Intuitively, you can build a cylinder out of a circle and a line segment. If you glue llines onto every point $b$ of a circle ^[b for base] (or alternatively glue a circle to every point of a line) you get a cylinder. The cylinder is the product space $S^1 \times [0,1]$ ![[_Media/Circle.png]] If you glue lines onto every point of a circle while continuously twisting each line you get a mobius strip. ![[_Media/Mobius.png]] ## Jargon - **bundle** (in topology) tends to be used in place of family e.g. a **vector bundle** as a way to make precise a family of vector spaces. - **fiber** is a component "line" $F_b$ of a larger total space $E$ s.t. $\coprod_{b:B} F_b = E$ - **fiber bundle** in the generalized element bundle (e.g. a generalized vector bundle) - **fibration** is the generalization of the notion of fiber bundles (where the base space, total space, and fiber don't have to be the space space or homeomorphic, just homotopy equivalent). Weak fibrations replace the homotopy equiv requirement for an even more general property. Fibrations do **not** have the locally cartesian product structure of fiber bundles ## Motivation and interpretations Fiber bundles provide a convenient way to take products of topological spaces (to build complex spaces from simple spaces). E.g. (vector bundles) we can think of a vector field $v$ as a map $v: \mathbb{R}^3 \to \mathbb{R}^3$. For every point in $\mathbb{R}^3$ we have a corresponding vector in $\mathbb{R}^3$ - for every point, we're gluing on another vector space to create a more complex vector space. A vector bundle over space $B:\mathbb{R}^3$ is a parameterized family of vector spaces $V_b$ for each point $b:B$ A fiber bundle is a space (that's locally a product space) but may have different global topological structure. Fiber bundles make precise the idea of one [[Topological Spaces |topological space]](called the fiber) being "parameterized" by another [[Topological Spaces |topological space]] (called the base) > A fiber bundle with base space B and fiber F can be viewed as a parameterized family of objects, each “isomorphic” to F, where the family is parameterized by points in B ^[ [The Topology of Fiber Bundles Lecture Notes - Ralph Cohen](http://math.stanford.edu/~ralph/fiber.pdf)] Fiber bundles are a way to make precise multi-valued functionsa ## Intuition ## Definition A fiber bundle with fiber $F$ consists of 2 [[Topological Spaces]], the total space $E$ and base space $B$, and a [[Projection map]] $\pi$ (a continuous surjection called the bundle projection) from the total space to its base space. For example with total space $E:MobiusStrip$, base space $B:BaseCircle$ where fiber $F:[0,1]$ ![[_Media/MobiusToCircle.png]] ## Resources - [Introduction to Bundles – good fibrations](http://rin.io/intro-to-bundles/) - [What is a Fiber Bundle in layman's terms? What is it used for? - Quora](https://www.quora.com/What-is-a-Fiber-Bundle-in-laymans-terms-What-is-it-used-for) - [Fiber Bundles | Pointing to the Moon](https://yk-liu.github.io/2019/Fiber-Bundles/) - [The Topology of Fiber Bundles Lecture Notes - Ralph Cohen](http://math.stanford.edu/~ralph/fiber.pdf) --- - Links: [[Math]] [[Type Theory]] [[Type Family]] [[Type-indexed family of types]] [[Spinner]] [[Spin Geometry]] [[Hopf Fibration]] [[Fibration]] [[Fibered Category]] [[Homotopy]] [[Vector Bundle]] [[Topology]] [[Guage Theory]] [[Vector Fields]] - Created at: [[2021-08-03]]