Topos

In mathematics, a Grothendieck topos ( , ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notion of localization. Grothendieck topoi find applications in algebraic geometry. They are generalized by elementary topoi, which are used in logic. The mathematical field that studies topoi is called topos theory.

Equivalent definitions

A Grothendieck topos is a category C which satisfies any one of the following three properties. (A theorem of Jean Giraud states that the properties below are all equivalent.) There is a small category D and an inclusion C\hookrightarrow \operatorname{Presh}(D) that admits a finite-limit-preserving left adjoint. C is the category of sheaves on a Grothendieck site. C satisfies Giraud's axioms, below. Here \operatorname{Presh}(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf.

Giraud's axioms

Giraud's axioms for a category C are: C has a small set of generators, and admits all small colimits. Furthermore, fiber products distribute over coproducts; that is, given a set I , an I -indexed coproduct mapping to A , and a morphism A'\to A , the pullback is an I -indexed coproduct of the pullbacks: \left(\coprod_{i\in I} B_i\right)\times_A A'\cong\coprod_{i\in I}(B_i\times_A A'). Sums in C are disjoint. In other words, the fiber product of X and Y over their sum is the initial object in C . All equivalence relations in C 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(Y, R) → Hom(Y, X) × Hom(Y, X) gives an ordinary equivalence relation on the set Hom(Y, X). 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 R \to X \times_{X/R} X \,\! 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.

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 (see below for the definition of points of a topos).

Geometric morphisms

If X and Y are topoi, a geometric morphism u:X\to Y is a pair of adjoint functors (u ∗ ,u ∗ ) (where u ∗ : Y → X is left adjoint to u ∗ : X → Y) 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 X and Y are topological spaces and u 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 \text{Open}(X),\text{Open}(Y) .

Points of topoi

A point of a topos X is defined as a geometric morphism from the topos of sets to X . 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 (the "skyscraper sheaf" functor), 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 (X)_{et} of a space X , a point is a bit more refined of an object. Given a point x:\text{Spec}(\kappa(x)) \to X of the underlying scheme X a point x' of the topos (X)_{et} is then given by a separable field extension k of \kappa(x) such that the associated map x':\text{Spec}(k) \to X factors through the original point x . Then, the factorization map \text{Spec}(k) \to \text{Spec}(\kappa(x)) 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 (the stage of definition) to X. There are enough of these to display the space-like aspect. For example, if X is the classifying topos S[T] for a geometric theory T, then the universal property says that its points are the models of T (in any stage of definition Y).

Essential geometric morphisms

A geometric morphism (u ∗ ,u ∗ ) is essential if u ∗ has a further left adjoint u ! , or equivalently (by the adjoint functor theorem) if u ∗ preserves not only finite but all small limits.

Ringed topoi

A ringed topos is a pair (X,R), 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

Michael Artin and Barry Mazur associated to the site underlying a topos a pro-simplicial set (up to homotopy). (It's better to consider it in Ho(pro-SS); see Edwards) 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 (if the scheme is Noetherian and geometrically unibranch), this pro-simplicial set is pro-finite.

Notes

References

Some gentle papers: A gentle introduction. Steven Vickers: "Toposes pour les nuls" and "Toposes pour les vraiment nuls." Elementary and even more elementary introductions to toposes as generalized spaces. The following texts provide introductions to toposes and the basics of category theory. Their intended audiences include those knowing little mathematical logic and set theory. An "introduction to categories for computer scientists, logicians, physicists, linguists, etc." (cited from cover text). Introduces the foundations of mathematics from a categorical perspective. Grothendieck foundational work on topoi: Tome 2 270 The following monographs include an introduction to some or all of topos theory, but do not cater primarily to beginning students. Listed in (perceived) order of increasing difficulty. A nice introduction to the basics of category theory, topos theory, and topos logic. Assumes very few prerequisites. A good start. Available online at Robert Goldblatt's homepage. Version available online at John Bell's homepage. More complete, and more difficult to read. (Online version). More concise than Sheaves in Geometry and Logic, but hard on beginners. Reference works for experts, less suitable for first introduction: The third part of "Borceux' remarkable magnum opus", as Johnstone has labelled it. Still suitable as an introduction, though beginners may find it hard to recognize the most relevant results among the huge amount of material given. For a long time the standard compendium on topos theory. However, even Johnstone describes this work as "far too hard to read, and not for the faint-hearted." As of early 2010, two of the scheduled three volumes of this overwhelming compendium were available. Books that target special applications of topos theory: Includes many interesting special applications.

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