on $\mathsf{Cart}$, but I would recommend David Carchedi's course on Topos theory, which is the quickest course that I could find that covered most of the relevant material.
After learning Grothendieck topos theory one must then learn how to combine homotopy theory with topos theory, this was originally achieved using simplicial sheaves or presheaves, and then was abstracted to the definition of a model topos. For this there is
Now one must learn Infinity Topos Theory. This is harder to recommend resources for as there are much fewer. There are many places to find recommendations for resources on learning infinity category theory, but honestly you don't need to delve into too much of the details to understand much for DCCT, you can really take much of infinity category theory as a black box that just works like usual category theory but with equivalences instead of isomorphisms and homotopy type mapping spaces instead of hom sets. I'd recommend Rezk's notes on quasicategories, get comfortable with the basics, and then watch Rezk's lectures on Youtube, as well as Joyal's.
I'm not saying you need to look at all of these resources, but what you need to know is contained in them. I would start off by first looking at some of Schreiber's previous papers that he mixed into DCCT, like Cech Cocycles for Differential Characterstic Classes and the Principal Infinity Bundles papers 1 and 2.
Addendum: Morally, the use of higher topos theory in DCCT is as a generalization of the nonabelian cohomology of Grothendieck and Giraud. In DCCT, these generalized cohomology classes are given by higher principal bundles. The use of higher category theory makes the formulation of this theory very elegant, but ultimately it is grounded in the theory of stacks and gerbes. Knowing this theory is not necessary for understanding DCCT or HTT, but it is a great way to build motivation and see what a lot of this theory is actually being used for. Here are some references in the differential geometry setting: