Permutative categories, multicategories, and algebraic K-theory

We show that the $K$-theory construction of arXiv:math/0403403, which preserves multiplicative structure, extends to a symmetric monoidal closed bicomplete source category, with the multiplicative structure still preserved. The source category of arXiv:math/0403403, whose objects are permutative categories, maps fully and faithfully to the new source category, whose objects are (based) multicategories.

💡 Research Summary

The paper builds on the algebraic K‑theory construction introduced by Elmendorf and Mandell (arXiv:math/0403403), which assigns a spectrum to a permutative category while preserving multiplicative structure. The author’s main contribution is to enlarge the source category from permutative categories to a much richer setting: the category of based multicategories equipped with a symmetric monoidal closed and bicomplete structure.

A based multicategory is a multicategory together with a distinguished “base” object that behaves like a unit for the tensor product. The author first defines a tensor product ⊗ on based multicategories by adapting Day convolution, and then constructs an internal hom