The third volume of Peter Johnstone's massive compendium of topos theory, "Sketches of an Elephant", is yet to be published. The volume is supposed to discuss cohomology and mathematical universes in the context of topos theory. While we wait for the publication of this volume, are there any alternative (hopefully just as comprehensive) references for the aforementioned subjects?
[Math] Alternatives to “Sketches of an Elephant” Volume 3
at.algebraic-topologyct.category-theorylo.logicreference-requesttopos-theory
Related Solutions
Many good books have already been mentioned; I like MacLane+Moerdijk as an introduction, and after that both books by Johnstone (in particular, Part C of the Elephant does a good job of connecting locale theory with topos theory). But I also wanted to mention Vickers' paper "Locales and Toposes as Spaces," which I think does a good job of connecting up the topology with the toposes and the logic in a way that isn't directly evident in many other introductions.
Here are quotes from three well-known sources.
Shoenfield, Mathematical Logic (1967), page 23:
We construct a model of $N$ by taking the universe to be the set of natural numbers and assigning the obvious individuals, functions, and predicates to the nonlogical symbols of $N$. This model is called the standard model of $N$, ...
(Emphasis in the original in all quotes.)
Kleene, Mathematical Logic (1967), page 200:
Since a formal system (usually) results in formalizing portions of existing informal or semiformal mathematics, its symbols, formulas, etc. will have meaning or interpretations in terms of that informal or semiformal mathematics. These meanings together we call the (intended or usual or standard) interpretation or interpretations of the formal system.
Kleene 1967 p. 207:
A we remarked in ยง 37, a formal system formalizing a portion of informal mathematics has an "intended" (or "usual" or "standard") interpretation. ... The informal mathematics that we aim to formalize in $N$ is elementary number theory. So for the intended interpretation, the variables range over the natural numbers $\{0, 1, 2, \ldots\}$, i.e. this set is the domain. ... The function symbol $'$ is interpreted as expressing the successor function $+1$, and $0$ ("zero"), $+$ ("plus"), $\cdot$ ("times") and $=$ (equals) have the same meanings as those symbols convey in informal mathematics.
Here Kleene explicitly speaks of the interpretation as referring to the numbers that were known informally before the axioms of $N$ were laid out.
Kaye, Models of Peano Arithmetic (1991), Chapter 1: "The standard model", p. 10:
The structure $\mathbb{N}$ (the standard model) is the $\mathcal{L}_A$ structure whose domain is the non-negative integers, $\{0, 1, 2, \ldots\}$ and where the symbols in $\mathcal{L}_A$ are given their obvious interpretation.
In contemporary practice, in formal arithmetic, it is normal practice to use the term "natural numbers" and the symbol $\mathbb{N}$ to refer to the standard natural numbers, i.e. to identify them with the informal counting numbers (e.g. this is Kaye's convention, and many others'). The need for a distinction between standard and nonstandard models is particularly evident in my own field of Reverse Mathematics; we have a different convention that $\omega$ refers to the standard numbers and $\mathbb{N}$ refers to an arbitrary model at hand (e.g. Simpson's Subsystems of Second Order Arithmetic).
Part of the issue here may be that the meaning of the term "standard model" $\mathbb{N}$ can be interpreted in several ways. From the perspective of a certain kind of realism, it refers to the "actual" counting numbers. From the point of view of a certain kind of formalism, it refers to the natural numbers in whatever metatheory is being used at the moment, so that the "standard numbers" are the ones that are metafinite and "nonstandard models" have numbers that do not correspond to numbers in the metatheory. In any case, the notation $\mathbb{N} = \{0, 1,2, \ldots\}$ is intended to convey that $\mathbb{N}$ is identified with the usual counting numbers $0$, $1$, $2$, $\ldots$ from basic arithmetic, whatever we think those are.
Best Answer
As I said in the comment, this would involve a very large number of different references! (almost one by subsection...)
But to some extent, contributors to the nLab already started doing that and it is probably the best place to start if you are interested in material covered by this third volume: https://ncatlab.org/nlab/show/Elephant
On this page you have the announced table of content of Sketches of an elephant including volume 3, with most subsection references to page on the nLab covering the idea suggested by the title of the subsection. The nLab page itself will generally not contains enough information to reconstruct the content of the subsection but will surely contains references to other papers and books.