# Research

## The internal language of toposes

Very briefly, a topos is a category 𝓔
which is sufficiently rich to support an *internal language*, a
device which allows to pretend that the objects of the topos are plain
old sets and that the morphisms of the topos are plain old maps between
those sets. We write “𝓔 ⊨ φ” to express that the statement φ holds from
the internal perspective of 𝓔. We can picture toposes as universes in
which we can do mathematics in, since the internal language supports
logical reasoning: Any (intuitionistically) provable statement holds in
the internal language of any topos.

The prototypical topos is Set, the category of sets. Its internal language is the usual mathematical language, in that 𝓔 ⊨ φ holds if and only if φ holds in the usual sense.

A further example is Eff, the effective
topos. A statement holds in Eff if and only if it can be
*witnessed* by a Turing machine. For instance, the statement “for
any number, there exists a prime larger than it” holds in Eff, since
there is a Turing machine which given a number computes a larger prime
number. In contrast, the statement “any Turing machine halts or doesn’t
halt” is false in Eff, since its external meaning is that there exists a
Turing machine which determines whether a given Turing machine halts or
doesn’t halt. But such a halting oracle doesn’t exist. This example
demonstrates that generally, toposes only support intuitionistic
reasoning.

You can learn more about the effective topos here.

## Applications in commutative algebra

Let *A* be a commutative ring with unit. The
topos Sh(Spec(*A*)) of set-valued sheaves over the spectrum
of *A* contains a mirror image of *A*, the structure
sheaf 𝓞_{Spec(A)}. From the point of view
of Sh(Spec(*A*)), this sheaf of rings is an ordinary ring. To a
first approximation, this ring can be thought of as a reification of all
stalks of *A*. For instance, 𝓞_{Spec(A)} is an
integral domain from the point of view of Sh(Spec(*A*)) if and
only if all stalks of *A* are integral domains.

However, the sheaf 𝓞_{Spec(A)} also has unique
properties not shared by *A* or its stalks, and this is where the
internal point of view derives most of its uses and its interest from.
One of the most important such properties is
that 𝓞_{Spec(A)} is almost a field, in the sense that,
from the point of view of Sh(Spec(*A*)), any element
of 𝓞_{Spec(A)} which is not invertible is nilpotent.
If *A* is assumed to be a reduced ring,
then 𝓞_{Spec(A)} is reduced as well, in which
case 𝓞_{Spec(A)} really is a field.

This observation can be exploited, for instance, to give a constructive one-paragraph proof of Grothendieck’s generic freeness lemma, basically because Grothendieck’s generic freeness lemma is trivial for fields. The previously known proofs are somewhat convoluted and proceed in a series of reduction steps, while the new proof is direct and even constructive.

## Applications in algebraic geometry

## Towards synthetic algebraic geometry

## Details

Details can be found in my PhD thesis. Large portions require only familiarity with scheme theory, not with topos theory.

**PhD thesis**- slides
- more slides
- 2015 talk at the IHÉS
- Paper: An elementary and constructive proof of Grothendieck’s generic freeness lemma

I’m always happy to answer questions and discuss topos-related matters by mail.

I’m keeping a research diary. It’s public, but probably of limited usefulness to others.