The Eckmann–Hilton argument and higher operads

By Michael Batanin. 2002

81 citations recorded by the source corpus.

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 Eckmann–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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c1c684d6e207d00c9ea85b26aa1a1eba97d1fc9c

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Citations in the corpus

Compact closed bicategories (2013, unknown): “In retrospect, we can see that this is the start of a more subtle collection of polytopes relevant to braided monoidal n-categories, which can be systematically obtained using Batanin’s approach to weak n-categories [9].”
Familial 2-functors and parametric right adjoints (2007, unknown): “The classical theory of operads in a good symmetric monoidal category A has four basic formal aspects.”
Operadic categories and Duoidal Deligne's conjecture (2014, background): “1] or [5, 18], generalizes to the realm of operadic categories.”
Iterated wreath product of the simplex category and iterated loop spaces (2005, background): “The assembly functor γn also occurs (in dual form) in Batanin’s n-operadic approach [7] to n-fold loop spaces, where it gives rise to an adjunction between n-operads and symmetric operads, the left adjoint of which sends contractible cofibrant n-operads to cofibrant En-operads, cf.”
Symmetrisation of n-operads and compactification of real configuration spaces (2006, methodology): “, however, that the main results of this paper are also true in the case of (n−1)-terminal n-operads but so far we have been unable to prove it in this generality. We apply the categorical methods of [4] to the category of pruned (n−1)- terminal n-operads and to the even smaller class of reduced (n−”
From operator categories to higher operads (2013, background): “As Clemens Berger has observed, the homotopy theory of weak O-operads is a homotopy-coherent variant of Michael Batanin’s notion of an n-operad [6].”
Homotopy theory for algebras over polynomial monads (2013, unknown): “It is one of the crucial observations of [6] that this left adjoint γ A (if it exists) can be computed as an internal left Kan extension.”