I’ve resurrected this post from my draft graveyard after chatting with Chas Leichner about the lightly related notion of domain theory, and the interaction between computation and topos theory.

What is fiber bundle dynamics?

A fiber bundle expresses global phenomenon in terms of the output of local data.
Analogously, a geometric multi-scale modeling technique, aptly named fiber bundle dynamics, expresses macroscale data in terms of the output of microscale models.

For example: if the microscale model is molecular dynamics and the macroscale model is continuum hydrodynamics, then this formula is the Irving-Kirkwood formula that expresses stress in terms of the atomistic data from molecular dynamics. If the microscale model is replaced by Brownian dynamics, then this link is replaced by Kramer’s expression, etc.

“…in concurrent coupling methods, one does not compute the constitutive relation within the full range of these variables – only the values that actually occur in the simulation are needed, and these might be a very small subset of the entire range.”
— Ren, Seamless Multiscale Modeling of Complex Fluids via Fiber Bundle Dynamics

This collection of notes resulted from my desire for an intuitive grasp of basic concepts in topos theory, and is meant tocomplement to a standard introduction to topoi (i.e. this is pedagogically unsound).

The mechanical definition of a topos (a category with finite limits and power objects) doesn’t tell us anything interesting about topos theory. The longer definition boils down to this: An (elementary) topos is the category of types in some world of intuitionistic logic.

My desire to understand topoi is mainly due to this: casting any object into it’s appropriate topos allows one to use topological intuition to reason about it.

Since Tarski, 1938 it has been known that topologies — specifically the lattices of open sets for topological spaces — can provide models for intuitionistic propositional logic. [Source]

For situations where a topological intuition is very effective but an honest topological space is lacking, it is sometimes possible to find a topos formalizing the intuition.

Why? The idea is this:

Constructive reasoning allows maps to be treated as generalized points.

Locales give a better constructive topology (better results hold) than ordinary spaces.

The constructive reasoning makes it possible to deal with locales as though they were spaces of points.

space ∼ logical theory
point ∼ model of the theory
open set ∼ propositional formula
sheaf ∼ predicate formula
continuous map ∼ transformation of models that is definable within geometric logic

This post assumes knowledge of fiber bundles, the group action functor, groupoids, and basic vector calculus. I am in the process of learning the topics discussed below, and I deeply appreciate constructive feedback.

How does a big space cover a little one?

Given a covering space $E \to B$ we can uniquely lift any path in the base space (once you choose a starting point) to a path in $E$. Conversely, we can create a covering space of $B$ by letting the fiber over $b$ be $F(b)$.

Treating spaces as fiber bundles allows us to tame twisted beasts. Most of spin geometry is phrased in the language of fiber bundles, and this post will begin to introduce that language — extremely powerful in its simplicity.

Introduction to Fiber Bundles

If we glue lines onto every point $b$ in a circle (or a circle to every point of a line), we get a cylinder. In other words, a cylinder is the product space $S^1 \times [0,1]$.

If we glue lines onto every point of a circle, progressively twisting each individual line, we get a Mobius strip.

A fiber bundle with fiber $F$ consists of: 2 topological spaces, and a projection map which projects the total space onto its base space.

If you flip the arrow around, $\pi^{-1}$, the inverse image of the projection map, maps every $b$ in the basespace to its corresponding fiber $\pi^{-1}(b)$ in the total space.

Similarly, the $\pi^{-1}(N)$ maps every point in the neighborhood $N$ of $b$ to their corresponding fibers $\pi^{-1}(N)$ in the total space.

We can locally treat the Mobius strip as a plane, in the same way that we can locally treat a cylinder as a plane.

This property allows us to vastly simplify calculations; it allows us to locally treat twisted spaces like their non-twisted counterparts. We can generalize this property as follows:

For every $b \in B$ there is a neighborhood $N$ of $b$ s.t. the following diagram commutes.

Formally:

A fiber bundle (with structure group $G$ and fiber $F$) over $B$ is a smooth surjection $\pi: E \to B$ together with a local triviality condition: every $b \in B$ has a neighborhood $N$ and a diffeomorphism $\phi: \pi^{-1}(N) \to N \times F$ s.t. the following commutes:

Another notation commonly used to represent the fiber in E over $b$ is $E_b$

As an aside: How can we formally construct a twisted space?

A Mobius strip := $[0,1] \times [0,1] /\sim$, where the equivalence relation is $(0,t) \sim (1, 1-t)$.

Basically, this equivalence relation gives us gluing instructions.

We must twist the plane an odd number of times s.t. $(0,t)$ are the same as $(1, 1-t)$.

Get Your Group On: Actions and Torsors

If you are unfamiliar with smooth groups and representations, I recommend reading Studying Symmetryfor context before venturing onward.

What does it mean for $G \curvearrowright X$ (a group $G$ to “act” on $X$)?

Suppose we write the group operation as multiplication and the identity element as $1$.

A $G$-action on $X$ takes any $g \in G$ and any $x \in X$, and returns $gx \in X$. In other words:

For it to be a $G$-action, we demand that it obeys:

and

These properties may look familiar to the categorically inclined. Every group $G$ is a category with a single object whose morphisms are the elements of $G$.

An “action” of $G$ on an object in the category $C$ is simply a functor $G \to C$.

If $C$ is a group, this action is a group homomorphism.

If $C$ is $\text{Vect}$, this action is a linear representation of $G$.

A $G$-torsor is a special type of $G$-action which satisfies the following: for any 2 elements $x_1,x_2$ in our $G$-torsor, $\exists! g \in G$ that satisfies $gx_1 = x_2$.

This means that for any two elements of our torsor, we can talk about their “ratio” $x_2/x_1$, which defines the unique element $g$ which satisfies the above equation.

Principal-Bundles

A principal-bundle is a bundle whose fibers are torsors.

A principal $G$-bundle over a base space $B$ is essentially a bundle of “affine $G$-spaces” over $B$.

To be more precise, it is a fiber bundle $E \xrightarrow{\pi} B$ together with a continuous right action of $G$ on $E \xrightarrow{\pi} B$ which preserves the fibers and acts freely and transitively on them. In other words, our fibers are the orbits of $G$.

Let $G$ be a Lie group, and $F$ be our typical fiber. We have a faithful smooth group action:
$\rho: G \times F \to F$, which we can curry into
$\rho: G \to (F \to F)$ and rewrite as
$\rho: G \to \text{Aut}(F)$

In the most general case, $\text{Aut}(F) = \text{Diff}(F)$ (the group of diffeomorphisms), but we will shortly be working with vector bundles, for which $F$ is a vector space and $\text{Auto}(F) = GL(F)$.

Let $(GL(E))_b$ be the set of orthonormal frames of the vector space $E_b$. Note that the set of all orthonormal frames is a right $\text{O}(n)$-torsor.

Given a vector bundle $E \xrightarrow{\pi} B$ with a vector space $F$ as the fiber, we can construct a it’s principal bundle $GL(E) \xrightarrow{\Pi}B$ by mapping each fiber $E_b$ to the bundle of orthonormal frames over that fiber $GL(E_b)$.

A Taste of What’s to Come

Fiber bundles are themselves merely the culmination of the centuries-long struggle to come fully and properly to grips with the idea of a multiple-valued function.

One of the purposes of cohomology is to specify how the typical fibers and the base may be combined to make a variety of fiber bundles (a way to classify and distinguish the different possibilities) — these are the so-called characteristic classes (such as Chern classes).