Topos

Topos ( pl. topoi in Greek ) is a specific concept of category theory, which occurs in two closely related forms, namely

• As an elementary topos is a generalized category of all sets, with the aim of a non- set-theoretic foundations of mathematics.
• As a Grothendieck topos is a generalized topological space and finds applications in algebraic geometry.

Elementary topos

Motivation

The idea of an elementary topos came originally from William Lawvere, which in 1963 the goal sat down, the math to put it on a purely theoretical foundation class ( instead of the usual up to now set theory ). In collaboration with Myles Tierney, he finally formulated in the late 60s, the axioms for an elementary topos. This is, in a kind of universe ( speaking informally ) in which it is possible to do mathematics simplified. An elementary topos has enough structure to define therein an abstract concept of a set and thus to do mathematics and logic. In particular, an elementary topos has a so-called internal logic that does not have to be necessarily classical.

Definition

An elementary topos is a category with

• (a) a pullback for each diagram;
• (b) a terminal object;
• ( c ) an object, called the subobject classifier (literally from engl subobject classifier. ) and a monomorphism, so that for every monomorphism a unique arrow (called the character of ) exists such that the following diagram is a pullback:

In which case the unique arrow call from the terminal object;

• ( d ) an exponential with associated evaluation arrow for two objects with the universal property that for each object and each arrow is exactly one arrow exists, so that the following diagram commutes:

Where the identity arrow of call.

The properties (a) and ( b ) can be summarized briefly by saying was finally complete (ie all finite Limites exist). The properties ( c ) and ( d ) seem to be extremely artificial and abstract at first, but both are motivated by the category of all sets. Less is often writes for ( d ) that the functor for all a right adjunction (usually referred to ) has.

The original definition of an elementary topos also contained the requirement that this should be finally kokomplett (ie, that all finite colimits exist). This requirement follows, however, for a non- trivial result of Mikkelsen.

Elementary topos as an abstraction of the category of all sets

As I said, a category theoretical foundation for mathematics should be placed with the help of topos theory. This means in particular that the category of all sets must be described by it. Accordingly, this is probably the most important example in terms of the motivation of the various concepts of topos theory. In is simply the set of all functions from to and act accordingly. Further (note that here to be understood as finite ordinal ), and the conventional characteristic function of a subset of the.

The property that contains only two elements means that it is in a so-called Boolean topos and is essential for classical mathematics (classical in the sense of non- intuitionistic ).

To abstract excel over general Elementartopoi, the following axioms are commonly used.

• There is an initial object ( non-triviality ).
• Are arrows, as is or there is a with ( well Dotted slope).
• There is an object of natural numbers.
• Exists an arrow with ( Axiom of Choice ) For every epimorphism.

Grothendieck topos

A Grothendieck topos is defined as a category that is equivalent to the category of sheaves ( of sets ) on a situs. By a theorem of Jean Giraud is a category if and only a Grothendieck topos if the following properties are satisfied:

• ( a) exist finite projective Limites.
• ( b ) In any coproducts exist, and they are disjoint universal.

• ( c ) equivalence relations in are universally effective.

• ( d) has a generating family of objects.

It should be noted that every Grothendieck topos is always also an elementary topos.