The Lumenary · Research pathways

Higher categories

Name: ______________________

Date: ______________________

You will learn to

Define strict 2-categories and bicategories, and show that a bicategory with one object is the same thing as a monoidal category
Compute in the 2-category $\mathbf{Cat}$ with vertical and horizontal composition, and prove the interchange law
Prove the Eckmann–Hilton argument and deduce that 2-endomorphisms of an identity 1-morphism commute
State the coherence theorem for bicategories and explain why strictification fails for tricategories (Gordon–Power–Street)
Describe the main approaches to weak $n$ -categories, separating established results from open questions

1.
A bicategory with exactly one object $\star$ is the same thing as which of the following?
- A.A monoidal category: the 1-cells $\star \to \star$ are its objects, the 2-cells its morphisms, and horizontal composition its tensor product
- B.A monoid
- C.A commutative monoid
- D.A groupoid
2.
How many ways are there to fully bracket a composite of six composable 1-cells $f_6 f_5 f_4 f_3 f_2 f_1$ ? (These are the vertices of the associahedron $K_6$ .)
3.
Let $\alpha : F \Rightarrow G$ (functors $\mathcal{C} \to \mathcal{D}$ ) and $\gamma : I \Rightarrow J$ (functors $\mathcal{D} \to \mathcal{E}$ ). Which formulas give the component $(\gamma \ast \alpha)_X : IFX \to JGX$ of the horizontal composite? (Choose all that apply.)
- A. $J(\alpha_X) \circ \gamma_{FX}$
- B. $\gamma_{GX} \circ I(\alpha_X)$
- C. $\gamma_{X} \circ \alpha_X$
- D. $I(\alpha_X) \circ \gamma_{GX}$
4.
In the proof of the interchange law $(\delta \cdot \gamma) \ast (\beta \cdot \alpha) = (\delta \ast \beta) \cdot (\gamma \ast \alpha)$ in $\mathbf{Cat}$ , after expanding components the two sides are $K(\beta_X) K(\alpha_X) \delta_{FX} \gamma_{FX}$ and $K(\beta_X)\, \delta_{GX} J(\alpha_X)\, \gamma_{FX}$ . Which fact finishes the proof?
- A.Naturality of $\delta$ at the morphism $\alpha_X$
- B.Naturality of $\alpha$
- C.Functoriality of $K$ alone
- D.Associativity of composition in $\mathcal{E}$
5.
The Eckmann–Hilton argument assumes that both operations are unital. Which pair of operations on a set with at least two elements satisfies the interchange law $(a \star b) \odot (c \star d) = (a \odot c) \star (b \odot d)$ but is not commutative, showing the unit hypothesis cannot be dropped?
- A. $a \odot b = a$ and $a \star b = a$ (both are 'left projection')
- B. $\odot = \star = +$ on $\Z$
- C. $\odot = \star =$ matrix multiplication on $2 \times 2$ matrices
- D. $a \odot b = a + b$ and $a \star b = ab$ on $\Z$
6.
Let B be a bicategory in Lean ([Bicategory B]) and a : B. Which statement formalises Corollary 3, 'the 2-endomorphisms of an identity 1-morphism commute'?
- A.∀ η θ : 𝟙 a ⟶ 𝟙 a, η ≫ θ = θ ≫ η
- B.∀ f g : a ⟶ a, f ≫ g = g ≫ f
- C.∀ (f : a ⟶ a) (η θ : f ⟶ f), η ≫ θ = θ ≫ η
- D.∀ η θ : 𝟙 a ⟶ 𝟙 a, η ▷ 𝟙 a = 𝟙 a ◁ θ
7.
Which of these are strict 2-categories, with composition of 1-cells associative and unital on the nose? (Choose all that apply.)
- A.Categories, functors and natural transformations
- B.Sets, spans of sets (composed by pullback) and maps of spans
- C.Rings, bimodules (composed by tensor product) and bimodule maps
- D.The one-object 2-category of a strict monoidal category, e.g. endofunctors of a category under composition
8.
Which statement about coherence in dimension 3 is correct?
- A.Every tricategory is triequivalent to a Gray-category, but not every tricategory is triequivalent to a strict 3-category.
- B.Every tricategory is triequivalent to a strict 3-category, as in dimension 2.
- C.No coherence theorem is known for tricategories.
- D.Coherence already fails for bicategories: not every bicategory is biequivalent to a strict 2-category.
9.
Write count_bracketings(n), the number of full bracketings of a word of n letters (use the recursion over the outermost bracket), and count_moves(n), the number of edges of the associahedron $K_n$ : pairs of bracketings that differ by one associator. Use the fact that every bracketing of $n \ge 2$ letters has exactly $n - 2$ internal brackets, each allowing exactly one associator move.
Computational — Practice

Answer key — Higher categories

1.
A monoidal category. Set $\mathcal{C} = \mathcal{B}(\star, \star)$ , $X \otimes Y := XY$ (horizontal composition), $I := 1_\star$ , and take the associator and unitors of $\mathcal{B}$ . The bicategory axioms (pentagon, triangle, naturality) are exactly the monoidal axioms. Conversely a monoidal category deloops to a one-object bicategory. (Making 'the same thing' precise for morphisms needs care: Cheng and Gurski, 2007.)
2.
$B(6) = B(1)B(5) + B(2)B(4) + B(3)B(3) + B(4)B(2) + B(5)B(1) = 14 + 5 + 4 + 5 + 14 = 42$ , the Catalan number $C_5 = \frac{1}{6}\binom{10}{5}$ . The experiment in the lesson confirms this count by enumeration.
3.
Both (a) and (b). The naturality square of $\gamma$ at $\alpha_X : FX \to GX$ has corners $IFX, IGX, JFX, JGX$ , and its two routes are $J(\alpha_X) \circ \gamma_{FX}$ and $\gamma_{GX} \circ I(\alpha_X)$ ; naturality says they are equal, and this common value is $(\gamma \ast \alpha)_X$ .
4.
The middle factors are $K(\alpha_X)\,\delta_{FX}$ and $\delta_{GX}\,J(\alpha_X)$ , and these are the two routes round the naturality square of $\delta$ at $\alpha_X : FX \to GX$ . In the Lean proof this is the single rewrite rw [δ.naturality_assoc].
5.
Left projection, $a \odot b = a \star b = a$ . Then $(a \star b) \odot (c \star d) = a = (a \odot c) \star (b \odot d)$ , so interchange holds, but the operation is not commutative. Every step of the Eckmann–Hilton proof inserts a unit ( $a = a \star e$ ), which is exactly what fails here.
6.
∀ η θ : 𝟙 a ⟶ 𝟙 a, η ≫ θ = θ ≫ η. The lesson's Lean proof derives it from the interchange axiom whisker_exchange, the fact that whiskering by 𝟙 a is conjugation by a unitor, and unitors_equal.
7.
$\mathbf{Cat}$ and the delooping of a strict monoidal category are strict. Spans and bimodules form bicategories: their composites are defined by universal properties (pullback, tensor product), hence only up to canonical isomorphism. By the coherence theorem each is biequivalent to some strict 2-category — but not with the same objects and 1-cells.
8.
Every tricategory is triequivalent to a Gray-category, and not every one to a strict 3-category (Gordon–Power–Street, 1995). In homotopy terms: strict interchange would make the braiding of 2-cells trivial, while the 3-type of $S^2$ has a non-trivial one (detected by the Hopf map).
9.
```
def count_bracketings(n):
    if n == 1:
        return 1
    return sum(count_bracketings(i) * count_bracketings(n - i) for i in range(1, n))


def count_moves(n):
    return (n - 2) * count_bracketings(n) // 2
```
Each bracketing is a binary tree with $n - 1$ internal nodes; every internal node except the root can be rotated with its parent in exactly one way, giving degree $n - 2$ . By the handshake lemma there are $(n-2)B(n)/2$ edges: $1, 5, 21, 84$ for $n = 3, 4, 5, 6$ .