The EckmanHilton argument and higher operads
Abstract
The classical EckmannHilton 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 2category, 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 EckmanHilton type argument in terms of the calculation of left Kan extensions in an appropriate 2category. 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 $(n1)$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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 July 2002
 arXiv:
 arXiv:math/0207281
 Bibcode:
 2002math......7281B
 Keywords:

 Mathematics  Category Theory;
 Mathematics  Algebraic Topology;
 18D05;
 18D50;
 55P48
 EPrint:
 57pp