Topos

A topos (plural: topoi) in mathematics is a type of category which allows to formulate 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 the law of the 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. If symmetry under a particular group G is of importance, one can use the topos consisting of all G-sets.

History

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 Grothendieck topos. F. W. Lavwere realized the logical content of this structure, and his axioms (elementary topos) lead to the current notion.

Formal definition

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

All limits and colimits taken over finite index sets exist.
Exponentials exist.
A subobject classifier exists.

References:

John Baez: Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html. A gentle introduction.
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.
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, Theories and Triples,Springer, 1985. Corrected online version at http://www.cwru.edu/artsci/math/wells/pub/ttt.html. More concise than Sheaves in Geometry and Logic