Topos

For discussion of topoi in literary theory, see literary topos. For discussion of topoi in rhetorical invention, see Inventio.

In mathematics, a topos (plural "topoi" or "toposes") is a type of category that behaves like the category of sheaves of sets on a topological space. For a discussion of the history of topos theory, see the article Background and genesis of topos theory.

Grothendieck topoi (topoi 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 that 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 intuition is very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the intuition. The greatest single success of this programmatic idea to date has been the introduction of the étale topos of a scheme.

Equivalent formulations

Let C be a category. A theorem of Giraud states that the following are equivalent:

There is a category D and an inclusion C $\hookrightarrow$ Presh(D) that admits a left adjoint.
C is the category of sheaves on a Grothendieck site.
C satisfies Giraud's axioms, below.

A category with these properties is called a "(Grothendieck) topos". Here 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 arbitrary colimits. Furthermore, colimits commute with base change.
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 $R \to X \times X$ in C such that all the maps $Hom(Y,R) \to Hom(Y,X) \times Hom(Y,X)$ are equivalence relations of sets. Since C has colimits we may form the coequalizer of the two maps $R \to X$ ; 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.

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.

More exotic examples, and the raison d'être of topos theory, come from algebraic geometry. To a scheme and even a stack one may associate an étale topos, an fppf topos, a Nisnevich topos...

=Counterexamples

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 Deligne of a nontrivial topos that has no points.

Geometric morphisms

If X and Y are topoi, a geometric morphism u:X $\to$ Y is a pair of adjoint functors (u^*,u_*) such that u^* preserves limits and colimits. Note that u^* automatically preserves colimits by virtue of having a right adjoint.

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.

Points of topoi

A point of a topos X is 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 it's stalk F_x has a right adjoint (the "skyscraper sheaf" functor), so an ordinary point of X also determines a topos-theoretic point.

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 etale topoi of Deligne-Mumford stacks.

Homotopy theory of topoi

Michael Artin and Barry Mazur associated to any 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 pro-simplicial set associated to the etale topos of a scheme is a pro-finite simplicial set. Its study is called etale homotopy theory.

Elementary topoi (topoi in logic)

Introduction

A traditional axiomatic foundation of mathematics is set theory, in which all mathematical objects are ultimately represented by sets (even functions which map between 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 alternate 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. 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, if not illuminating:

A topos is a category which has the following two properties:

All limits taken over finite index categories exist.
Every object has a power object.

From this one can derive that

All colimits taken over finite index categories exist.
The category has a subobject classifier.
Any two objects have an exponential object.
The category is cartesian closed.

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

Further examples

If C is a small category, then the functor category Set^C (consisting of all covariant functors from C to sets, with natural transformations as morphisms) is a topos. For instance, the category of all directed graphs is a topos. A graph consists of two sets, an arrow set and a vertex set, and two functions between those sets, assigning to every arrow its start and end vertex. The category of graphs is thus equivalent to the functor category Set^C, where C is the category with two objects joined by two morphisms.

The categories of finite sets, of finite G-sets and of finite directed graphs are also topoi.

References

John Baez: Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html. A gentle introduction.
Stephen Vickers: Toposes pour les nuls and Toposes pour les vraiment nuls. Available at Vickers’ website. Elementary and even more elementary introductions to toposes as generalized spaces.

The following textbooks provide easy paced first introductions (including basics of category theory). They should be suitable for students of various—even non-mathematical—disciplines:

F. William Lawvere and Stephen H. Schanuel: Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, Cambridge, 1997. An "introduction to categories for computer scientists, logicians, physicists, linguists, etc." (cited from cover text).
F. William Lawvere and Robert Rosebrugh: Sets for Mathematics, Cambridge University Press, Cambridge, 2003. Discusses the foundations of mathematics from a categorical perspective. A book "for students who are beginning the study of advanced mathematical subjects".

The original work of Grothendieck

Grothendieck and Verdier: Théorie des topos et cohomologie étale des schémas (known as SGA4)". New York/Berlin: Springer, ??. (Lecture notes in mathematics, 269–270)

Interesting research books that are provide introductions to topos theory (or to a specific aspect of it), but which do not primarily cater to students. The given order roughly (!) reflects the difficulty of the level of exposition:

Colin McLarty: Elementary Categories, Elementary Toposes, Clarendon Press, Oxford, 1992. Includes a nice introduction of the basic notions of category theory, topos theory, and topos logic. Assumes very few prerequisites.
Robert Goldblatt: Topoi, the Categorial Analysis of Logic. North-Holland, New York, 1984. (Studies in logic and the foundations of mathematics, 98.). A good start.

This book is now out of print and the copyright has reverted to the author. It can be accessed freely on Robert Goldblatt's homepage: Topoi, the Categorical Analysis of Logic.

John L. Bell: The development of categorical logic. http://publish.uwo.ca/~jbell/catlogprime.pdf (PDF)
Saunders Mac Lane and Ieke Moerdijk: Sheaves in Geometry and Logic: a First Introduction to Topos Theory, Springer, New York, 1992. More complete, and more difficult to read.
Michael Barr and Charles Wells: Toposes, Triples and Theories, Springer, 1985. Corrected online version at http://www.cwru.edu/artsci/math/wells/pub/ttt.html. More concise than Sheaves in Geometry and Logic, but not an easy reading for the beginner.

Works which serve as a reference for experts in the field rather than as a treatment suitable for first introduction:

Francis Borceux: Handbook of Categorical Algebra 3: Categories of Sheaves, Volume 52 of the Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1994. 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.
Peter T. Johnstone: Topos Theory, L. M. S. Monographs no. 10, Academic Press, 1977. For a long time the standard compendium on topos theory. However, it has also been described as "far too hard to read, and not for the faint-hearted", as quoted by Johnstone himself.
Peter T. Johnstone: Sketches of an Elephant: A Topos Theory Compendium, Oxford Science Publications, Oxford, 2002. Johnstone’s overwhelming compendium. As of early 2006, two of the scheduled three volumes were available.

Books that target special applications of topos theory:

Maria Cristina Pedicchio and Walter Tholen (editors): Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory. Volume 97 of the Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2004. Includes many interesting special applications.

Categorical logic | Sheaf theory

Topos (Mathematik) | Topos

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