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:

  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:

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