ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 641An indexed V -functor F : A → B consists of V X -enriched functors F X : A X → B Xtogether with isomorphismsF X ◦ f ∗ ∼ = f ∗ ◦ (f ∗ ) • (F Y )such that the following diagrams of isomorphisms commute:F X ◦ (gf) ∗(gf) ∗ ◦ ((gf) ∗ ) • (F Z )F X ◦ f ∗ ◦ (f ∗ ) • (g ∗ ) f ∗ ◦ (f ∗ ) • (F Y ) ◦ (f ∗ ) • (g ∗ )f ∗ ◦ (f ∗ ) • (F Y ◦ g ∗ )f ∗ ◦ (f ∗ ) • (g ∗ ) ◦ (f ∗ ) • (g ∗ ) • (F Z ) f ∗ ◦ (f ∗ ) • (g ∗ ◦ (g ∗ ) • (F Z ))F X ◦ (1 X ) ∗ (1 X ) ∗ ◦ ((1 X ) ∗ ) • (F X )F X .Finally, an indexed V -natural transformation consists of V X -natural transformationsα X : F X → G X such that the following diagram of isomorphisms commutes:F X ◦ f ∗f ∗ ◦ (f ∗ ) • (F Y )G X ◦ f ∗ f ∗ ◦ (f ∗ ) • (G Y ).We denote the resulting 2-category by V -Cat.The required fully-faithfulness of f ∗ : (f ∗ ) • A Y → A X may seem odd. The followingexample should help clarify the intent.4.2. Example. Let V be an ordinary monoidal category and A a (large) V-enrichedcategory. We define an indexed Fam(V)-category Fam(A) where, for a set X, Fam(A) Xis the V X -enriched category of X-indexed families of objects of A. That is, for (A x ) x∈Xand (B x ) x∈X with each A x , B x ∈ A, the hom-object in V X is defined byFam(A) X (A, B) x = A(A x , B x ).For a function f : Y → X, the V Y -enriched category (f ∗ ) • A has hom-objects in V Y :Fam(A) X (A, B) y = A(A f(y) , B f(y) ).Finally, the functor f ∗ : (f ∗ ) • A X → A Y sends an X-indexed family (A x ) x∈X to theY -indexed family defined by (f ∗ A) y = A f(y) . Thus we haveFam(A) Y (f ∗ A, f ∗ B) y = A((f ∗ A) y , (f ∗ B) y ) = A(A f(y) , B f(y) ) = Fam(A) X (A, B) f(y)