(For discussion of topoi in literary theory, see literary topos.
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2 History 3 Formal definition 4 Further examples 5 References |
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 groupss, 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.
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 lead 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".
A topos is a category which has the following two properties:
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 SetC (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 SetC, 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.
Introduction
History
Main article: Background and genesis of topos theoryFormal definition
From this one can derive that
Further examples
References
A gentle introduction.