What are topoi?

I have been hearing a lot about the concept of "topos". I asked a friend of mine in the know and he said that topoi are a generalization of sheaves on a topological space. In particular, topoi were usefull when an actual topology was not available. Can anyone elaborate on this or make this idea more clear?

Solution 1:

Topoi can be looked at from many points of view.

• Topoi can be seen as categories of sheaves on (generalized) spaces. Indeed, the premier example of a (Grothendieck) topos is the category $\mathrm{Sh}(X)$ of set-valued sheaves on a topological space $X$. Instead of spaces, also sites work.

• Topoi can be seen as generalized spaces. For instance we have a functor from the category of topological spaces to the category of topoi, namely the functor $X \mapsto \mathrm{Sh}(X)$. This functor is fully faithful if we restrict to sober topological spaces. (Soberness is a very weak separation axiom. Every Hausdorff space is sober and so is every scheme from algebraic geometry.)

  Many geometrical concepts generalize to topoi, for instance there are: point of a topos, open and closed subtopos, connected topos, continuous map between topoi, coverings of topoi, ...

• Topoi can be seen as alternate mathematical universes. The special topos $\mathrm{Set}$, the category of sets and maps, is the usual universe. Any topos admits an "internal language" which can be used for working inside of a topos as if it consisted of plain sets. Any theorem which admits an intuitionistic proof (a proof not using the law of excluded middle or axiom of choice) is valid in any topos.

  For instance the statement "For any short exact sequence $0 \to M' \to M \to M'' \to 0$ of modules, the module $M$ is finitely generated if $M'$ and $M''$ are" is such a theorem and therefore also holds in the topos of sheaves on a ringed space. In this way it automatically yields the statement "For any short exact sequence $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F''} \to 0$ of sheaves of $\mathcal{O}_X$-modules, the sheaf $\mathcal{F}$ is of finite type if $\mathcal{F}'$ and $\mathcal{F}''$ are".

  In the internal language of some topoi, exotic statements such as "any map $\mathbb{R} \to \mathbb{R}$ is smooth" or "there exists a real number $\varepsilon$ such that $\varepsilon^2 = 0$ but $\varepsilon \neq 0$" hold. This is useful for synthetic differential geometry.

• Topoi can be seen as embodiments of logical theories: For any (so-called "geometric") theory $\mathbb{T}$ there is a classifying topos $\mathrm{Set}[\mathbb{T}]$ whose points are precisely the models of $\mathbb{T}$ in the category of sets, and conversely any (Grothendieck) topos is the classifying topos of some theory. The classifying topoi of two theories are equivalent if and only if the theories are Morita-equivalent.

I learned this from the nLab entry on topoi. The main examples for topoi are:

• The category $\mathrm{Set}$ of sets and maps.

• The category $\mathrm{Sh}(X)$ of set-valued sheaves on any site. Grothendieck conceived topoi because of this example – he needed it for etalé cohomology. The "etalé topology" on a scheme is not an honest topology, but a Grothendieck site.

• The effective topos associated to any model of computation. In the internal language of such a topos, the statement "for any natural number $n$, there is a prime number $p > n$" holds if and only of there is a program in the given model of computation which computes, given any number $n$, a prime number $p > n$. The statement "any map $\mathbb{N} \to \mathbb{N}$ whatsoever is given by a Turing machine" is true in many of those topoi.

Topoi can for instance be used

• in algebraic geometry to work with generalized topologies like the etalé topology,

• in logic to construct interesting models of theories,

• in computer science to compare models of computation,

• as tools to build bridges between different subjects of mathematics.

Very fine resources for learning about topoi include:

• Tom Leinster's informal introduction to topos theory. Start here!

• The textbook Sheaves in Geometry and Logic by Saunders Mac Lane and Ieke Moerdijk.

• The reference Sketches of an Elephant: A Topos Theory Compendium by Peter Johnstone.

If you are in a hurry, then enjoy Luc Illusie's two-page note in the AMS series "What is …?".