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Stream: deprecated: topos theory

Topic: Applications of Topos Theory to bundles

Francesco Bilotta (Nov 18 2020 at 15:25):

While reading the chapter on toposes in the beautiful Spivak and Fong's "Seven Sketches in compositionality", I bumped into the following:

Example 7.61. For a vector bundle $π : E → X$ over a space $X$, the corresponding sheaf
is $Sec(π)$ corresponding to its sections: to each open set $i_U : U ⊆ X$, we associate the set
of functions $s : U → E$ for which $\pi\circ s=i_U$. For example, in the case of the tangent
bundle $π : TM → M$, the corresponding sheaf, call it $VF$, associates
to each $U$ the set $VF(U)$ of vector fields on $U$.

The internal logic of the topos can then be used to consider properties of vector
fields. For example, one could have a predicate $Grad : VF → Ω$ that asks for the largest
subspace $Grad(v)$ on which a given vector field $v$ comes from the gradient of some
scalar function. One could also have a predicate that asks for the largest open set on
which a vector field is non-zero. Logical operations like $∧$ and $∨$ could then be applied
to hone in on precise submanifolds throughout which various desired properties hold,and to reason logically about what other properties are forced to hold there.

I would be particularly interested in knowing something more about the second part of the example I reported here. In particular if such an apporach to study submanifolds of interest (for instance associated to differential equations, as suggested) has indeed been carried out or this is "just" some suggestive intuition.
Any reference would be most appreciated.

Thanks in advance and pardon me if this question may sound inappropriate or trivial: I never studied Topos Theory!

sarahzrf (Nov 18 2020 at 16:54):

(tip: use double dollar signs if you want the latex to render properly—like here's some math: $$x + y$$)

sarahzrf (Nov 18 2020 at 16:58):

i know about some related things—i can tell you for sure that it's possible to formulate a bunch of algebraic geometry in terms of sheaves and topoi and internal logic (although i couldn't tell you much as far as nitty gritty details, only some beginning broad strokes)

sarahzrf (Nov 18 2020 at 17:00):

modal hott mmmmight be relevant to this? i know it can supposedly be used to reason about smooth stuff, although I don't know much detail

Eduardo Ochs (Nov 18 2020 at 17:23):

Hi @Francesco Bilotta,
The section II.3 in Mac Lane and Moerdijk's "Sheaves in Geometry and Logic" is about sheaves and manifolds... that book is considered to be relatively easy, but I think it is quite hard for people who know little CT... but it's worth a look anyway,

Peter Arndt (Nov 18 2020 at 18:44):

There is a wonderful article telling you how to use internal language of Sh(M) (M a manifold) for saying something about vector bundles on M: Mulvey, Intuitionistic algebra and representations of rings. It doesn't go into the direction of submanifolds determined by vector fields, but it is great to read. I will leave a link here, but it will stop working eventually: https://uni-duesseldorf.sciebo.de/s/l0qtvlzZWKLzw7f

Jens Hemelaer (Nov 18 2020 at 19:44):

The idea of looking where a vector field is nonzero is a good example, because it is not too difficult, and it still illustrates the basic ideas.

For a vector field, having no zeroes is a local property, or in other words:

• if a vectorfield has no zeroes on $U$, then it also does not have zeroes on $V$, for each $V \subseteq U$,
• if a vectorfield has no zeroes on $U_i$ for all $i \in I$, then it also does not have zeroes on $\bigcup_{i \in I} U_i$.

In sheaf language, this says that the vector fields $s : U \to E$ that have no zeroes form a subsheaf of the sheaf of all vector fields.
This makes "having no zeroes" a local property.

For $M$ a manifold, the topos $\mathbf{Sh}(M)$ has enough points (the points of the topos are the points of $M$). This means that you can check local properties by checking them in each point. In this case, this means that a vector field $s : U \to E$ has no zeroes if the stalks $s_p$ for $p \in U$ all do not have zeroes.

You don't need topos theory to prove any of the above properties, but it's a different way of looking at it.

Francesco Bilotta (Nov 18 2020 at 22:07):

Thanks to all for the refernces! I will surely look into them

Matteo Capucci (he/him) (Nov 19 2020 at 14:04):

Spivak has published a book (co-authored by Schultz) on using the internal language of topoi to speak about time-evolving systems, including differential equations. The book is titled 'Temporal Type Theory'

Matteo Capucci (he/him) (Nov 19 2020 at 14:05):

The topos they use though is not the topos of sheaves on a manifold. A closer approach is pursued by Blechschmidt in its thesis 'Using the internal language of topoi in algebraic geometry', though this time he's not talking (explicitly) about differential equations

Francesco Bilotta (Nov 21 2020 at 08:33):

Thanks! I will look at it. The fact is that I am following a course on bundle theory where we were presented with a little of analysis on manifolds. I never really liked differential equations but I think that identifying them with submanifolds of the jet bundle is a beautiful move. So I somehow recalled this example. Maybe I could as more in general about connections between topos theory and (differential) fiber bundles. I have heard about synthetic differential geometry, but that seems to deal with objects which are not the classical ones