Closed categories vs. closed multicategories

We prove that the 2-category of closed categories of Eilenberg and Kelly is equivalent to a suitable full 2-subcategory of the 2-category of closed multicategories.

💡 Research Summary

The paper establishes a precise 2‑categorical equivalence between the classical notion of closed categories, as introduced by Eilenberg and Kelly, and the more recent concept of closed multicategories. After recalling that a closed category is a monoidal category equipped with an internal hom functor satisfying the usual adjunction with the tensor product, the authors formalize the 2‑category ClosedCat: objects are closed categories, 1‑cells are strong monoidal functors, and 2‑cells are monoidal natural transformations.

In parallel, a closed multicategory is defined as a multicategory in which every multimorphism (M(X_{1},\dots,X_{n};Y)) is representable by an internal hom object (