Practice: Monoidal categories

Monoidal categories

Warm-upHuman

In the monoidal category $(\mathbf{Set}, \sqcup, I)$ whose tensor product is disjoint union, what must the unit object $I$ be (up to isomorphism)?

Choose one

The empty set

\varnothing

A one-element set

\{*\}

A two-element setThere is no unit;

\sqcup

is only a semigroup operation

Monoidal categories

Warm-upHuman

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.

Monoidal categories

CoreHuman

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?

Choose one

Functoriality of

\otimes

(the interchange law): both composites equal

f \otimes g

The braiding

c_{I,I}

The pentagon axiomNaturality of the unitors

Monoidal categories

CoreHuman

What is a monoid object in $(\mathbf{Ab}, \otimes_{\mathbb{Z}}, \mathbb{Z})$ ?

Choose one

A ring (with identity, not necessarily commutative)A commutative ringJust an abelian group: every abelian group has a unique such structureA monoid whose underlying set is an abelian group, with no relation between the two

Monoidal categories

CoreHuman

Which of these braided monoidal categories is not symmetric?

Choose one

The braid category, with

\sigma \in B_2

as the braiding on

1 \otimes 1

(\mathbf{Set}, \times, 1)

with

c(x, y) = (y, x)

Super vector spaces with the Koszul sign rule

c(v \otimes w) = (-1)^{|v||w|} w \otimes v

(\mathbf{Vect}_k, \otimes, k)

with

c(v \otimes w) = w \otimes v

Monoidal categories

StretchHuman

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?

Choose one

\ast = \cdot

, and

M

is commutativeNothing more:

\ast

can be any second associative operation

M

must be a group

M

must be trivial

Monoidal categories

CoreFormal

Which Lean statement (in Mathlib's notation, ≫ = composition in diagrammatic order) formalises " $\mathrm{End}(I)$ is commutative"?

Choose one

∀ f g : 𝟙_ C ⟶ 𝟙_ C, f ≫ g = g ≫ f∀ (X : C) (f g : X ⟶ X), f ≫ g = g ≫ f(λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom∀ f g : 𝟙_ C ⟶ 𝟙_ C, f ⊗ₘ g = g ⊗ₘ f

Monoidal categories

StretchComputational

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 $n$ ).

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