The Lumenary · Advanced undergraduate

Products, coproducts, initial and terminal objects

Name: ______________________

Date: ______________________

You will learn to

Identify initial and terminal objects in $\cat{Set}$ , $\cat{Grp}$ , rings and posets, or show that none exist
Compute products and coproducts in $\cat{Set}$ (disjoint union), $\cat{Grp}$ (free product), $\cat{Ab}$ (direct sum), $\cat{Top}$ and a poset (meet and join)
Prove that products are unique up to unique isomorphism, that a product is a terminal cone, and that coproducts in $\mathcal{C}$ are products in $\mathcal{C}^{\mathrm{op}}$
Use universal properties to prove structural facts, such as that $\Z * \Z$ is not abelian and that the box topology is not the product topology
Recognise when a product or coproduct does not exist, for example in a poset or in the category of fields

1.
In $\cat{Set}$ , which statement is correct?
- A. $\varnothing$ is initial and every one-point set is terminal
- B. $\varnothing$ is terminal and $\{\ast\}$ is initial
- C. $\varnothing$ is a zero object
- D. $\cat{Set}$ has no terminal object, since there are many one-point sets
2.
Which of these categories have a zero object (an object that is both initial and terminal)? Select all that apply.
- A. $\cat{Grp}$
- B. $\cat{Vect}_k$
- C.rings with identity
- D. $\cat{Set}$
- E.the poset $(\Z, \le)$
3.
Regard the positive integers, ordered by divisibility ( $a \le b$ iff $a \mid b$ ), as a category. What is the coproduct of $12$ and $18$ ?
4.
In $\cat{Set}$ , let $|X| = 2$ , $|A| = 2$ and $|B| = 3$ . How many morphisms $X \to A \times B$ are there? (Use the universal property rather than listing them.)
5.
In the proof that two products $(P, \pi_i)$ and $(Q, \rho_i)$ of $A$ and $B$ are isomorphic, we have $\varphi : P \to Q$ and $\psi : Q \to P$ with $\rho_i \varphi = \pi_i$ and $\pi_i \psi = \rho_i$ . Why is $\psi \circ \varphi = \mathrm{id}_P$ ?
- A.Both $\psi \circ \varphi$ and $\mathrm{id}_P$ satisfy $\pi_i \circ h = \pi_i$ for $i = 1, 2$ , and by the uniqueness clause for $P$ (applied to the pair $(\pi_1, \pi_2)$ ) there is only one such $h$
- B.Because $\varphi$ is injective and $\psi$ is surjective
- C.Because $P$ and $Q$ have the same number of elements
- D.By the existence clause of the universal property of $Q$
6.
What is the coproduct of $\Z/2$ and $\Z/3$ in the category of commutative rings with identity?
- A.the zero ring
- B. $\Z/6 \cong \Z/2 \times \Z/3$
- C.it does not exist
- D. $\Z$
7.
Which statements are true? Select all that apply.
- A.The coproduct of $\Z$ and $\Z$ in $\cat{Grp}$ is non-abelian.
- B.The coproduct of $\Z$ and $\Z$ in $\cat{Ab}$ is $\Z \oplus \Z$ .
- C.Since $\cat{Ab}$ is a full subcategory of $\cat{Grp}$ , the two coproducts above must agree.
- D.The product of $\Z$ and $\Z$ is $\Z \times \Z$ in both $\cat{Grp}$ and $\cat{Ab}$ .
8.
Mathlib writes h ≫ π₁ for $\pi_1 \circ h$ (first h, then π₁). Which Lean field correctly expresses the uniqueness clause of the universal property of a product with projections π₁ : P ⟶ A, π₂ : P ⟶ B?
- A.uniq : ∀ {X} (f : X ⟶ A) (g : X ⟶ B) (h : X ⟶ P), h ≫ π₁ = f → h ≫ π₂ = g → h = lift f g
- B.uniq : ∀ {X} (f : X ⟶ A) (g : X ⟶ B) (h : X ⟶ P), π₁ ≫ h = f → π₂ ≫ h = g → h = lift f g
- C.uniq : ∀ {X} (f : X ⟶ A) (g : X ⟶ B), ∃ h : X ⟶ P, h ≫ π₁ = f ∧ h ≫ π₂ = g
- D.uniq : ∀ {X} (h k : X ⟶ P), h = k
9.
Write glb(elements, leq, a, b) returning the product of a and b in a finite poset — the greatest lower bound — or None if it does not exist. Here elements is a list and leq(x, y) decides $x \le y$ .
Computational — Practice

Answer key — Products, coproducts, initial and terminal objects

1.
For any set $X$ there is exactly one function $\varnothing \to X$ (the empty function), so $\varnothing$ is initial; and exactly one function $X \to \{\ast\}$ , so every one-point set is terminal. $\varnothing$ is not terminal (no map $\{\ast\} \to \varnothing$ ), so $\cat{Set}$ has no zero object.
2.
$\cat{Grp}$ (trivial group) and $\cat{Vect}_k$ (zero space) have zero objects. In rings with identity, $\Z$ is initial and $0$ is terminal; in $\cat{Set}$ , $\varnothing$ is initial and $\{\ast\}$ terminal; $(\Z, \le)$ has neither.
3.
The coproduct is the least upper bound: a common multiple of $12$ and $18$ that divides every other common multiple, namely $\operatorname{lcm}(12, 18) = 36$ . (The product is $\gcd(12, 18) = 6$ .)
4.
$|\Hom(X, A \times B)| = |\Hom(X, A)| \cdot |\Hom(X, B)| = 2^2 \cdot 3^2 = 36$ . Check: $|A \times B| = 6$ , and $6^2 = 36$ .
5.
$\pi_i \circ \psi \circ \varphi = \rho_i \circ \varphi = \pi_i = \pi_i \circ \mathrm{id}_P$ . So $\psi\varphi$ and $\mathrm{id}_P$ are two morphisms $P \to P$ with the same composites with both projections; by the uniqueness clause of the universal property of $P$ they are equal. In Lean this is IsBinaryProduct.hom_ext.
6.
Any ring $R$ with homomorphisms $\Z/2 \to R$ and $\Z/3 \to R$ satisfies $2 \cdot 1_R = 0$ and $3 \cdot 1_R = 0$ , so $1_R = 0$ and $R = 0$ . So the only cocone is the zero ring, and the zero ring (with its unique maps from $\Z/2$ and $\Z/3$ ) is the coproduct: for every pair of maps into some $Y$ , $Y$ is itself zero and the identity is the unique map. Equivalently, $\Z/2 \otimes_\Z \Z/3 \cong \Z/\gcd(2, 3) = 0$ .
7.
In $\cat{Grp}$ the coproduct is the free product $\Z * \Z$ , which is not abelian (Theorem 4). In $\cat{Ab}$ it is $\Z \oplus \Z$ . Being a full subcategory does not make coproducts agree, because the universal property in $\cat{Grp}$ quantifies over all groups. Products agree in both: $\Z \times \Z$ with the projections.
8.
Option (a). In diagrammatic order, $\pi_1 \circ h = f$ is h ≫ π₁ = f. Option (c) is existence; (b) does not type-check; (d) forgets the role of the projections.
9.
```
def glb(elements, leq, a, b):
    lower = [x for x in elements if leq(x, a) and leq(x, b)]
    best = [x for x in lower if all(leq(y, x) for y in lower)]
    return best[0] if best else None
```
In a poset a greatest lower bound is unique when it exists (antisymmetry), so best has at most one element. In $\{2, 3, 12, 18\}$ the pair $12, 18$ has two maximal lower bounds, $2$ and $3$ , neither above the other — so no product.