• Math 250A: Groups, Rings, and Fields
• Will conduct class like a “very” advanced undergraduate class, including an (in-class) midterm and final
• Suggested exercies on course website. No GSI yet, no grading
• No office hours yet
• Lang’s algebra is a guide
• Class is going to be very much like 113. Hope to find sweet spot between “enriched 113” and completely novel
• Umbrella: Groups, Rings, Fields, Modules, focus on noncommutative
• cat $$\to$$ groups $$\to$$ rings $$\to$$ galois $$\to$$ representation theory

Some life advice: a trait of good scientists is that they always look for new facets in what they already know. Nadler gets this through teaching.

# Category Theory

Categories are immensely useful and generalizable objects. The reason we do category theory is that it helps us organize a lot of disparate phenomena among the algebraic objects that we already know and love.

Note that we talk about collections in the definition of a category. This is not formally defined, and we can talk about the required set theory, but it’s not important to understand the point of categories.

A category $$\mathscr{C}$$ consists of:

• A collection of objects formally denoted $$\text{Ob} \ \mathscr{C}$$, informally just $$\mathscr{C}$$
• For any $$x, y \in \mathscr{C}$$, a set of morphisms (also called arrows) $$\text{Hom}_\mathscr{C}(x, y) = \mathscr{C}(x, y)$$
• For any $$x, y, z \in \mathscr{C}$$, composition of morphisms:

$\text{Hom}_\mathscr{C}(y, z) \times \text{Hom}_\mathscr{C}(x, y) \to \text{Hom}_\mathscr{C}(x, z)$ $(g, f) \mapsto g \circ f$

Such that:

1. Composition of arrows is associative (add diagram): $$h \circ (g \circ f) = (h \circ g) \circ f$$
2. There exist identity morphisms $$e_x \in \text{Hom}_\mathscr{C}(x, x)$$ so that $$e_x \circ g = g, f \circ e_x = f$$

Categories were invented in the 1950s for algebraic topology. Categories are so pervasive that today they seem fundamental to lots of math.

Examples

• $$\mathscr{C} = \text{Set}$$: Objects are sets, Homs are set maps, compositions are function composition
• $$\mathscr{C} = \text{Finset}$$, finite sets
• $$\mathscr{C} = \text{Top}$$, topological spaces and continuous maps
• $$\mathscr{C} = \text{Grp}, \text{AbGrp}$$ groups and abelian groups
• $$\mathscr{C} = \text{Ring}, \text{CommRings}$$, rings and commutative rings

The structure of each category should be evident “culturally”, based on what kind of morphisms help us classify objects within the category.

Example. Suppose $$\mathscr{C}$$ has a single object $$x$$. (diagram) Then there exists some identity morphism $$id_x$$, and all other morphisms compose with each other with associativity. It turns out this kind of structure has a name, a monoid, with $$M = \text{Hom}_\mathscr{C}(x, x)$$.

A monoid is a set $$M$$ along with a binary operator $$\circ$$ such that:

• there is an identity $$e \in M$$ (where $$e \circ x = x$$ and $$x \circ e = x$$ for all $$x \in M$$), and
• $$\circ$$ is associative ($$(x \circ y) \circ z = x \circ (y \circ z)$$).

Conversely, given a monoid $$M$$, look at the classifying category $$BM$$ (we will talk about this name later). It consists of a single object $$x$$, with $$\text{Hom}_\mathscr{C}(x, x) = M$$.

Then in a specific way, the theory of monoids is a special case of the theory of categories.

1. A morphism $$f \in \text{Hom}_\mathscr{C}(x, y)$$ is an isomorphism/invertible if:
$$\exists g \in \text{Hom}_\mathscr{C}(y, x)$$ s.t. $$g \circ f = \text{id}_x$$ and $$f \circ g = \text{id}_y$$. (draw diagram)
2. A category $$\mathscr{C}$$ is called a groupoid if all morphisms are invertible.

Examples

• All categories, if we restrict to their isomorphisms, are groupoids.
• Let $$X$$ be a finite set. $$\text{Hom}_\text{Set}(X, X)$$ is all maps from $$X$$ to itself, with composition is function composition
• Let $$X$$ be a finite dimension vector space over $$K$$. Then $$\text{Hom}_{\text{Vect}_K}(X, X)$$ is all $$n \times n$$ matrices over $$K$$
• Let $$X$$ be a topological space. We can associate the Fundamental/Poncaire Groupoid $$\Pi(X)$$
• the objects are points $$x \in X$$
• The morphisms
$$\text{Hom}_{\Pi(X)}(x, y) = \{ \text{ homotopy classes of paths } \gamma \colon [0, 1] \to X \text{ s.t. } \gamma(0) = x, \gamma(1) = y \}$$
• compositions: path concatenations of equivalence classes, with the identity being the constant path
• (draw diagram) this is an interesting example, but the details are not too relevant to our discussion of categories
• Suppose $$\mathscr{C}$$ has a single object and is a groupoid. Then $$G = \text{Hom}_\mathscr{C}(x, x)$$ is a group. For the converse, given a group $$G$$, we can construct the classifying category $$BG$$.
• Suppose in the context of the Fundamental Groupoid we fix $$x \in X$$, then $$\text{Hom}_{\Pi(X)}(x, x) = G = \Pi_1(X, x)$$ is the fundamental group

Exercise. (for those well-versed in algebraic topology) Show any group $$G$$ is $$\Pi_1(X, x)$$ for some $$X$$ and $$x \in X$$.

Mention of Jean-Pierre Serre saying the best homework for a category theory class is to re-prove all the theorems.

# Constructing new categories from old ones

Let’s go over some ways to build new categories from ones we already know.

1. Dual/opposite category, $$\mathscr{C}^{op}$$. The objects are the same, but all the morphisms are flipped, i.e.:

$\text{Hom}_{\mathscr{C}^{op}}(x, y) = \text{Hom}_{\mathscr{C}}(y, x)$

Note that $$\left(\mathscr{C}^{op}\right)^{op} = \mathscr{C}$$.

2. Full subcategory $$\mathscr{C}' \subset \mathscr{C}$$. This is some subcollection of the objects, with all morphisms between them. (Verify that this is still a category)

3. Over/under categories. (draw diagrams). Fix $$x \in \mathscr{C}$$.

• Over: Denoted $$\quot{\mathscr{C}}{x}$$

The objects are $$y \stackrel{f}{\to} x$$, i.e. $$(y, f)$$ where $$y \in \mathscr{C}, f \in \text{Hom}_\mathscr{C}(y, x)$$.
Let $$y \stackrel{f}{\to} x$$ and $$z \stackrel{g}{\to} x$$. Then if there is $$h \in \text{Hom}_\mathscr{C}(y, z)$$ so that $$g \circ h = f$$, then $$h$$ is a homomorphism in $$\quot{\mathscr{C}}{x}$$.

Check that this forms a category, i.e. that composition and identity are preserved.

• Under: Denoted $$\quot{x}{\mathscr{C}}$$

Exercise: Show that $$(\quot{x}{\mathscr{C}})^{op} = \quot{\mathscr{C}^{op}}{x}$$.

# Functors and natural transformations

A functor $$F \colon \mathscr{C} \to \mathscr{D}$$ consists of:

• map of objects $$F \colon \mathscr{C} \to \mathscr{D}$$
• For each $$x, y \in \mathscr{C}$$, map of morphisms (draw diagram):

$F \colon \text{Hom}_{\mathscr{C}}(x, y) \to \text{Hom}_\mathscr{D}(Fx, fy)$

such that $$F(f \circ g) = F(f) \circ F(g)$$ and $$F(e_x) = e_{F(x)}$$ (in analogy to group homomorphisms)

Examples

• “Forgetful” functors, which simply forget whatever structure there may be:
1. $$F \colon \text{Top} \to \text{Set}$$ “forgets” topology
2. $$F \colon \text{Grp} \to \text{Set}$$
• One specific example of many:

$F \colon \text{Top}^{op} \to \text{Vect}_\mathbb{R}$ $X \mapsto \mathscr{C}^0(X, \mathbb{R})$