An enriched category is a category with generalized morphisms.

1 Enriched category

A category $C = (Obj, Hom, \circ, id)$ enriched in a monoidal category $(V, \otimes, I, α, λ, ρ)$ ( $V$ -enriched category) consists of

a collection of objects $Obj$
a hom-object $Hom (A, B) \in {Obj}_{V}$ between objects
a composition morphism $\circ_{A, B, C} : Hom (B, C) \otimes Hom (A, B) \to Hom (A, C)$ for each triple of objects
an identity element ${id}_{A} : I \to Hom (A, A)$ for each object

subject to

associativity: $\circ (id \otimes \circ) α = \circ (\circ \otimes id)$
identity: $\circ ({id}_{B} \otimes id) = λ$ and $\circ (id \otimes {id}_{A}) = ρ$

/assets/diagrams/enriched_category_associativity.svg

/assets/diagrams/enriched_category_identity.svg

2 Enriched functor

For $V$ -enriched categories $C$ and $D$ , a $V$ -enriched functor $F : C \to D$ consists of

a function sending $C$ -objects to $D$ -objects $F : {Obj}_{C} \to {Obj}_{D} := A \mapsto F A$
a collection of $V$ -morphisms indexed by pairs of $C$ -objects $F_{A, B} : {Hom}_{C} (A, B) \to {Hom}_{D} (F A, F B)$

in such a way that $F$ preserves

composition: $F_{A, C} \circ_{C} = \circ_{D} (F_{B, C} \otimes F_{A, B})$
identity: $F_{A, A} {id}_{C : A} = {id}_{D : F A}$

/assets/diagrams/enriched_functor_composition.svg

/assets/diagrams/enriched_functor_identity.svg

3 Enriched natural transformation

For $V$ -enriched functors $F, G : C \to D$ , a $V$ -enriched natural transformation $α : F \Rightarrow G$ is a collection of $V$ -morphisms indexed by $C$ -objects $α_{A} : I \to {Hom}_{D} (F A, G A)$ that satisfies

naturality: $\circ_{D} (α_{B} \otimes F_{A, B}) λ^{- 1} = \circ_{D} (G_{A, B} \otimes α_{A}) ρ^{- 1}$

/assets/diagrams/enriched_natural_transformation_naturality.svg

$V Cat$ : 2-category of $V$ -enriched categories, $V$ -enriched functors, and $V$ -enriched natural transformations

4 Construction

$V$ -product category $C \otimes D$ :

${Obj}_{C \otimes D} := {Obj}_{C} \times {Obj}_{D}$
${Hom}_{C \otimes D} ((A, B), (A^{'}, B^{'})) := {Hom}_{C} (A, A^{'}) \otimes {Hom}_{D} (B, B^{'}) \in {Obj}_{V}$

The unit of $V$ -product is $I := ({*}, I, \circ, id)$ .

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

$V$ -bifunctor:

$\begin{array}{r} \begin{array}{cccc} F : & C \otimes D & \to & E \\ (A, B) & \mapsto & F (A, B) \\ \begin{matrix} {Hom}_{C \otimes D} ((A, B), (A^{'}, B^{'})) \\ {Hom}_{C} (A, A^{'}) \otimes {Hom}_{D} (B, B^{'}) \end{matrix} & \mapsto & {Hom}_{E} (F (A, B), F (A^{'}, B^{'})) \end{array} \end{array}$

5 Base change

A lax monoidal functor $(F : V \to W, μ, η)$ induces a 2-functor $F_{*} : V Cat \to W Cat$ between enriched categories, where

${Obj}_{F_{*} C} := {Obj}_{C}$
${Hom}_{F_{*} C} (A, B) := F {Hom}_{C} (A, B) \in {Obj}_{W}$
$\circ_{F_{*} C : A, B, C} := \begin{matrix} {Hom}_{F_{*} C} (B, C) \otimes {Hom}_{F_{*} C} (A, B) \\ F {Hom}_{C} (B, C) \otimes F {Hom}_{C} (A, B) \end{matrix} \overset{μ}{\to} F ({Hom}_{C} (B, C) \otimes {Hom}_{C} (A, B)) \overset{F \circ_{C : A, B, C}}{\to} \begin{matrix} F {Hom}_{C} (A, C) \\ {Hom}_{F_{*} C} (A, C) \end{matrix}$
${id}_{F_{*} C : A} := I_{W} \overset{η}{\to} F I_{V} \overset{F {id}_{C : A}}{\to} \begin{matrix} F {Hom}_{C} (A, A) \\ = {Hom}_{F_{*} C} (A, A) \end{matrix}$

6 Examples

6.1 Category

6.1.1 Locally small category

$(Set, \times, {*})$
$Hom (A, B) \in Set$
$\circ : Hom (B, C) \times Hom (A, B) \to Hom (A, C)$ (morphism composition)
${id}_{A} : {*} \to Hom (A, A)$ (identity morphism)
$Set$ -functor: functor
$Cat$ : small categories and functors

6.1.2 Strict 2-category

$(Cat, \times, 1)$
$C (A, B) \in Cat$ (1-cells $f, g : A \to B$ , 2-cells $α : f \Rightarrow g$ , and vertical composition)
$C (B, C) \times C (A, B) \to C (A, C)$ (horizontal composition)
$1 \to C (A, A)$ (identity 1-cell ${id}_{A}$ and identity 2-cell ${id}_{{id}_{A}}$ )
$Cat$ -functor: 2-functor
$Cat Cat$ ?

6.2 Metric

6.2.1 Preorder

$({⊥, ⊤}, ⊥ ⊢ ⊤, \land, ⊤)$
$a \leq b$
$(b \leq c) \land (a \leq b) ⊢ (a \leq c)$ (transitivity)
$⊤ \to (a \leq a)$ (reflexivity)
${⊥, ⊤}$ -functor: monotone map $(a \leq_{C} b) ⊢ (F a \leq_{D} F b)$
$Pos$ : posets and monotones

6.2.2 Metric space

$(Q, \geq, \otimes, I)$ monoidal preorder
$d (a, b) \in Q$
$d (b, c) \otimes d (a, b) \geq d (a, c)$ (triangle inequality)
$I \geq d (a, a)$ (indiscernibility of identicals)
$Q$ -functor: metric map $d_{C} (a, b) \geq d_{D} (F a, F b)$
${Met}_{Q}$ : $Q$ -metric spaces and $Q$ -metric maps

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

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

$d_{P A} (X, Y) := \underset{x \in X}{⋀} \underset{y \in Y}{⋁} d_{A} (x, y)$

6.2.3 Metric category

$({Met}_{Q}, \otimes, I)$
$d_{[A, B]} \in {Met}_{Q}$
$d_{[B, C]} \otimes d_{[A, B]} \to d_{[A, C]}$ (metric composition)

$d_{[B, C]} \otimes d_{[A, B]} ((g, f), (g^{'}, f^{'})) := d_{[B, C]} (g, g^{'}) \otimes d_{[A, B]} (f, f^{'}) \geq d_{[A, C]} (g \circ f, g^{'} \circ f^{'})$
$I \to d_{[A, A]}$ (identity morphism)

$I \geq d_{[{*}, {*}]} ({id}_{{*}}, {id}_{{*}}) \geq d_{[A, A]} ({id}_{A}, {id}_{A})$

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

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

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

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

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

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