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Stream: learning: questions

Topic: how are Grothendieck toposes useful?

Peiyuan Zhu (Dec 23 2021 at 06:03):

Is there an example of why Grothendick topos is useful? Can it be used to describe general relativity, as an example?

Peiyuan Zhu (Dec 23 2021 at 06:58):

I'm reading this post of Andrej Bauer http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/. I don't quite understand the point he's trying to make of why 'law of excluded middle' doesn't hold under 'local truth' in the meteorologist example. Why exclude these isolated points? It sounds like something discussed in measure theory where null sets doesn't matter. What does it have to do with modality?

Matteo Capucci (he/him) (Dec 23 2021 at 09:58):

Peiyuan Zhu said:

Is there an example of why Grothendick topos is useful?

All of modern algebraic geometry is based on the idea of sheaf topoi. I'm no expert in this area so I won't delve into it.

Matteo Capucci (he/him) (Dec 23 2021 at 09:58):

Peiyuan Zhu said:

Can it be used to describe general relativity, as an example?

I don't know about GR, but people definitely do QM with it. See [[Bohr topos]] and here

Matteo Capucci (he/him) (Dec 23 2021 at 10:03):

Peiyuan Zhu said:

I'm reading this post of Andrej Bauer http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/. I don't quite understand the point he's trying to make of why 'law of excluded middle' doesn't hold under 'local truth' in the meteorologist example. Why exclude these isolated points? It sounds like something discussed in measure theory where null sets doesn't matter. What does it have to do with modality?

The isolated points exclude themselves!
It's because the interior of a set and the interior of its complement are not complimentary, their union misses the boundary of said set.
Indeed $\neg\{T > 273\} = int(\{T \leq 273\}) = \{T < 273\}$ and thus $\neg A \lor A \neq X$ in general.

Matteo Capucci (he/him) (Dec 23 2021 at 10:03):

Here it's crucial to distinguish $\neg$, whose semantics is 'interior of the complement', from $-^C$, which is just the set-theoretic complement

Matteo Capucci (he/him) (Dec 23 2021 at 10:04):

The reason $\neg$ is not $-^C$ is because the complement of an open set is, in general, not open, so you pcik the 'best open approximation' you have of it, namely its interior

Matteo Capucci (he/him) (Dec 23 2021 at 10:08):

The reason you want an open set is because you're doing logic with truth values being 'open sets of a space'. Ideally, $U \Vdash \phi(x)$ iff the predicate $\phi(x)$ is true on the open set $U$. For instance, if $\phi(x) =$ 'temperature is above 273K at location $x$', then $U \Vdash \phi(x)$ iff at every point $x \in U$, temperature is above 273K.
Hence

$$\text{Sahara desert} \Vdash \phi(x)$$

but

$$\text{Antarctica} \not\Vdash \phi(x)$$

Matteo Capucci (he/him) (Dec 23 2021 at 10:10):

The connection with modal logic comes from the fact in this context, truth is local, i.e. $\phi$ is true if it holds on every $U_i$ of an open cover $\mathcal U = \{U_i | i \in I\}$ that covers $X$ (i.e. $X=\bigcup_i U_i$)

Matteo Capucci (he/him) (Dec 23 2021 at 10:10):

This might seems unwarranted but there are good reasons to do so, mainly coming from geometry but also from logic

Matteo Capucci (he/him) (Dec 23 2021 at 10:11):

The first thing to realize is that, with this definition of truh, more things are true! This is because every $\phi$ which holds on $X$ ($X \Vdash \phi$) still holds (because $\{X\}$ is an open cover of $X$), but there might be some predicates $\psi$ that do not hold on the whole $X$ (globally) but do hold locally.

Peiyuan Zhu (Dec 23 2021 at 18:32):

Matteo Capucci (he/him) said:

This might seems unwarranted but there are good reasons to do so, mainly coming from geometry but also from logic

What is the geometric reason of doing so? Can you give an example?

Peiyuan Zhu (Dec 23 2021 at 18:40):

Matteo Capucci (he/him) said:

The reason you want an open set is because you're doing logic with truth values being 'open sets of a space'. Ideally, $U \Vdash \phi(x)$ iff the predicate $\phi(x)$ is true on the open set $U$. For instance, if $\phi(x) =$ 'temperature is above 273K at location $x$', then $U \Vdash \phi(x)$ iff at every point $x \in U$, temperature is above 273K.
Hence

$$\text{Sahara desert} \Vdash \phi(x)$$

but

$$\text{Antarctica} \not\Vdash \phi(x)$$

What's the meaning of the $\Vdash$ operator? Is it implication? Or 'turnstile' in sequent calculus?

Peiyuan Zhu (Dec 23 2021 at 18:43):

Matteo Capucci (he/him) said:

Here it's crucial to distinguish $\neg$, whose semantics is 'interior of the complement', from $-^C$, which is just the set-theoretic complement

So is intuitionistic negation an operator defined on the space now, which is different from complement. Is there a reason why we consider 'interior of complement' as negation of an event?

Patrick Nicodemus (Dec 24 2021 at 02:57):

Peiyuan Zhu said:

Is there an example of why Grothendick topos is useful? Can it be used to describe general relativity, as an example?

Topos theory is a framework for doing sheaf cohomology theory. So I will start by answering the question - why is sheaf cohomology useful?

Some theorems in geometry that were proved using sheaf cohomology were Cartan's theorem A and B. More generally Cartan and Oka developed a powerful theory based on sheaf cohomology that was used to resolve many problems in several complex variables.

This did not use Grothendieck topoi. Complex analytic manifolds are very nice spaces. Grothendieck invented topos theory to try and generalize these methods to attack problems in algebraic geometry where the spaces are less nice. I am not a number theorist so I cannot explain the Weil conjectures to you (which were Grothendieck's direct motivation) but the Oka-Cartan theory seems like it is an accessible place to begin. The book "Analytic functions of several complex variables" by Gunning and Rossi describes this theory in whole.

In algebraic geometry the paper "Faisceaux Algebriques Coherents" by Serre showed that these techniques could be used in algebraic geometry as well. Perhaps more relevant because this paper is more directly related to Grothendieck's work in topos theory.

Matteo Capucci (he/him) (Dec 24 2021 at 11:24):

Peiyuan Zhu said:

Matteo Capucci (he/him) said:

This might seems unwarranted but there are good reasons to do so, mainly coming from geometry but also from logic

What is the geometric reason of doing so? Can you give an example?

The geometric reason is that often geometric objects are only locally defined or satisfy property only locally. For instance, the tangent bundle $TX \to X$ of a smooth manifold $X$ is locally trivial (there is an open cover such that on each open set the bundle is isomorphic to one of the form $E\vert_U \times U \to U$) but it's not (globally) trivial (it's not of the from $E \times X \to X$), and so on.
Blechschmidt's Using the internal language of toposes in algebraic geometry is a very accessible introduction to these ideas, in that he discusses locality and the use of internal language explicitly (but the geometric objects themselves are far older). He also bring much more examples.

Matteo Capucci (he/him) (Dec 24 2021 at 11:25):

Peiyuan Zhu said:

What's the meaning of the $\Vdash$ operator? Is it implication? Or 'turnstile' in sequent calculus?

It's exactly what I defined here
Matteo Capucci (he/him) said:

$U \Vdash \phi(x)$ iff the predicate $\phi(x)$ is true on the open set $U$.

The symbol $\Vdash$ is read 'forces' sometimes, but it's not necessary to do so.

Matteo Capucci (he/him) (Dec 24 2021 at 11:28):

Peiyuan Zhu said:

So is intuitionistic negation an operator defined on the space now, which is different from complement. Is there a reason why we consider 'interior of complement' as negation of an event?

Yes:
Matteo Capucci (he/him) said:

The reason $\neg$ is not $-^C$ is because the complement of an open set is, in general, not open, so you pcik the 'best open approximation' you have of it, namely its interior

In other words, you want negation to be an operator on your algebra of truth values $\Omega$. So if $\Omega = \mathcal O(X)$, defining $\neg = -^C$ doesn't typecheck: in general $U^C \not\in \Omega$. So you define $\neg = int(-^C)$, the 'closest thing to $-^C$ which is in $\Omega$'

Zhen Lin Low (Dec 24 2021 at 11:45):

I think all this talk of "defining $\lnot$" is misleading. $\lnot$ already has a good, general definition: given an element $u$ of a lattice, $\lnot u$ is (if it exists) the largest element $v$ such that $u \land v = \bot$. For the lattice of all subsets of a set, this recovers the classical/boolean complement. For the lattice of open subsets of a topological space, it is the interior of the set-theoretic complement. For the lattice of logical propositions it is the logical negation. And so on.

Fawzi Hreiki (Dec 24 2021 at 14:09):

That’s the same definition as above no?

Zhen Lin Low (Dec 24 2021 at 23:31):

No. Making "interior of the complement" the definition makes it sound like $\lnot$ is being defined arbitrarily. It is a (easy) theorem that "interior of the complement" satisfies the definition,.

John Baez (Dec 25 2021 at 20:15):

Peiyuan Zhu said:

Is there an example of why Grothendick topoi are useful?

Grothendieck used Grothendieck topoi in algebraic geometry. He used them to help invent etale cohomology, a cohomology theory for algebraic varieties over finite fields that resembles the usual cohomology for algebraic varieties over $\mathbb{C}$. He wanted this to prove the Weil conjectures, which are a baby version of the Riemann Hypothesis.

Can they be used to describe general relativity, as an example?

Maybe they can be, but nobody has done it, so that's not a very good example to think about if you're starting to learn topos theory.

John Baez (Dec 25 2021 at 20:19):

I recommend learning a bit of the history:

• Colin McLarty, The uses and abuses of the history of topos theory.

John Baez (Dec 25 2021 at 20:19):

(If you register you can log in and read this paper for free.)

John Baez (Dec 25 2021 at 20:21):

@Patrick Nicodemus has already given a nice sketch of some of the algebraic geometry surrounding Grothendieck's invention of topoi.

Later, when Lawvere and Tierney invented 'elementary topoi', a lot of people started using topos theory to think about logic in new ways.

So, if you want to study applications of topos theory, two obvious places are algebraic geometry and logic. Another is [[synthetic differential geometry]], meaning roughly geometry with infinitesimals.

John Baez (Dec 25 2021 at 20:22):

If you're just trying to get oriented, you might also try this:

• John Baez, Topos theory in a nutshell.

John Baez (Dec 25 2021 at 20:26):

In general I recommend learning topos theory after you're comfortable with basic category theory, like limits, colimits, adjunctions, etc. Basic category theory is useful in many more ways than topos theory - and if you don't understand it, topos theory will seem hard.