# Math 250A Lecture 1: Administrivia, Categories

# Administriva

- Math 250A: Groups, Rings, and Fields
- Instructor: David Nadler
- 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:

- Composition of arrows is associative (add diagram): \(h \circ (g \circ f) = (h \circ g) \circ f\)
- 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.

- 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) - 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.

*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 \(\left(\mathscr{C}^{op}\right)^{op} = \mathscr{C}\).

*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)*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:
- \(F \colon \text{Top} \to \text{Set}\) “forgets” topology
- \(F \colon \text{Grp} \to \text{Set}\)

One specific example of many:

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