The Eckman-Hilton argument and higher operads

TL;DR

This paper extends the classical Eckmann-Hilton argument to higher dimensions using categorical schemes, applying it to $n$-operads and symmetric operads to relate algebra categories.

Contribution

It introduces a categorical framework for higher-dimensional Eckmann-Hilton arguments and constructs a symmetrisation functor for $n$-operads, linking them to symmetric operads.

Findings

Established a categorical scheme for Eckmann-Hilton type arguments in higher dimensions.

Constructed a symmetrisation functor from $n$-operads to symmetric operads.

Provided an explicit formula for the symmetrisation functor under mild conditions.

Abstract

The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its $Hom$-set is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is `the same' as a braided monoidal category. In this paper we begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the Eckman-Hilton type argument in terms of the calculation of left Kan extensions in an appropriate 2-category. Then we apply this scheme to the case of $n$-operads in the author's sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation $Sym_n$ from a certain subcategory of $n$-operads to the category of symmetric operads such that the category of one object, one arrow, . . ., one $(n-1)$-arrow algebras of $A$ is isomorphic to the category of algebras of $Sym_n(A)$. Under some mild conditions, we present an explicit formula for $Sym_n(A)$ which involves taking the colimit over a remarkable categorical symmetric operad. We will consider some applications of the methods developed to the theory of $n$-fold loop spaces in the second paper of this series.