The Lumenary · First-year graduate

Monoidal categories

Name: ______________________

Date: ______________________

You will learn to

Verify that $(\cat{Set}, \times, 1)$ , $(\cat{Set}, \sqcup, \varnothing)$ and $(\cat{Vect}_k, \otimes, k)$ are symmetric monoidal categories
State Mac Lane's coherence theorem, compute with string diagrams, and prove that $\mathrm{End}(I)$ is commutative
Prove the Eckmann–Hilton argument and use it to explain why braidings exist only "one dimension up"
Distinguish braided from symmetric monoidal categories, with the braid groups and quantum-group representations as examples
Define monoid objects and identify them in examples — monoids, rings, $k$ -algebras, and monads as monoids of endofunctors

1.
In the monoidal category $(\mathbf{Set}, \sqcup, I)$ whose tensor product is disjoint union, what must the unit object $I$ be (up to isomorphism)?
- A.The empty set $\varnothing$
- B.A one-element set $\{*\}$
- C.A two-element set
- D.There is no unit; $\sqcup$ is only a semigroup operation
2.
How many different full bracketings are there of the word $A \otimes B \otimes C \otimes D \otimes E$ (keeping the order of the letters)? These are the vertices of the 3-dimensional associahedron.
3.
The proof that $\mathrm{End}(I)$ is commutative uses that $I \otimes g$ and $f \otimes I$ commute, for $f, g : I \to I$ . Which fact gives this?
- A.Functoriality of $\otimes$ (the interchange law): both composites equal $f \otimes g$
- B.The braiding $c_{I,I}$
- C.The pentagon axiom
- D.Naturality of the unitors
4.
What is a monoid object in $(\mathbf{Ab}, \otimes_{\mathbb{Z}}, \mathbb{Z})$ ?
- A.A ring (with identity, not necessarily commutative)
- B.A commutative ring
- C.Just an abelian group: every abelian group has a unique such structure
- D.A monoid whose underlying set is an abelian group, with no relation between the two
5.
Which of these braided monoidal categories is not symmetric?
- A.The braid category, with $\sigma \in B_2$ as the braiding on $1 \otimes 1$
- B. $(\mathbf{Set}, \times, 1)$ with $c(x, y) = (y, x)$
- C.Super vector spaces with the Koszul sign rule $c(v \otimes w) = (-1)^{|v||w|} w \otimes v$
- D. $(\mathbf{Vect}_k, \otimes, k)$ with $c(v \otimes w) = w \otimes v$
6.
A monoid object in $(\mathbf{Mon}, \times, 1)$ is a monoid $(M, \cdot, 1)$ together with a multiplication $\ast : M \times M \to M$ and unit that are monoid homomorphisms. What can you conclude?
- A. $\ast = \cdot$ , and $M$ is commutative
- B.Nothing more: $\ast$ can be any second associative operation
- C. $M$ must be a group
- D. $M$ must be trivial
7.
Which Lean statement (in Mathlib's notation, ≫ = composition in diagrammatic order) formalises " $\mathrm{End}(I)$ is commutative"?
- A.∀ f g : 𝟙_ C ⟶ 𝟙_ C, f ≫ g = g ≫ f
- B.∀ (X : C) (f g : X ⟶ X), f ≫ g = g ≫ f
- C.(λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom
- D.∀ f g : 𝟙_ C ⟶ 𝟙_ C, f ⊗ₘ g = g ⊗ₘ f
8.
An associator on $\mathbb{Z}/n$ -graded vector spaces is given by exponents $w(a, b, c) \in \mathbb{Z}/n$ of $\zeta = e^{2\pi i/n}$ . Write is_cocycle(w, n) that returns True exactly when the pentagon (3-cocycle) condition

$w(b, c, d) + w(a, b + c, d) + w(a, b, c) \equiv w(a + b, c, d) + w(a, b, c + d) \pmod n$

holds for all $a, b, c, d \in \{0, \dots, n - 1\}$ (sums of grades taken mod $n$ ).
Computational — Practice

Answer key — Monoidal categories

1.
We need $I \sqcup X \cong X$ naturally in $X$ ; counting elements for finite $X$ forces $|I| = 0$ , so $I = \varnothing$ , and indeed $\lambda_X : \varnothing \sqcup X \to X$ is the evident bijection. More generally, a category with finite coproducts is monoidal under $\sqcup$ with the initial object as unit.
2.
With $b_1 = b_2 = 1$ , $b_3 = 2$ , $b_4 = 5$ : $b_5 = 1 \cdot 5 + 1 \cdot 2 + 2 \cdot 1 + 5 \cdot 1 = 14$ , the Catalan number $C_4$ . The experiment monoidal_categories_pentagon lists them and finds the 21 associator moves between them.
3.
Because $\otimes$ is a functor on $\mathcal{C} \times \mathcal{C}$ , $(f \otimes I) \circ (I \otimes g) = (f \circ \mathrm{id}_I) \otimes (\mathrm{id}_I \circ g) = f \otimes g = (I \otimes g) \circ (f \otimes I)$ . Combined with $u \circ (I \otimes g) = g \circ u$ and $u \circ (f \otimes I) = f \circ u$ for $u = \lambda_I = \rho_I$ , this gives $f \circ g = g \circ f$ . It is the Eckmann–Hilton argument in categorical form.
4.
By the universal property of $\otimes_{\mathbb{Z}}$ , $\mu : M \otimes M \to M$ is a biadditive map $M \times M \to M$ , i.e. a multiplication distributing over addition; $\eta : \mathbb{Z} \to M$ is determined by $1 = \eta(1)$ . The associativity and unit diagrams say the multiplication is associative with identity $1$ . So monoid objects are rings, and morphisms of monoid objects are ring homomorphisms.
5.
In $\mathbf{Set}$ , $\mathbf{Vect}_k$ and super vector spaces, braiding twice is the identity (for the Koszul rule because the sign squares to $1$ ). In the braid category $c_{1,1} \circ c_{1,1} = \sigma^2$ , which is not the identity braid since $B_2 \cong \mathbb{Z}$ is infinite cyclic on $\sigma$ . Representations of $U_q(\mathfrak{sl}_2)$ at generic $q$ give another non-symmetric example.
6.
$\ast$ being a homomorphism means $(a \cdot c) \ast (b \cdot d) = (a \ast b) \cdot (c \ast d)$ , and its unit must be the unit of $M$ (it is a homomorphism from the trivial monoid). By Eckmann–Hilton the two units agree, $\ast = \cdot$ , and the operation is commutative. Conversely every commutative monoid is such a monoid object. The same argument shows $\pi_n(X)$ is abelian for $n \ge 2$ .
7.
∀ f g : 𝟙_ C ⟶ 𝟙_ C, f ≫ g = g ≫ f quantifies over endomorphisms of the unit object and compares the two composites. It is proved in the Lean file from naturality of the unitors, the interchange law whisker_exchange and Kelly's identity unitors_equal.

def is_cocycle(w, n):
    for a in range(n):
        for b in range(n):
            for c in range(n):
                for d in range(n):
                    lhs = w(b, c, d) + w(a, (b + c) % n, d) + w(a, b, c)
                    rhs = w((a + b) % n, c, d) + w(a, b, (c + d) % n)
                    if (lhs - rhs) % n:
                        return False
    return True

The carry cochains pass (the lesson proves the identity for every $n$ using the fact that the number of carries in $b + c + d$ does not depend on the bracketing); trilinear forms such as $abc$ are cocycles too; carry(a, b) and b are not. The search checks finitely many cases; for each fixed $n$ that is a complete proof, but only the general argument covers all $n$ .