What does the connection to topology provide to topoi?

Question

What does the connection to topology provide to topoi? I've been told by a professor that the connection to topology is one of the key features of topoi, and it provides a lot of the intuition and motivation for the subject, but topoi would still provide a powerful tool for abstracting away from the specific details of a particular system in order to focus on its general properties (because of group homomorphism?).

I am asking this question, because I didn't develop an intuition to what exactly topology seem to bring to the table as I didn't learn topology and only have a vague understanding of it. Is there a simple example that could perhaps show why topology is extremely useful to topoi?

Answer 1

Topology provides quite a few connections with topos theory.

First, topological spaces provide the original examples of toposes: sheaves on a topological space. These toposes have interesting properties in their own right, and they are some of the first examples one learns about in topos theory.

Second, topology inspires the notion of a geometric morphism, which is one of the kinds of maps between toposes that is heavily studied in topos theory. When we have two sober spaces $X$ and $Y$, the geometric morphisms between the two toposes $Sh(X) \to Sh(Y)$ are (almost) in bijection with the continuous maps $X \to Y$ in a canonical and natural way. But geometric morphisms are useful when studying all sorts of toposes, not just sheaves on a space.

Third, there are all kinds of applications of topos theory to algebraic geometry. Algebraic geometry perhaps isn’t strictly topology, but it uses many topological tools (and many algebraic ones, as its name implies).

You may also be interested in this question on MathOverflow.

Answer 2

At some point in our mathematics education, we learn that:

• (A) The characteristic polynomial of every real symmetric matrix has a zero.
• (B) Every real symmetric matrix has an eigenvector.

We may then move on to continuous families of real symmetric matrices, i.e. continuous maps $M : \Omega \to \mathrm{Sym}(n)$ from some parameter space $\Omega$ (for instance some subset of $\mathbb{R}^k$) to the set of real symmetric $n \times n$ matrices. For a concrete example of such a continuous family, see this answer.

Given such a continuous family of real symmetric matrices, we may then ask:

• (A') Can zeroes of the characteristic polynomials of the matrices $M(\theta)$ locally be picked continuously? (That is: Does every point $\theta \in \Omega$ have an open neighborhood $U \subseteq \Omega$ such that there is a continuous function $f : U \to \mathbb{R}$ such that for every point $\theta \in U$, $f(\theta)$ is a zero of the characteristic polynomial of $M(\theta)$?)

• (B') Can eigenvectors of the matrices $M(\theta)$ locally be picked continuously? (That is: Does every point $\theta \in \Omega$ have an open neighborhood $U \subseteq \Omega$ such that there is a continuous function $f : U \to \mathbb{R}^n$ such that for every point $\theta \in U$, $f(\theta)$ is an eigenvector of $M(\theta)$?)

The answer is: Yes for (A'), no for (B').

Why is that? The connection between topos theory and topology gives a conceptual explanation for this fact.

Namely: Unlike statement (B), which is only provable in classical mathematics, statement (A) also has a proof in the context of constructive mathematics, where finer distinctions (such as between $\neg\neg\varphi$ and $\varphi$) become visible. It is then a fact of topos theory that constructively provable statements always entail their globalized version, referring to continuous families of the objects in question.

The story does not end here. While statement (B) does not admit a constructive proof, the following statement does:

• (C) Every real symmetric matrix does not not have an eigenvector.

By the connection between topos theory and topology, we also have:

• (C') On a dense open subset of $\Omega$, eigenvectors of the matrices $M(\theta)$ can locally be picked continuously.

So topos theory teaches us that in some way, double negation from constructive mathematics corresponds to restricting to dense open subsets in topology.

This is just the tip of an iceberg. To open up a different example, consider the following statements.

• (D) Every finitely generated vector space has a basis.
• (E) Every finitely generated vector space does not not have a basis.

Statement (D) is only available in classical mathematics, while statement (E) is also provable in constructive mathematics. Hence the globalization (E') of (E) holds, while the globalization (D') of (D) does not.

• (D') Every finitely generated module $M$ over a reduced ring $A$ is locally finite free. (That is, there exists a partition $1 = f_1 + \ldots + f_n$ of unity such that for each index $i$, the localized module $M[f_i^{-1}]$ is finite free over $A[f_i^{-1}]$.)

• (E') Every finitely generated module $M$ over a reduced ring $A$ is locally finite free on a dense open. (That is, if $f = 0$ is the only element of $A$ such that $M[f^{-1}]$ is finite free over $A[f^{-1}]$, then already $1 = 0$ in $A$. Phrased contrapositively: If $1 \neq 0$ in $A$, then there is a nonzero element $f \in A$ such that $M[f^{-1}]$ is finite free over $A[f^{-1}]$.)

Statement (E') is known as (a version of) Grothendieck's Generic Freeness Lemma. Hence topos theory allows us to conclude from the rather simple observation that statement (E) has a constructive proof a nontrivial theorem in algebraic geometry.