Topos

In mathematics, a topos is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization. The Grothendieck topoi find applications in algebraic geometry, and more general elementary topoi are used in logic.
The mathematical field that studies topoi is called topos theory.

Grothendieck topos (topos in geometry)

Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic. An important example of this programmatic idea is the étale topos of a scheme. Another illustration of the capability of Grothendieck topoi to incarnate the “essence” of different mathematical situations is given by their use as "bridges" for connecting theories which, albeit written in possibly very different languages, share a common mathematical content.

Equivalent definitions

A Grothendieck topos is a category which satisfies any one of the following three properties.

There is a small category and an inclusion that admits a finite-limit-preserving left adjoint.
is the category of sheaves on a Grothendieck site.
satisfies Giraud's axioms, below.

Here denotes the category of contravariant functors from to the category of sets; such a contravariant functor is frequently called a presheaf.

Giraud's axioms

Giraud's axioms for a category are:

has a small set of generators, and admits all small colimits. Furthermore, fiber products distribute over coproducts; that is, given a set, an -indexed coproduct mapping to, and a morphism, the pullback is an -indexed coproduct of the pullbacks:
Sums in are disjoint. In other words, the fiber product of and over their sum is the initial object in.
All equivalence relations in are effective.

The last axiom needs the most explanation. If X is an object of C, an "equivalence relation" R on X is a map R → X × X in C
such that for any object Y in C, the induced map Hom → Hom × Hom gives an ordinary equivalence relation on the set Hom. Since C has colimits we may form the coequalizer of the two maps R → X; call this X/''R''. The equivalence relation is "effective" if the canonical map
is an isomorphism.

Examples

Giraud's theorem already gives "sheaves on sites" as a complete list of examples. Note, however, that nonequivalent sites often give
rise to equivalent topoi. As indicated in the introduction, sheaves on ordinary topological spaces motivate many of the basic definitions and results of topos theory.

Category of sets and G-sets

The category of sets is an important special case: it plays the role of a point in topos theory. Indeed, a set may be thought of as a sheaf on a point since functors on the singleton category with a single object and only the identity morphism are just specific sets in the category of sets.
Similarly, there is a topos for any group which is equivalent to the category of -sets. We construct this as the category of presheaves on the category with one object, but now the set of morphisms is given by the group. Since any functor must give a -action on the target, this gives the category of -sets. Similarly, for a groupoid the category of presheaves on gives a collection of sets indexed by the set of objects in, and the automorphisms of an object in has an action on the target of the functor.

Topoi from ringed spaces

More exotic examples, and the raison d'être of topos theory, come from algebraic geometry. The basic example of a topos comes from the Zariski topos of a scheme. For each scheme there is a site whose category of presheaves forms the Zariski topos. But once distinguished classes of morphisms are considered, there are multiple generalizations of this which leads to non-trivial mathematics. Moreover, topoi give the foundations for studying schemes purely as functors on the category of algebras.
To a scheme and even a stack one may associate an étale topos, an fppf topos, or a Nisnevich topos. Another important example of a topos is from the crystalline site. In the case of the étale topos, these form the foundational objects of study in anabelian geometry, which studies objects in algebraic geometry that are determined entirely by the structure of their étale fundamental group.

Pathologies

Topos theory is, in some sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances of pathological behavior. For instance, there is an example due to Pierre Deligne of a nontrivial topos that has no points.

Geometric morphisms

If and are topoi, a geometric morphism is a pair of adjoint functors such that u^∗ preserves finite limits. Note that u^∗ automatically preserves colimits by virtue of having a right adjoint.
By Freyd's adjoint functor theorem, to give a geometric morphism X → Y is to give a functor u^∗: Y → X that preserves finite limits and all small colimits. Thus geometric morphisms between topoi may be seen as analogues of maps of locales.
If and are topological spaces and is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated topoi for the sites.

Points of topoi

A point of a topos is defined as a geometric morphism from the topos of sets to.
If X is an ordinary space and x is a point of X, then the functor that takes a sheaf F to its stalk F_x has a right adjoint
, so an ordinary point of X also determines a topos-theoretic point. These may be constructed as the pullback-pushforward along the continuous map x: 1 → X.
For the etale topos of a space, a point is a bit more refined of an object. Given a point of the underlying scheme a point of the topos is then given by a separable field extension of such that the associated map factors through the original point. Then, the factorization map is an etale morphism of schemes.
More precisely, those are the global points. They are not adequate in themselves for displaying the space-like aspect of a topos, because a non-trivial topos may fail to have any. Generalized points are geometric morphisms from a topos Y to X. There are enough of these to display the space-like aspect. For example, if X is the classifying topos S for a geometric theory T, then the universal property says that its points are the models of T.

Essential geometric morphisms

A geometric morphism is essential if u^∗ has a further left adjoint u_!, or equivalently if u^∗ preserves not only finite but all small limits.

Ringed topoi

A ringed topos is a pair, where X is a topos and R is a commutative ring object in X. Most of the constructions of ringed spaces go through for ringed topoi. The category of R-module objects in X is an abelian category with enough injectives. A more useful abelian category is the subcategory of quasi-coherent R-modules: these are R-modules that admit a presentation.
Another important class of ringed topoi, besides ringed spaces, are the étale topoi of Deligne–Mumford stacks.

Homotopy theory of topoi

and Barry Mazur associated to the site underlying a topos a pro-simplicial set. Using this inverse system of simplicial sets one may sometimes associate to a homotopy invariant in classical topology an inverse system of invariants in topos theory. The study of the pro-simplicial set associated to the étale topos of a scheme is called étale homotopy theory. In good cases, this pro-simplicial set is pro-finite.

Elementary topoi (topoi in logic)

Introduction

Since the early 20th century, the predominant axiomatic foundation of mathematics has been set theory, in which all mathematical objects are ultimately represented by sets. More recent work in category theory allows this foundation to be generalized using topoi; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set-theoretic mathematics. But one could instead choose to work with many alternative topoi. A standard formulation of the axiom of choice makes sense in any topos, and there are topoi in which it is invalid. Constructivists will be interested to work in a topos without the law of excluded middle. If symmetry under a particular group G is of importance, one can use the topos consisting of all G-sets.
It is also possible to encode an algebraic theory, such as the theory of groups, as a topos, in the form of a classifying topos. The individual models of the theory, i.e. the groups in our example, then correspond to functors from the encoding topos to the category of sets that respect the topos structure.

Formal definition

When used for foundational work a topos will be defined axiomatically; set theory is then treated as a special case of topos theory. Building from category theory, there are multiple equivalent definitions of a topos. The following has the virtue of being concise:
A topos is a category that has the following two properties:

All limits taken over finite index categories exist.
Every object has a power object. This plays the role of the powerset in set theory.

Formally, a power object of an object is a pair with, which classifies relations, in the following sense.
First note that for every object, a morphism induces a subobject. Formally, this is defined by pulling back along. The universal property of a power object is that every relation arises in this way, giving a bijective correspondence between relations and morphisms.
From finite limits and power objects one can derive that

All colimits taken over finite index categories exist.
The category has a subobject classifier.
The category is Cartesian closed.

In some applications, the role of the subobject classifier is pivotal, whereas power objects are not. Thus some definitions reverse the roles of what is defined and what is derived.