Homotopy Equivalent Algebraic Structures in Multicategories and Permutative Categories

TL;DR

This paper demonstrates that the free construction from multicategories to permutative categories induces an equivalence of homotopy theories, with applications to ring categories, advancing the understanding of algebraic structures in category theory.

Contribution

It establishes that the functor from multicategories to permutative categories is a categorically-enriched non-symmetric multifunctor and proves the induced functor between algebra categories is a homotopy equivalence.

Findings

The free construction induces an equivalence of homotopy theories.

The functor is a categorically-enriched non-symmetric multifunctor.

Application to ring categories demonstrates practical relevance.

Abstract

We show that the free construction from multicategories to permutative categories is a categorically-enriched non-symmetric multifunctor. Our main result then shows that the induced functor between categories of algebras is an equivalence of homotopy theories. We describe an application to ring categories.