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## Über dieses Buch

We dedicate this book to the memory of J. Frank Adams. His clear insights have inspired many mathematicians, including both of us. In January 1989, when the first draft of our book had been completed, we heard the sad news of his untimely death. This has cast a shadow on our subsequent work. Our views of topos theory, as presented here, have been shaped by continued study, by conferences, and by many personal contacts with friends and colleagues-including especially O. Bruno, P. Freyd, J.M.E. Hyland, P.T. Johnstone, A. Joyal, A. Kock, F.W. Lawvere, G.E. Reyes, R Solovay, R Swan, RW. Thomason, M. Tierney, and G.C. Wraith. Our presentation combines ideas and results from these people and from many others, but we have not endeavored to specify the various original sources. Moreover, a number of people have assisted in our work by pro­ viding helpful comments on portions of the manuscript. In this respect, we extend our hearty thanks in particular to P. Corazza, K. Edwards, J. Greenlees, G. Janelidze, G. Lewis, and S. Schanuel.

## Inhaltsverzeichnis

### Prologue

Abstract
A startling aspect of topos theory is that it unifies two seemingly wholly distinct mathematical subjects: on the one hand, topology and algebraic geometry, and on the other hand, logic and set theory. Indeed, a topos can be considered both as a “generalized space” and as a “generalized universe of sets”. These different aspects arose independently around 1963: with A. Grothendieck in his reformulation of sheaf theory for algebraic geometry, with F. W. Lawvere in his search for an axiomatization of the category of sets and that of “variable” sets, and with Paul Cohen in the use of forcing to construct new models of Zermelo-Frwnkel set theory.
Saunders Mac Lane, Ieke Moerdijk

### Categorical Preliminaries

Abstract
Before embarking on the actual topic of this book, we wish to review briefly the basic notions that will be used from category theory. Many readers will be familiar with these preliminaries; they should immediately start with Chapter I, referring back to these preliminaries whenever necessary. On the other hand, these preliminaries do not present sufficiently many examples and are by no means enough to constitute a proper introduction to category theory, and the reader who lacks sufficient categorical background is advised to first read some of the relevant parts of Mac Lane’s [CWM—Categories for the Working Mathematician 1971] (or some other such text), perhaps using the following pages as a guideline.
Saunders Mac Lane, Ieke Moerdijk

### I. Categories of Functors

Abstract
Many constructions on various mathematical objects depend not just on the elements of those objects but also on the morphisms between them. Such constructions can thus be effectively formulated in the corresponding category of objects. A “topos” is a category in which a number of the most basic such constructions (product, pullback, exponential, characteristic function,…) are always possible. With these constructions available, many other properties can be efficiently developed. Superficially quite different categories, arising in geometry, topology, algebraic geometry, group representations, and set theory, all turn out to satisfy the axioms defining such a topos.
Saunders Mac Lane, Ieke Moerdijk

### II. Sheaves of Sets

Abstract
This chapter starts with the notion of a sheaf F on a topological space X. Such a sheaf is a way of describing a class of functions on X- especially classes of “good” functions, such as the functions on (parts of) X which are continuous or which are differentiable. The description tells the way in which a function f defined on an open subset U of X can be restricted to functions f ∣v on open subsets V ⊂ U and then can be recovered by piecing together (collating) the restrictions to the open subsets Vi of a covering of U. This restriction-collation description applies not just to functions, but also to other mathematical structures defined “locally” on a space X.
Saunders Mac Lane, Ieke Moerdijk

### III. Grothendieck Topologies and Sheaves

Abstract
The notion of a sheaf, presented as in Chapter II in terms of coverings, restrictions, and collation, can be defined and used (e.g., for cohomology) not just on the usual topological spaces but also on more general “topologies”. This section is meant to provide some informal background and motivation for the development of these general topological ideas.
Saunders Mac Lane, Ieke Moerdijk

### IV. First Properties of Elementary Topoi

Abstract
In this chapter we present elementary conditions (or axioms) that make a category £ a topos, and then develop from these conditions and in a suitable order certain other basic properties. Most of these properties have already been seen to hold for our typical categories discussed in Chapter I, and for the categories of sheaves on a space (Chapter II) or on a site (Chapter III).
Saunders Mac Lane, Ieke Moerdijk

### V. Basic Constructions of Topoi

Abstract
In this chapter we present some basic ways to construct new topoi from old ones.
Saunders Mac Lane, Ieke Moerdijk

### VI. Topoi and Logic

Abstract
Topos theory involves both geometry, especially sheaf theory, and logic, especially set theory. This chapter will develop some of the connections with set theory and illustrate how geometric constructions such as sheafification are deeply involved in independence proofs for the axioms of set theory.
Saunders Mac Lane, Ieke Moerdijk

### VII. Geometric Morphisms

Abstract
In this chapter, we begin the study of the maps between topoi: the so-called geometric morphisms. The definition is modeled on the case of topological spaces, where a continuous map XY gives rise to an adjoint pair Sh(X) ⇄Sh(Y) of functors between sheaf topoi. The first two sections of this chapter are concerned mainly with a number of examples, and with the construction of the necessary adjunctions by analogues of the ®-Hom adjunction of module theory. In a third section, we consider two special types of geometric morphisms: the embeddings and the surjections. For these two types, there is a factorization theorem, parallel to the familiar factorization of a function as a surjection followed by an injection. Moreover, we prove that the embeddings FEε of topoi correspond to Lawvere-Tierney topologies in the codomain ε, while surjections FE correspond to left exact comonads on the domain.F.
Saunders Mac Lane, Ieke Moerdijk

### VIII. Classifying Topoi

Abstract
The idea of “classifying” geometric or algebraic structures or spaces by maps into a given space is familiar from topology. For example, for any abelian group 7 and any n, there is a classifying space K(π, n) for cohomology: for each space X, cohomology classes α ∈ H n (X, π) correspond to (“are classified by”) maps X→ K(π, n). After reviewing some of these topological examples in more detail, we introduce a similar notion of a classifying topos. Again, the idea is to classify structures over topoi by maps into one suitably constructed topos. For example, a topos R. is said to be a classifying topos for commutative rings when for any topos ε there is a natural equivalence between ring objects in E and geometric morphisms ER. An application of the results on continuous filtering functors from the previous chapter will construct such a classifying topos R; it will turn out to be the topos of set-valued functors on the familiar category of finitely presented commutative rings. This will follow from the fact that this category is “freely generated” by the polynomial ring Z[X], in a suitable sense to be formulated below (see Proposition 5.1).
Saunders Mac Lane, Ieke Moerdijk

### IX. Localic Topoi

Abstract
Among the Grothendieck topoi those of the form Sh(S) for some topological space S play a special (and motivating) role. In this chapter we consider a related class of topoi those of the sheaves on a so-called “locale”. In the case of a topological space S, a sheaf is a suitable functor on the lattice O(S) of open sets of S, where the lattice order is defined by the inclusion relation between open sets. Thus the notion of a sheaf can be explained just in terms of the open sets of S, without any use of its points. Any suitable such lattice (one which is complete, with an infinite distributive law) may be taken as defining a modified sort of topological space, a so-called “locale”. The beginning sections of this chapter provide an introduction to the study of such locales, motivated by the topological examples. It will turn out that a topological space is essentially determined by its lattice of open sets when that space S has the property of “sobriety”, but, beyond that point, spaces and locales diverge.
Saunders Mac Lane, Ieke Moerdijk

### X. Geometric Logic and Classifying Topoi

Abstract
A first-order formula q5(xl,xn)is called“geometric”if it is built up from atomic formulas by using conjunction,disjunction,and existential quantification,Geometric logic is the logic of the implications between geometric formulas:
$$\forall x(\phi (x) \to \psi (x))$$
(1)
where the arrow here is for “implication” and and z/) are geometric. Many mathematical structures can be axiomatized by formulas of this form (1).For instance, local rings are axiomatized by the usual equations for a commutative ring with unit, together with the axiom
$$\forall x,y \in R(x + y = 1 \to \exists z(x \cdot z = 1) \vee \exists z(y \cdot z = 1))$$
(2)
which states that the ring is local; this axiom (2) is indeed of the form (1).
Saunders Mac Lane, Ieke Moerdijk

### Backmatter

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