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The Lumenary · Advanced undergraduate

Representable functors

Name: ______________________

Date: ______________________

You will learn to

Show that the forgetful functor $\cat{Grp} \to \cat{Set}$ is represented by $\Z$ and that the contravariant power-set functor is represented by $\{0, 1\}$
Define universal elements and prove that representations of a functor correspond exactly to its universal elements
Prove that a representing object, together with its universal element, is unique up to unique isomorphism
Express products, free objects, quotients and limits as representing objects of suitable functors
Recognise functors that are not representable, and keep covariant and contravariant representability apart

1.
Which pair $(A, u)$ is a universal element of the forgetful functor $U : \cat{Ab} \to \cat{Set}$ from abelian groups to sets?
- A. $(\Z, 1)$
- B. $(\Z/2, 1)$
- C. $(\{0\}, 0)$
- D. $(\Q, 1)$
2.
The group $\Z^2 = \langle a, b \mid ab = ba \rangle$ represents the functor sending a group $G$ to the set of commuting pairs $\{(x, y) \in G^2 \mid xy = yx\}$ . How many group homomorphisms $\Z^2 \to S_3$ are there?
3.
How many different natural isomorphisms $\Hom(\Z, -) \Rightarrow U$ are there, where $U : \cat{Grp} \to \cat{Set}$ is the forgetful functor?
4.
Which pairs $(A, u)$ with $u \subseteq A$ are universal elements of the contravariant power-set functor $\mathcal{P} : \cat{Set}^{\mathrm{op}} \to \cat{Set}$ (acting by preimage)? Choose all that apply.
- A. $(\{0, 1\}, \{1\})$
- B. $(\{0, 1\}, \{0\})$
- C. $(\{0, 1\}, \varnothing)$
- D. $(\{0, 1, 2\}, \{1\})$
5.
In the proof of Theorem 2 we have $\varphi : A \to A'$ with $F(\varphi)(u) = u'$ and $\psi : A' \to A$ with $F(\psi)(u') = u$ . Why is $\psi \circ \varphi = \id_A$ ?
- A.Both $\psi \circ \varphi$ and $\id_A$ send $u$ to $u$ , and by universality of $(A, u)$ only one arrow $A \to A$ does.
- B.Because $F$ is faithful.
- C.Because $\varphi$ and $\psi$ are isomorphisms.
- D.Because $F(\psi \circ \varphi) = F(\id_A)$ as functions on all of $F(A)$ .
6.
Which of these functors $\cat{Set} \to \cat{Set}$ is not representable?
- A. $X \mapsto X \times X \times X$
- B. $X \mapsto \{\ast\}$ (constant one-point set)
- C. $X \mapsto \mathcal{P}(X)$ , with $f$ acting by direct image $S \mapsto f(S)$
- D. $X \mapsto X$ (the identity functor)
7.
Let $\mathcal{F} : \cat{Top}^{\mathrm{op}} \to \cat{Set}$ send a space to its set of closed subsets (a continuous map acts by preimage). Which pair represents $\mathcal{F}$ ? Here $\mathbb{S} = \{0, 1\}$ is the Sierpiński space with open sets $\varnothing, \{1\}, \{0, 1\}$ .
- A. $(\mathbb{S}, \{0\})$
- B. $(\{0, 1\} \text{ discrete}, \{0\})$
- C. $(\{0, 1\} \text{ indiscrete}, \{0\})$
- D. $(\mathbb{S}, \{1\})$
8.
In the lesson's Lean file, imagePowerset : Type ⥤ Type is the covariant power-set functor (direct images). Which Lean statement says that it is not representable, i.e. that no type $A$ represents it?
- A.∀ A : Type, IsEmpty (imagePowerset.CorepresentableBy A)
- B.Nonempty (powerset.RepresentableBy Bool)
- C.∀ A : Type, Nonempty (imagePowerset.CorepresentableBy A)
- D.Subsingleton (imagePowerset.obj PUnit)
9.
Since $\Z/n = \langle a \mid a^n \rangle$ represents $G \mapsto \{g \in G \mid g^n = 1\}$ , the number of homomorphisms $\Z/n \to \Z/m$ equals the number of $k \in \Z/m$ with $nk \equiv 0 \pmod m$ . Write hom_count(n, m) computing this number (for $n, m \ge 1$ ). Then compare your answers with $\gcd(n, m)$ .
Computational — Practice

Answer key — Representable functors

1.
$(\Z, 1)$ . Given $g \in G$ , the map $n \mapsto ng$ is a homomorphism with $1 \mapsto g$ , and any homomorphism $\varphi$ with $\varphi(1) = g$ satisfies $\varphi(n) = ng$ by induction (up and down from $0$ ). So $\Hom(\Z, G) \cong U(G)$ , naturally in $G$ , by $\varphi \mapsto \varphi(1)$ .
2.
A homomorphism $\Z^2 \to G$ is determined by the images $x, y$ of $a, b$ , which can be any commuting pair. In $S_3$ : the identity commutes with all $6$ elements; each of the $3$ transpositions commutes with $2$ elements (itself and $e$ ); each of the $2$ three-cycles commutes with $3$ elements (the rotations). Total $6 + 3 \cdot 2 + 2 \cdot 3 = 18$ . In general the number of commuting pairs is $|G| \cdot k(G)$ , with $k(G)$ the number of conjugacy classes: $6 \cdot 3 = 18$ .
3.
Representations of $U$ by $\Z$ correspond to universal elements $u \in \Z$ (Theorem 1). If $u$ is universal, there is a homomorphism $n \mapsto kn$ with $ku = 1$ , so $u = \pm 1$ . Conversely $-1$ is universal: $g$ is the image of $-1$ under exactly one homomorphism, $n \mapsto g^{-n}$ . So there are exactly $2$ natural isomorphisms, $\varphi \mapsto \varphi(1)$ and $\varphi \mapsto \varphi(-1)$ ; the unique isomorphism relating them is $n \mapsto -n$ (Theorem 2).
4.
A universal element needs, for every $S \subseteq X$ , exactly one $f : X \to A$ with $f^{-1}(u) = S$ . For $A = \{0, 1\}$ and $u = \{1\}$ this is $\chi_S$ ; for $u = \{0\}$ it is $1 - \chi_S$ . Both are universal — consistent with Theorem 2, which gives the unique isomorphism (the swap) relating them. With $u = \varnothing$ nothing nonempty is reached; with $A = \{0, 1, 2\}$ the maps are not unique.
5.
$F(\psi \circ \varphi)(u) = F(\psi)(F(\varphi)(u)) = F(\psi)(u') = u$ , and also $F(\id_A)(u) = u$ . A universal element $(A, u)$ allows exactly one arrow $f : A \to A$ with $F(f)(u) = u$ , so $\psi \circ \varphi = \id_A$ . The same argument with $(A', u')$ gives $\varphi \circ \psi = \id_{A'}$ .
6.
If $F \cong \Hom(A, -)$ then $F(\{\ast\}) \cong \Hom(A, \{\ast\})$ has exactly one element. The covariant power set has $\mathcal{P}(\{\ast\}) = \{\varnothing, \{\ast\}\}$ , so it is not representable. The others are represented by $\{1,2,3\}$ , $\varnothing$ and $\{\ast\}$ . Note the contravariant power set (acting by preimage) is representable, by $\{0, 1\}$ .
7.
A function $f : X \to \mathbb{S}$ is continuous exactly when $f^{-1}(\{1\})$ is open, i.e. when $f^{-1}(\{0\}) = X \setminus f^{-1}(\{1\})$ is closed. So closed sets $C$ correspond bijectively to continuous $f$ with $f^{-1}(\{0\}) = C$ , naturally in $X$ : $(\mathbb{S}, \{0\})$ is a universal element. The same object $\mathbb{S}$ represents open sets via $\{1\}$ : the universal element is part of the data.
8.
Mathlib's F.CorepresentableBy A is the type of natural bijections (A ⟶ Y) ≃ F.obj Y. Non-representability says this type is empty for every A: ∀ A, IsEmpty (imagePowerset.CorepresentableBy A). The proof applies subsingleton_of_corepresentable: a corepresentable functor has at most one element over PUnit, but ∅ ≠ Set.univ in Set PUnit.
9.
```
def hom_count(n, m):
    return sum(1 for k in range(m) if (n * k) % m == 0)
```
The count is always $\gcd(n, m)$ : the solutions of $nk \equiv 0 \pmod m$ are the multiples of $m / \gcd(n, m)$ in $\Z/m$ , and there are $\gcd(n, m)$ of them. Representability turned a question about homomorphisms into a question about solving one equation.