Exercises 1 : Introduction

EX 1: Categories and functors

1.

• $Set$: sets and functions
• $Top$: topological spaces and continuous maps
• $Vect$: vector spaces and linear maps
• $Grp$: groups and homomorphisms
• $Mon, Fields, Ring$, etc…
• $Poset$: posets and monotonous maps
• $(E, ≤)$: the poset category
• categories with a single object are monoids
• logical formuals and implications

EX 2: Cartesian categories

1.

• terminal object: greatest element
• product: greatest lower bound

2.

Products of sets are cartesian products, terminal objects are singletons.

3.

If $1$ and $1’$ are two terminal objects:

• there exist unique maps $u: 1 ⟶ 1’$ and $v: 1’ ⟶ 1$ by terminality
• $v \circ u ∈ 𝒞_1(1, 1) = \lbrace id_1 \rbrace$ by universality of $1$
• idem for $u \circ v = id_{1’}$

4 / 5.

It’s “the” (up to unique isomorphism) terminal object in the category of cones.

6.

There exist: $π_1: 1 × A ⟶ 1$, $π_2: 1 × A ⟶ A$

We check that \(⟨π_2, π_1⟩: 1 × A ⟶ A × 1\)

is an isomorphism:

\[⟨π_2, π_1⟩ \circ ⟨π_1, π_2⟩ = ⟨π_2 \circ π_1, π_1 \circ π_2⟩ = ⟨id_A, id_1⟩\]

indeed, by universality: $⟨id_A, id_1⟩$ is the only arrow in $𝒞_1(A × 1, A × 1)$.

---

\[A ≃ 1 × A\]

since $A$ is a cartesian product of $1$ and $A$ (easy to check).

7.

Same demonstration as before.

8.

We show that $(A×B)×C$ is a product of $A$ and $B×C$.

M2.

With the Yoneda lemma: $𝒞_1(y((A×B)×C), y(A×(B×C))) ≃ 𝒞_1((A×B)×C, A×(B×C))$, so:

\[(A×B)×C ≃ A×(B×C) ⟺ \underbrace{y((A×B)×C) ≃ y(A×(B×C))}_{𝒞_1(\_, (A×B)×C) \, ≃ \, 𝒞_1(\_, A×(B×C))}\]

9.

Coproducts: disjoint union of sets

Initial object in $Set$: the empty set $∅$.

10.

Category of relations $Rel$:

• objects: sets
• morphisms: relations $⊆ A × B$

It is cartesian:

1. terminal object: $∅$ since the only morphism $⊆X × ∅$ is $∅$
2. products: the disjoint union: $π_{X}⊆ X \sqcup Y × X$ is defined as $\lbrace (ι_X(x), x) \mid x∈X \rbrace$, and if $f ⊆ C × X, \; g ⊆ C × Y$, $⟨f, g⟩ ⊆ \lbrace (c, ι_X(x)) \mid (c, x) ⊆ f\rbrace ∪ \lbrace (c, ι_Y(y)) \mid (c, y) ⊆ g\rbrace$

What are coproducts in this category? A product in the opposite category.

But as

\[Rel^{op} = Rel\]

They are the same.

11.

Products are direct sums, whose basis are disjoint unions of basis of $A$ and $B$.

\[π_A: \begin{cases} A \oplus B ⟶ A \\ a ∈ Basis_A ⟼ a\\ b ∈ Basis_B ⟼ 0 \end{cases}\]

12.

$Cat$ is cartesian:

• terminal objet: unit category
• product: product category

EX 3: Pullbacks

1.

It is a product.

2.

It is the set \(\lbrace (x,y) ∈ A × B \mid f(x) = g(y) \rbrace\)

EX 4: Dual notions

1.

Coproducts in:

• $Set$: disjoint unions
• $Rel$: disjoint unions
• $Top$: product of spaces
• $Vect$: product of spaces

EX 5: (Co)monoids in cartesian categories

1.

Monoid:

A category with one single object.

The monoid is commutative whenever for all $f, g: X ⟶ X$, $f \circ g = g \circ f$

2.

Homomorphism of monoids:

a functor from the first category seen as a monoid to the second one

3.

A comonoid