Notes on Covering Spaces as Extensions

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 $F(b)$.

unnamed (9) unnamed (8) Continue reading Notes on Covering Spaces as Extensions

A Gentle Introduction to Tensors and Monoids

There are at least three distinct conceptual roles which vectors and vector spaces play in mathematics:

  • A vector is a column of numbers. This is the way vector spaces appear in quantum mechanics, sections of line bundles, elementary linear algebra, etc.
  • A vector is a weighted direction in space. Vector spaces of this kind are often the infinitesimal data of some global structure, such as tangent spaces to manifolds, Lie algebras of Lie groups, and so on.
  • A vector is an element of a module over the base ring/field.

What is a module? The basic idea is that a module $V$ is an object equipped with an action by a monoid $A$. (This is closely related to the concept of a representation of a group.)

Let’s take an example that you’re familiar with, vector spaces, and generalize it to get some intuition for working with modules.

Fields $\hookrightarrow$ Rings

If $K$ is a field, then a $K$-vector space (a vector space over $K$) $\equiv$ $K$-module.

$K$-Vector Spaces $\hookrightarrow$ $K$-modules

For the categorically minded: a familiar example of a module is a vector space $V$ over a field $K$; this is a module over $K$ in the category of abelian groups: every element of $K$ acts on the vector space by a multiplication of vectors, and this action respects the addition of vectors.

Tensors play analogous conceptual roles.

  • A tensor is a multidimensional array of numbers.
  • A tensor is multiple weighted directions in space.
  • A tensor is an element of a free monoid over the base ring. (If you don’t know what a free monoid is, don’t worry. I’ll go over them later in this post.)

An explanation of tensors as type constructors is postscript for fellow Haskell enthusiasts. Continue reading A Gentle Introduction to Tensors and Monoids

An Informal Categorical Introduction to Lie’s Theorems

This quick post assumes basic knowledge of Lie algebras and category equivalence. I am new to category theory, and appreciative of constructive feedback.

We commonly study smooth manifolds, e.g. Lie groups, by studying their tangent spaces. Since the product map induces a map from one tangent space to another, we can oftentimes just consider the tangent space to a Lie group at the identity.  This tangent space can be equipped with a structure induced by the group structure, called a Lie algebra structure.

The Theorems of Lie

The assignment Lie : $G \to $Lie$(G)$ is functorial. The theorems of Lie in their modern incarnation emerge out of the attempt to see how close this functor is to being an equivalence of categories. Note that we are working in Diff.

Lie proved that the category of local real Lie groups is equivalent to the category of finite-dimensional real Lie algebras.

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This equivalence was extended to global cases by Cartan: the category of real Lie algebras is equivalent to the category of simply-connected Lie Groups.

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Note: We cannot drop the condition of being simply connected for $G$, as, for example, $G = S^1$ and $G = \mathbb{R}$ have the same Lie algebras, but are not isomorphic.

Lie I: Groups $\to$ Algebras

The assignment $G \mapsto$ Lie$(G)$ induces a functor Lie: LieGrp $\to$ LieAlg and for each morphism $g:G \to H$ of Lie groups the following diagram commutes:

Screenshot from 2014-09-15 23:22:51

The Lie algebra homomorphism Lie$(g)$ is equivalent to the first order infinitesimal of the group homomorphism.

Lie II: Do You Even Lift, Lie?
Alegbras $\to$ Groups

Let $G$ and $H$ be Lie groups with Lie algebras Lie$(G)$ and Lie$(H)$, with a Lie algebra homomorphism $f:$Lie$(G) \to$Lie$(H)$.

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For notational convenience, we denote Lie$(G)$ as the lowercase gothic letter $\mathfrak{g}$.

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Lie II states that there exists a unique morphism $F$ lifting $f$ Screenshot from 2014-09-15 17:41:44

such that $f=$Lie$(F)$.

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Lie-Cartan III: Groups $\leftrightarrow$ Algebras

The functor Lie cannot be inverted because locally isomorphic Lie groups have isomorphic Lie algebras. However, we can invert Lie on the subcategory of simply connected Lie groups. The essential surjectivity of this functor is the third theorem.

For every finite-dimensional real Lie algebra $\mathfrak{g}$ there exists a Lie group $G$ with Lie algebra $\mathfrak{g}$.

Screenshot from 2014-09-15 20:02:05 Note that $G$ is not necessarily unique.

Sources

Lie’s Three Theorems 
The Lie group-Lie algebra correspondence
Sophus Lie’s Fundamental Theorem – Categorical Aspects
Lie’s Third Theorem
Lie Theory of Differential Equations and Computer Algebra
What is a Dynkin Diagram, and how does it work?

A Unifying Language

Mathematics is a huge subject.

Category theory is one area of mathematics dedicated to exploring the commonality of structure between different branches of mathematics.

Categorical language allows us to ascend a layer of abstraction, and recognize the obvious underlying principles that guide seemingly unrelated concepts. Generality facilitates connections.

What is a category?

A category $C$ consists of:

  1. a class of objects $Ob(C)$
  2. For every ordered pair of objects $X$ and $Y$, a set $C(X,Y)$ of morphisms with domain $X$ and range $Y$ [$C(X,Y)$ is possibly empty] Note: $f \in C(X,Y)$  $\equiv$  $f : X\rightarrow Y$   $\equiv$  $X \overset{f}{\rightarrow} Y$.
  3. For every object an identity morphism $Id_x \in C(X,X)$.
  4. A composition law $$C(X,Y) \times C(Y,Z) \rightarrow C(X,Z)$$ $$(g,f) \rightarrow f\cdot g$$

The concept of composition follows naturally from the definition of path equivalence in graph theory: Two paths with the same source as destination are equal. Screenshot from 2014-08-11 00:10:36   Additionally, categories must satisfy the laws of associativity and identity.

Category Laws

Categories must obey 2 laws:

  1. Composition must be associative:Screenshot from 2014-08-11 00:09:55
  2.  Every object $a$ in $C$ has a morphism $id_a$ which is equivalent to a loop in graph theory. 431px-Self-loopThe identity morphism $id_a$ connects the object $a$ to itself, $$id_a: a \rightarrow a$$.

Screenshot from 2014-08-11 00:10:15

Two paths are equal if the source and destination of the paths are equal. With this in mind, we can represent the category laws of identity and associativity with diagrams.

Follow the arrows, recall that we write function composition backward! If we traverse $f$ then $g$, it is the convention to write $g \circ f$

Screenshot from 2014-08-11 10:06:43                                     Screenshot from 2014-08-11 10:06:54

Examples:

Groups, together with group homomorphisms, form a category (we will discuss these next lecture for those who have not danced with abstract algebra).

Each of the natural numbers is a category:

Screenshot from 2014-08-18 16:34:05

Categories have some nice properties

Any property which can be expressed in terms of (category, objects, morphism, and composition):

  • Dual: $D$ is $C$ with reversed morphisms
  • Initial: $Z \in obj(C)$ s.t. $\forall Y \in obj(C)$, #$hom(Z,Y) =1$. In other words: an object is initial if there exists a unique morphism from that object to any other object in $C$.
  • Terminal: $T \in obj(C)$ s.t. $T$ is initial in the dual of $C$
  • Functor: Structure preserving mapping between categories

Homomorphims Between Categories: What the func is a functor?

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An example of associating morphisms from $C$ to $D$ with the functor $F(C)$.

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A Reflection on the Unfication of Familiar Concepts

A few categories you have likely encountered before without recognizing it:

  • Set (sets and functions)
  • Vec (vectorial spaces and linear transformations)
  • Top (topological spaces and continuous maps)
  • Grp (groups and homomorphisms) — we will be discussing these next lecture for those who have not danced with abstract algebra.
  • Ab (abelian groups and homomorphisms)
  • Hask (Haskell types and functions)
  • Cat (categories and functors)

A categorical key to highlight some of the relationships between structure preserving maps:

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Here are some more advanced examples for those who feel groovy and algebraic:

  • $R$-Mod (R-modules and homomorphisms)
  • $Gr_R$ ($\mathbb{Z}$-graded $R$-modules and graded $R$-module homomorphisms)

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Sources

Introduction to Haskell