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The Lumenary · First-year graduate

Enriched categories

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Date: ______________________

You will learn to

Define categories, functors and natural transformations enriched in a monoidal category $\mathcal{V}$ , and the underlying ordinary category
Show that Lawvere metric spaces are exactly the categories enriched in $([0, \infty], \ge, +, 0)$ , with the triangle inequality as composition, and that enriched functors are the non-expansive maps
Show that locally small categories, preorders and rings are the categories enriched in sets, truth values and abelian groups (one object)
Explain why additive categories are $\cat{Ab}$ -enriched and strict 2-categories are $\cat{Cat}$ -enriched, and derive the interchange law
Use closedness of $\mathcal{V}$ to prove the enriched Yoneda embedding for Lawvere metric spaces

1.
In a category enriched in $([0, \infty], \ge, +, 0)$ , the hom-object from $x$ to $y$ is a number $d(x, y)$ . What does the composition morphism $X(y, z) \otimes X(x, y) \to X(x, z)$ say?
- A. $d(x, z) \le d(x, y) + d(y, z)$
- B. $d(x, x) = 0$
- C. $d(x, y) = d(y, x)$
- D. $d(x, z) \ge d(x, y) + d(y, z)$
2.
Directed footpaths between four places take these times (minutes): $A \to B$ : 4, $B \to C$ : 3, $C \to D$ : 5, $A \to C$ : 9, $D \to A$ : 2, $B \to D$ : 10. In the free Lawvere metric space on this weighted graph (shortest-path distance), what is $d(A, D)$ ?
3.
Replace $+$ by $\max$ : let $\mathcal{V} = ([0, \infty], \ge, \max, 0)$ . What is a $\mathcal{V}$ -category?
- A.A set with $d(x, x) = 0$ and $d(x, z) \le \max(d(x, y), d(y, z))$ — a generalised ultrametric space
- B.An ordinary metric space
- C.A preorder
- D.Nothing: $\max$ is not a monoidal structure
4.
Which statement about one-object enriched categories is correct?
- A.A one-object $\mathbf{Ab}$ -category is a ring
- B.A one-object $\mathbf{Vect}_k$ -category is a field extension of $k$
- C.A one-object $\mathbf{Set}$ -category is a group
- D.A one-object $[0, \infty]$ -category is any number $d \ge 0$
5.
The category with abelian groups as objects and all functions as morphisms has an abelian group $B^A$ of morphisms from $A$ to $B$ (pointwise addition). Why is it not an $\mathbf{Ab}$ -enriched category?
- A.Composition is not bilinear: $h \circ (f + g) \neq h \circ f + h \circ g$ when $h$ is not additive
- B. $B^A$ is not an abelian group
- C.There are no identity morphisms
- D.There are too many objects
6.
Let $f, g : X \to Y$ be non-expansive maps between Lawvere metric spaces, viewed as $[0, \infty]$ -functors. When is there a $[0, \infty]$ -natural transformation $f \Rightarrow g$ ?
- A.Exactly when $d(f x, g x) = 0$ for every $x$ , and then it is unique
- B.Always, since naturality is automatic
- C.Exactly when $\sup_x d(fx, gx) < \infty$
- D.Only when $f = g$
7.
In the Lean file, Dist is the order dual of ℝ≥0∞, so a ⟶ b holds when a ≥ b as distances, and Mathlib writes the composition of an enriched category as comp x y z : Hom x y ⊗ Hom y z ⟶ Hom x z. In lawvereEquivEnriched, what does homOfLE (M.triangle x y z) provide?
- A.The composition morphism, from the triangle inequality M.d x z ≤ M.d x y + M.d y z
- B.The identity morphism 𝟙_ Dist ⟶ Hom x x
- C.A proof of the associativity axiom
- D.A proof that M.d is symmetric
8.
Theorem 4 says $d(x, y) = \sup_z \big(d(z, y) \mathbin{\dot-} d(z, x)\big)$ in any Lawvere metric space. Which $z$ attains the supremum, and why is no term larger?
- A. $z = x$ ; each term is $\le d(x, y)$ by the triangle inequality $d(z, y) \le d(z, x) + d(x, y)$
- B. $z = y$ ; by symmetry of $d$
- C.No single $z$ ; the supremum is a limit
- D.Every $z$ attains it, because the terms are all equal
9.
Write free_metric(w): given a square matrix w (list of lists) of direct path lengths, with 0 on the diagonal and float('inf') where there is no path, return the shortest-path distance matrix — the free Lawvere metric. Use repeated min-plus squaring $D \mapsto D \otimes D$ , $(D \otimes D)_{ik} = \min_j (D_{ij} + D_{jk})$ , until the matrix stops changing.
Computational — Practice

Answer key — Enriched categories

1.
The composition morphism is an arrow $d(y, z) + d(x, y) \to d(x, z)$ in $([0, \infty], \ge)$ , i.e. $d(y, z) + d(x, y) \ge d(x, z)$ : the triangle inequality. The identity $0 \to d(x, x)$ is $d(x, x) = 0$ , and the axioms hold automatically because there is at most one arrow between two numbers.
2.
Routes: $A \to B \to D$ takes $4 + 10 = 14$ ; $A \to C \to D$ takes $9 + 5 = 14$ ; $A \to B \to C \to D$ takes $4 + 3 + 5 = 12$ . So $d(A, D) = 12$ , while $d(D, A) = 2$ . This min-plus computation is exactly what the experiment enriched_categories_shortest_paths does with tropical matrix multiplication.
3.
$\max$ is associative, commutative and monotone with unit $0$ , so it makes $([0, \infty], \ge)$ monoidal. An identity says $d(x, x) = 0$ and composition says $\max(d(x, y), d(y, z)) \ge d(x, z)$ : these are generalised ultrametric spaces, such as the $p$ -adic numbers with $d(x, y) = |x - y|_p$ .
4.
A one-object $\mathcal{V}$ -category is a monoid object in $\mathcal{V}$ : in $\mathbf{Set}$ a monoid, in $\mathbf{Ab}$ a ring, in $\mathbf{Vect}_k$ a $k$ -algebra, and in $[0, \infty]$ the single number $0$ .
5.
Precomposition is additive, $(f + g) \circ h = f \circ h + g \circ h$ , but postcomposition with a non-additive $h$ is not: with $h(x) = x^2$ and $f = g = \mathrm{id}_{\mathbb{Z}}$ , $h \circ (f + g)$ sends $x$ to $4x^2$ while $h \circ f + h \circ g$ sends it to $2x^2$ . So composition is not bilinear and the hom-groups are not an enrichment.
6.
A component $\theta_x : 0 \to d(fx, gx)$ exists iff $0 \ge d(fx, gx)$ . The naturality condition is an equation between parallel arrows of a preorder, so it holds automatically, and the transformation is unique when it exists. So $f \Rightarrow g$ exists iff $d(fx, gx) = 0$ for all $x$ — i.e. iff the hom-object $\sup_x d(fx, gx)$ of the functor category is $0$ .
7.
homOfLE turns the inequality M.d x z ≤ M.d x y + M.d y z (which, in the order dual, reads d x y + d y z ≤ d x z) into the arrow Hom x y ⊗ Hom y z ⟶ Hom x z of Dist — the composition. The identity comes from M.self, and the axioms from Subsingleton.elim.
8.
At $z = x$ the term is $d(x, y) \mathbin{\dot-} 0 = d(x, y)$ . For every $z$ , the triangle inequality $d(z, y) \le d(z, x) + d(x, y)$ is equivalent to $d(z, y) \mathbin{\dot-} d(z, x) \le d(x, y)$ , by the closed structure of $[0, \infty]$ . This is the enriched Yoneda lemma; Lean proves it as yoneda_isometry.
9.
```
def free_metric(w):
    n = len(w)
    d = w
    while True:
        nxt = [[min(d[i][j] + d[j][k] for j in range(n)) for k in range(n)]
               for i in range(n)]
        if nxt == d:
            return d
        d = nxt
```
Because the diagonal is $0$ , each squaring can only decrease entries, and after the $m$ -th squaring the matrix records shortest routes with at most $2^m$ steps; so it stabilises after about $\log_2 n$ squarings. The result has zero diagonal and satisfies $D \otimes D = D$ — precisely a Lawvere metric, the free $[0, \infty]$ -category on the weighted graph.