Department of Mathematics, University of Chicago, 5734 University av., Chicago IL 60637, USA
Universiteit Utrecht, Fac. Wiskunde & Informatica, Postbus 80010, 35 08 TA Utrecht, The Netherlands e-mail: moerdijk@math.ruu.nl
Contents
This chapter will introduce topos theory, which arose from two separate
explorations; first, Lawvere's proposal to replace the membership axioms
for set theory by axioms on the composition of functions; and second, the
Grothendieck initiatives in algebraic geometry. Indeed, Grothendieck and
his collaborators [] needed to use cohomological ideas for
algebraic varieties, not just for topological spaces, and in this context
they introduced the notion of a topos as a generalized topological space.
In topology the cohomology groups of a space, originally defined for a
``constant'' coefficient group, soon required coefficients which varied;
these were codified by Leray and Cartan as sheaves on the space X; they
were sheaves of abelian groups or of modules, but they could be described
in terms of the more general (and simpler) sheaves of sets on the space.
The category 
of all such sheaves of sets on X is an example of a
topos. For algebraic geometry Grothendieck needed more general
``tolopogies'', defined not by open sets but by more general
``coverings'', and the sheaves of sets defined by such coverings provided
the general notion of a ``Grothendieck topos'' (and the slogan: a topos is
what a topologist needs to study, [], p. 301).
For many purposes, a space X can be replaced by the corresponding
topos 
. For example, continuous mappings
between spaces
correspond exactly to ``geometric'' morphisms 
of topoi.
The generalization, from spaces X and their sheaves to Grothendieck
topoi, has proved extremely useful in algebraic geometry, as for example in
P. Deligne's 1974 solution of the famous Weil conjectures about the
solutions of Diophantine equations. These geometric ideas will be
introduced in Section (cohomologies for a topos)
and Section (the general fundamental group).
But a topos is not only a generalized space; it can also be viewed as a generalized ``universe'' of sets -- as indeed the sheaves on a one-point space form the classical category of sets. This viewpoint appeared when Lawvere and Tierney developed an axiomatic treatment of sheaf theory without explicit reference to the specific Grothendieck topology used to define these sheaves. They also discovered that the process of turning a ``presheaf'' into a sheaf was implicitly involved in the notion of forcing used by Cohen in his proof of the independence of the Continuum Hypothesis -- and by Scott and Solovay in the corresponding ``Boolean valued'' models of set theory. This led to the discovery of the more general ``elementary'' topoi described by first order (elementary) axioms. Any such topos is a universe in which one can do mathematics, classical except for the restriction that the logic in such a topos is in general ``constructive'' or ``intuitionistic''.
This chapter will begin with the Lawvere--Tierney definition of an elementary topos, followed by the definition of Grothendieck topologies and their sheaves. The third section of the chapter then describes the mappings between topoi, called geometric morphisms. Since topologies can be described in terms of open sets, with little mention of points, it is possible to discuss ``pointless'' spaces, or locales, and the corresponding topoi of sheaves (Section ). The next section discusses representations of topoi in terms of such pointless spaces, including the theorem of Freyd showing how every topos can be embedded, in a suitably nice way, in the category of equivariant sheaves on some locale. In Section the sheaf cohomology groups of an arbitrary topos are introduced. Certain basic spectral sequences relate these groups to Cech cohomology and to Verdier's cohomology of ``hypercovers''. These can be employed to define certain pro-groups, which are the ``étale'' homotopy groups of a topos, matching the Grothendieck fundamental group, to be discussed in Section . The final section describes the intimate relation between a topos and its ``logic'' -- with Heyting (not Boolean) algebras of subobjects and with quantifiers described as adjoints -- plus the ``typed'' language associated with a topos.
Our presentation is necessarily brief, and the interested reader is urged to consult further references. The material in Sections -- and is treated in detail in our recent book [], which also contains an extensive bibliography. Homotopy and cohomology of topoi are discussed extensively in [] (for the fundamental group), [] (vol. 2) and []. For basic notions the reader may consult [] for category theory, [] for homological algebra, and [] for algebraic geometry.