An enriched category is a category with generalized morphisms.

# 1 Enriched category

A category $$\cC = (\Obj, \Hom, \compL, \id)$$ enriched in a monoidal category $$(\cV, \otimes, I, \alpha, \lambda, \rho)$$ ($$\cV$$-enriched category) consists of

• a collection of objects $$\Obj$$
• a hom-object $$\Hom(A, B) \in \Obj_\cV$$ between objects
• a composition morphism $$\compL_{A, B, C}: \Hom(B, C) \otimes \Hom(A, B) \to \Hom(A, C)$$ for each triple of objects
• an identity element $$\id_A: I \to \Hom(A, A)$$ for each object

subject to

• associativity: $$\compL (\id \otimes \compL) \alpha = \compL (\compL \otimes \id)$$
• identity: $$\compL (\id_B \otimes \id) = \lambda$$ and $$\compL (\id \otimes \id_A) = \rho$$

# 2 Enriched functor

For $$\cV$$-enriched categories $$\cC$$ and $$\cD$$, a $$\cV$$-enriched functor $$F: \cC \to \cD$$ consists of

• a function sending $$\cC$$-objects to $$\cD$$-objects $$F: \Obj_\cC \to \Obj_\cD:= A \mapsto FA$$
• a collection of $$\cV$$-morphisms indexed by pairs of $$\cC$$-objects $$F_{A, B}: \Hom_\cC(A, B) \to \Hom_\cD(FA, FB)$$

in such a way that $$F$$ preserves

• composition: $$F_{A, C} \compL_\cC = \compL_\cD (F_{B, C} \otimes F_{A, B})$$
• identity: $$F_{A, A} \id_{\cC:A} = \id_{\cD:FA}$$

# 3 Enriched natural transformation

For $$\cV$$-enriched functors $$F, G: \cC \to \cD$$, a $$\cV$$-enriched natural transformation $$\alpha: F \nat G$$ is a collection of $$\cV$$-morphisms indexed by $$\cC$$-objects $$\alpha_A: I \to \Hom_\cD(FA, GA)$$ that satisfies

• naturality: $$\compL_\cD (\alpha_B \otimes F_{A, B}) \lambda\inv = \compL_\cD (G_{A, B} \otimes \alpha_A) \rho\inv$$

$$\cV\cCat$$: 2-category of $$\cV$$-enriched categories, $$\cV$$-enriched functors, and $$\cV$$-enriched natural transformations

# 4 Construction

$$\cV$$-product category $$\cC \otimes \cD$$:

• $$\Obj_{\cC \otimes \cD} := \Obj_\cC \times \Obj_\cD$$
• $$\Hom_{\cC \otimes \cD}((A, B), (A', B')) := \Hom_\cC(A, A') \otimes \Hom_\cD(B, B') \in \Obj_\cV$$

The unit of $$\cV$$-product is $$\cI := (\singleton, I, \compL, \id)$$.

A closed monoidal category is enriched over itself, but not vice versa.

$$\cV$$-bifunctor:

\begin{aligned} \begin{array}{cccc} F: & \cC \otimes \cD & \to & \cE \\ & (A, B) & \mapsto & F(A, B) \\ & \substack{ \Hom_{\cC \otimes \cD}((A, B), (A', B')) \\ \Hom_\cC(A, A') \otimes \Hom_\cD(B, B') } & \mapsto & \Hom_\cE(F(A, B), F(A',B')) \end{array} \end{aligned}

# 5 Base change

A lax monoidal functor $$(F: \cV \to \cW, \mu, \eta)$$ induces a 2-functor $$F_*: \cV\cCat \to \cW\cCat$$ between enriched categories, where

• $$\Obj_{F_*\cC} := \Obj_\cC$$
• $$\Hom_{F_*\cC}(A, B) := F \Hom_\cC(A, B) \in \Obj_\cW$$
• $$\compL_{F_*\cC: A, B, C} := \substack{ \Hom_{F_*\cC}(B, C) \otimes \Hom_{F_*\cC}(A, B) \\ F \Hom_\cC(B, C) \otimes F \Hom_\cC(A, B) } \xto{\mu} F (\Hom_\cC(B, C) \otimes \Hom_\cC(A, B)) \xto{F \compL_{\cC: A, B, C}} \substack{ F \Hom_\cC(A, C) \\ \Hom_{F_*\cC}(A, C) }$$
• $$\id_{F_* \cC: A} := I_\cW \xto{\eta} F I_\cV \xto{F \id_{\cC: A}} \substack{ F \Hom_\cC(A, A) \\ = \Hom_{F_* \cC}(A, A) }$$

# 6 Examples

## 6.1 Category

### 6.1.1 Locally small category

• $$(\cSet, \times, \singleton)$$
• $$\Hom(A, B) \in \cSet$$
• $$\compL: \Hom(B, C) \times \Hom(A, B) \to \Hom(A, C)$$ (morphism composition)
• $$\id_A: \singleton \to \Hom(A, A)$$ (identity morphism)
• $$\cSet$$-functor: functor
• $$\cCat$$: small categories and functors

### 6.1.2 Strict 2-category

• $$(\cCat, \times, \cOne)$$
• $$\cC(A, B) \in \cCat$$ (1-cells $$f, g: A \to B$$, 2-cells $$\alpha: f \nat g$$, and vertical composition)
• $$\cC(B, C) \times \cC(A, B) \to \cC(A, C)$$ (horizontal composition)
• $$\cOne \to \cC(A, A)$$ (identity 1-cell $$\id_A$$ and identity 2-cell $$\id_{\id_A}$$)
• $$\cCat$$-functor: 2-functor
• $$\cCat\cCat$$?

## 6.2 Metric

### 6.2.1 Preorder

• $$\cBool = (\set{\lfal, \ltru}, \lfal \lproves \ltru, \lcon, \ltru)$$
• $$a \leq b$$
• $$(b \leq c) \lcon (a \leq b) \lproves (a \leq c)$$ (transitivity)
• $$\ltru \limp (a \leq a)$$ (reflexivity)
• $$\cBool$$-functor: monotone map $$(a \leq_\cC b) \lproves (Fa \leq_\cD Fb)$$
• $$\cPos$$: posets and monotones

### 6.2.2 Metric space

• $$(\cQ, \geq, \otimes, I)$$ monoidal preorder
• $$d(a, b) \in \cQ$$
• $$d(b, c) \otimes d(a, b) \geq d(a, c)$$ (triangle inequality)
• $$I \geq d(a, a)$$ (indiscernibility of identicals)
• $$\cQ$$-functor: metric map $$d_\cC(a, b) \geq d_\cD(Fa, Fb)$$
• $$\cMet_\cQ$$: $$\cQ$$-metric spaces and $$\cQ$$-metric maps

A Lawvere metric space is enriched in $$\cCost = ([0, \infty], \geq, +, 0)$$.

The Hausdorff metric is a metric on the powerset $$(PA, d_{PA})$$ induced by a metric on a set $$(A, d_A)$$.

$d_{PA}(X, Y) := \bigmeet_{x \in X} \bigjoin_{y \in Y} d_A(x, y)$

### 6.2.3 Metric category

• $$(\cMet_\cQ, \otimes, \cI)$$

• $$d_{[A, B]} \in \cMet_\cQ$$

• $$d_{[B, C]} \otimes d_{[A, B]} \to d_{[A, C]}$$ (metric composition)

$d_{[B, C]} \otimes d_{[A, B]}((g, f), (g', f')) := d_{[B, C]}(g, g') \otimes d_{[A, B]}(f, f') \geq d_{[A, C]}(g \compL f, g' \compL f')$

• $$\cI \to d_{[A, A]}$$ (identity morphism)

$I \geq d_{[\singleton, \singleton]}(\id_\singleton, \id_\singleton) \geq d_{[A, A]}(\id_A, \id_A)$

$d_{[A, D]}(b \compL f, c) \otimes d_{[B, D]}(h \compL g, b) \otimes d_{[A, C]}(a, g \compL f) \geq d_{[A, D]}(h \compL a, c)$

$d_{[A, D]}(b \compL f, c) \otimes d_{[A, D]}(h \compL g \compL f, b \compL f) \otimes d_{[A, D]}(h \compL a, h \compL g \compL f) \geq d_{[A, D]}(h \compL a, c)$

$d_{[A, C]}(a, g \compL f) \geq d_{[C, D]}(h, h) \otimes d_{[A, C]}(a, g \compL f) \geq d_{[A, D]}(h \compL a, h \compL g \compL f)$

$d_{[B, D]}(h \compL g, b) \geq d_{[B, D]}(h \compL g, b) \otimes d_{[A, B]}(f, f) \geq d_{[A, D]}(h \compL g \compL f, b \compL f)$

$d_{[B, D]}(h \compL g, b) \otimes d_{[A, C]}(a, g \compL f) \otimes d_{[A, D]}(c, h \compL a) \geq d_{[A, D]}(c, b \compL f)$

$d_{[A, D]}(h \compL g \compL f, b \compL f) \otimes d_{[A, D]}(h \compL a, h \compL g \compL f) \otimes d_{[A, D]}(c, h \compL a) \geq d_{[A, D]}(c, b \compL f)$

$d_{[A, C]}(a, g \compL f) \geq d_{[C, D]}(h, h) \otimes d_{[A, C]}(a, g \compL f) \geq d_{[A, D]}(h \compL a, h \compL g \compL f)$

$d_{[B, D]}(h \compL g, b) \geq d_{[B, D]}(h \compL g, b) \otimes d_{[A, B]}(f, f) \geq d_{[A, D]}(h \compL g \compL f, b \compL f)$

The supremum metric is a metric on the function space $$(\Hom(A, B), d_{[A, B]})$$ induced by a metric on the codomain $$(B, d_B)$$:

$d_{[A, B]}(f, g) := \bigmeet_{a \in A} d_B(fa, ga)$

• triangle inequality:

\begin{aligned} & d_{[A, B]}(g, h) \otimes d_{[A, B]}(f, g) \\ = & \bigmeet_{a \in A} d_B(ga, ha) \otimes \bigmeet_{a \in A} d_B(fa, ga) \overset{\angles{p_a \otimes p_a}}{\geq} \bigmeet_{a \in A} (d_B(ga, ha) \otimes d_B(fa, ga)) \overset{\bigmeet \compL_{fa, ga, ha}}{\geq} \bigmeet_{a \in A} d_B(fa, ha) \\ = & d_{[A, B]}(f, h) \end{aligned}

• indiscernibility of identicals:

$I \overset{\angles{\id_{fa}}}{\geq} \bigmeet_{a \in A} d_B(fa, fa) = d_{[A, B]}(f, f)$