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The Lumenary · Advanced undergraduate

Natural transformations

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Define natural transformations and natural isomorphisms, and verify naturality for the double-dual map $V \to V^{**}$ and the determinant $\mathrm{GL}_n(R) \to R^\times$ , natural in the commutative ring $R$
Explain why $V \cong V^{**}$ is natural while $V \cong V^*$ cannot even be stated as a natural transformation between functors of the same variance, and prove that it fails even on the groupoid of isomorphisms over a field with more than three elements
Prove that functors $\mathcal{C} \to \mathcal{D}$ and natural transformations form a category $[\mathcal{C}, \mathcal{D}]$ , in which the isomorphisms are exactly the natural transformations with invertible components
Use naturality as a uniformity constraint, for example to show that the only natural maps $X \times X \to X$ on sets are the two projections
Define equivalence of categories and prove that a functor is an equivalence if and only if it is full, faithful and essentially surjective

1.
Let $F, G : \mathcal{C} \to \mathcal{D}$ be functors and $\alpha_A : F(A) \to G(A)$ a family of morphisms. Which equation must hold for every $f : A \to B$ for $\alpha$ to be natural?
- A. $G(f) \circ \alpha_A = \alpha_B \circ F(f)$
- B. $F(f) \circ \alpha_A = \alpha_B \circ G(f)$
- C. $\alpha_B \circ G(f) = G(f) \circ \alpha_A$
- D. $\alpha_A = \alpha_B$ whenever there is an arrow $A \to B$
2.
Let $T : \cat{Set} \to \cat{Set}$ be the functor $T(X) = X \times X \times X$ , $T(f) = f \times f \times f$ . How many natural transformations $T \Rightarrow \mathrm{id}_{\cat{Set}}$ are there?
3.
Which of these families are natural transformations (between the functors indicated)? Select all that apply.
- A. $\mathrm{rev}_X : L(X) \to L(X)$ , reversing a finite list (on $\cat{Set}$ , with $L$ the list functor)
- B. $X \to L(X)$ , $x \mapsto [x, x]$ (from $\mathrm{id}_{\cat{Set}}$ to $L$ )
- C. $q_G : G \to G/[G, G]$ (from $\mathrm{id}_{\cat{Grp}}$ to abelianisation)
- D.sorting a list of integers into increasing order, as a map $L(\Z) \to L(\Z)$ “natural in the set of entries”
- E. $\det : \mathrm{GL}_n(R) \to R^\times$ (from $\mathrm{GL}_n$ to $(-)^\times$ , functors $\cat{CRing} \to \cat{Grp}$ )
4.
Let $\alpha : \mathrm{id}_{\cat{Ab}} \Rightarrow \mathrm{id}_{\cat{Ab}}$ be a natural transformation with $\alpha_{\Z}(1) = 7$ . What is $\alpha_{\Z/12}(5)$ ? Give the representative in $\{0, 1, \dots, 11\}$ .
5.
In the proof that an equivalence $F$ (with $\eta : \mathrm{id} \cong GF$ , $\varepsilon : FG \cong \mathrm{id}$ ) is full, we set $f = \eta_B^{-1} \circ G(h) \circ \eta_A$ for a given $h : F(A) \to F(B)$ and showed $GF(f) = G(h)$ . What is needed to conclude $F(f) = h$ ?
- A.That $G$ is faithful, which follows from the same argument applied to $G$ (using $\varepsilon^{-1}$ )
- B.That $F$ is faithful
- C.The axiom of choice
- D.Nothing: $GF(f) = G(h)$ already means $F(f) = h$
6.
Which statement about $\mathrm{ev}_V : V \to V^{**}$ on the category of all vector spaces over a field $k$ is true?
- A. $\mathrm{ev}$ is natural, every component is injective, but for infinite-dimensional $V$ it is not surjective
- B. $\mathrm{ev}$ is a natural isomorphism on all vector spaces
- C. $\mathrm{ev}$ is natural only on finite-dimensional spaces
- D. $\mathrm{ev}_V$ fails to be injective for some infinite-dimensional $V$
7.
Mathlib writes composition in diagrammatic order: f ≫ g means “first f, then g”. For α : F ⟶ G between functors C ⥤ D and f : X ⟶ Y in C, which Lean statement is the naturality square?
- A.F.map f ≫ α.app Y = α.app X ≫ G.map f
- B.α.app Y ≫ F.map f = G.map f ≫ α.app X
- C.F.map f ≫ G.map f = α.app X ≫ α.app Y
- D.α.app X = α.app Y
8.
Which pairs of categories are equivalent? Select all that apply.
- A. $\cat{FinVect}_{\R}$ and $\cat{Mat}_{\R}$ (objects $n \in \N$ , morphisms matrices)
- B. $\cat{FinSet}$ and its full subcategory on the sets $\{1, \dots, n\}$ , $n \ge 0$
- C. $\cat{Set}$ and $\cat{Set}^{\mathrm{op}}$
- D.the category with two objects and exactly one arrow between any two objects, and the category $\mathbf{1}$
- E. $\mathbf{B}\Z/2$ and $\mathbf{B}\Z/3$ (one-object categories of the groups)
9.
A binary operation on $X = \{0, \dots, n-1\}$ is stored as a tuple t with t[a * n + b] $= \alpha(a, b)$ . Write natural_tables(n) returning the list of all such tables that commute with every function $f : X \to X$ , i.e. t[f[a] * n + f[b]] == f[t[a * n + b]] for all $a, b$ (with f a tuple of values). Theorem 3 predicts the answer.
Computational — Practice

Answer key — Natural transformations

1.
The naturality square has corners $F(A), G(A), F(B), G(B)$ . The two paths from $F(A)$ to $G(B)$ are $G(f) \circ \alpha_A$ (across, then down) and $\alpha_B \circ F(f)$ (down, then across), and naturality says they are equal. The other options either do not type-check or compare morphisms with different domains.
2.
Let $B = \{0, 1, 2\}$ and $c = \alpha_B(0, 1, 2) \in B$ . For any $X$ and $(a, b, d) \in X^3$ , naturality at $f : B \to X$ , $f(0) = a, f(1) = b, f(2) = d$ , gives $\alpha_X(a, b, d) = f(c)$ , which is the $c$ -th coordinate. So $\alpha$ is one of the three projections, and each projection is natural. Answer: $3$ . (In Yoneda terms: $X^3 \cong \Hom(B, X)$ and $\mathrm{Nat}(\Hom(B, -), \mathrm{id}) \cong B$ , a three-element set.)
3.
Reversal, duplication, the abelianisation maps and the determinant are natural; in each case the square commutes for every morphism because the construction uses no extra information about the objects. Sorting is not: it needs an order on the entries, and on $L(\Z)$ it fails to commute with negation.
4.
With $f : \Z \to \Z/12$ , $f(1) = 5$ , naturality gives $\alpha_{\Z/12}(5) = \alpha_{\Z/12}(f(1)) = f(\alpha_\Z(1)) = f(7) = 35 \equiv 11 \pmod{12}$ . In general $\alpha_A(a) = 7a$ : natural endomorphisms of the identity of $\cat{Ab}$ are multiplications by integers.
5.
From $G(F(f)) = G(h)$ we may cancel $G$ if $G$ is faithful. $G$ is faithful by the faithfulness argument applied to $G$ with the natural isomorphism $\varepsilon^{-1} : \mathrm{id}_{\mathcal{D}} \cong FG$ : every $k : X \to Y$ in $\mathcal{D}$ equals $\varepsilon_Y \circ FG(k) \circ \varepsilon_X^{-1}$ . In the Lean proof this is the line have hG : G.Faithful := faithful_of_unitIso G F ε.symm.
6.
Naturality holds on all of $\cat{Vect}_k$ (the calculation uses no dimension). Injectivity holds for all $V$ (extend a non-zero vector to a basis). Surjectivity uses the dimension count $\dim V^{**} = \dim V$ , valid only in finite dimension; for infinite-dimensional $V$ , $\dim V^{**} > \dim V$ . So $\mathrm{ev}$ is a natural transformation everywhere and a natural isomorphism exactly on $\cat{FinVect}_k$ .
7.
The square $G(f) \circ \alpha_X = \alpha_Y \circ F(f)$ reads, in diagrammatic order, α.app X ≫ G.map f = F.map f ≫ α.app Y, which is option (a) with the sides swapped. Mathlib's field NatTrans.naturality states exactly F.map f ≫ α.app Y = α.app X ≫ G.map f.
8.
Equivalent: $\cat{FinVect}_\R \simeq \cat{Mat}_\R$ ; $\cat{FinSet}$ and its skeleton; the two-object codiscrete category and $\mathbf{1}$ . Not equivalent: $\cat{Set}$ and $\cat{Set}^{\mathrm{op}}$ (the initial object of $\cat{Set}$ is strict, the terminal object is not); $\mathbf{B}\Z/2$ and $\mathbf{B}\Z/3$ (an equivalence of one-object categories is a group isomorphism).
9.
```
from itertools import product

def natural_tables(n):
    tables = product(range(n), repeat=n * n)
    maps = list(product(range(n), repeat=n))
    return [t for t in tables
            if all(t[f[a] * n + f[b]] == f[t[a * n + b]]
                   for f in maps for a in range(n) for b in range(n))]
```
For $n = 2$ and $n = 3$ only the two projections survive, $(a, b) \mapsto a$ and $(a, b) \mapsto b$ . (For $n = 1$ both projections are the same table.) The search is evidence for small $n$ ; Theorem 3 proves the statement for every set, using only the test pair $(0, 1)$ in a two-element set.