For discussion of topoi in literary theory, see literary topos.

In mathematics, a topos (plural: topoi or toposes - this is a contentious topic) is a type of category which allows the formulation of all of mathematics inside it.

Introduction

Traditionally, mathematics is built on set theory, and all objects studied in mathematics are ultimately sets and functions. It has been argued that category theory could provide a better foundation for mathematics. By analyzing precisely which properties of the category of sets and functions are needed to express mathematics, one arrives at the definition of topoi, and one can then formulate mathematics inside any topos. Of course, the category of sets forms a topos, but that is boring. In more interesting topoi, the axiom of choice may no longer be valid, or the law of excluded middle (every proposition is either true or false) may break down. It is thus of some interest to collect those theorems which are valid in all topoi, not just in the topos of sets.

One may also work in a particular topos in order to concentrate only on certain objects. For instance, constructivists may be interested in the topos of all "constructible" sets and functions in some sense. If symmetry under a particular group G is of importance, one can use the topos consisting of all G-sets. Another important example of a topos (and historically the first) is the category of all sheaves of sets on a given topological space.

It is also possible to encode a logical theory, such as the theory of all groups, in 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.

History

Main article: Background and genesis of topos theory

The historical origin of topos theory is algebraic geometry. Alexander Grothendieck generalized the concept of a sheaf. The result is the category of sheaves with respect to a Grothendieck topology - also called a Grothendieck topos. F. W. Lawvere realized the logical content of this structure, and his axioms led to the current notion. Note that Lawvere's notion, initially called elementary topos, is more general than Grothendieck's, and is the one that's nowadays simply called "topos".

Formal definition

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.

Further examples

There is one major class of examples of topoi that wasn't listed in the introduction: 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.

Example from logic should go here

References

• John Baez: Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html. A gentle introduction.

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".

Below follows a list of interesting research books that are providing introductions to topos theory (or to a specific aspect of it), but which do not primarily focus on 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 (Studies in logic and the foundations of mathematics vol. 98.), North-Holland, New York, 1984. 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 (http://www.mcs.vuw.ac.nz/~rob/): Topoi, the Categorical Analysis of Logic (http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3).

• 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.

The following are 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. Johnstones overwhelming compendium. Currently two of the scheduled three volumes are available.

Finally, a number of books 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.