ENRICHED INDEXED CATEGORIES Contents 1. Introduction

ENRICHED INDEXED CATEGORIES 6252.14. Theorem. Let V be an S-indexed symmetric monoidal category with indexed products,and indexed coproducts preserved by ⊗. Then the following are equivalent.(i) Each fiber V X is closed symmetric monoidal and each restriction functor f ∗ is closedsymmetric monoidal. This means that for B, C ∈ V X we have V X (B, C) ∈ V Xand isomorphismsV X (A ⊗ X B, C) ∼ = V X ( A, V X (B, C) ) ,natural in A, and moreover the canonical mapsare isomorphisms.f ∗ V Y (B, C) −→ V X (f ∗ B, f ∗ C) (2.15)(ii) For any B ∈ V Yand C ∈ V X×Y , we have a V [Y ] (B, C) ∈ V X and isomorphismsV X×Y (A ⊗ B, C) ∼ )= V(A, X V [Y ] (B, C)natural in A, and moreover the resulting canonical mapsare isomorphisms.f ∗ V [Y ] (B, C) −→ V [Y ] (B, (f × 1) ∗ C)(iii) For any B ∈ V Y and C ∈ V X we have a V (B, C) ∈ V X×Y and isomorphismsV X (A ⊗ [Y ] B, C) ∼ = V X×Y (A, V (B, C)) (2.16)natural in A, and moreover the resulting canonical mapsare isomorphisms.(f × g) ∗ V (B, C) → V (g ∗ B, f ∗ C)When these conditions hold, we say that V is closed.Proof. The relationships between the three kinds of hom-functors areV X (B, C) ∼ ( = V[X]B, ∆ X∗ C ) ∼ = ∆∗X V (B, C)V (B, C) ∼ ( = VY ×XπX ∗ B, π∗ Y C) ∼ ( = V[Y ]B, ∆ Y ∗ πY ∗ C)V [Y ] (B, C) ∼ = πY ∗ V ( X×Y πX ∗ B, C) ∼ = πY ∗ ∆ ∗ Y V (B, C).Checking that the canonical maps coincide is an exercise in diagram chasing. The equivalenceof (i) and (ii) can be found in [Shu08].