Weak cat-operads

Abstract

An operad (this paper deals with non-symmetric operads) may be conceived as a partial algebra with a family of insertion operations, which correspond to substitution of an operation within an operation. These insertion operations are Gerstenhaber’s circle-i products, and they satisfy two kinds of associativity, one of them involving commutativity. A Cat-operad is an operad enriched over the category Cat of small categories, as a 2-category with small hom-categories is a category enriched over Cat. This means that the operadic operations of the same arity in a Cat-operad do not make just a set, but they are the objects of a small category. The notion of weak Cat-operad is to the notion of Cat-operad what the notion of bicategory is to the notion of 2-category. This means that the equations of operads like associativity of insertions are replaced by isomorphisms in a category. The goal of this paper is to formulate conditions concerning these isomorphisms that ensure coherence, in the sense that all diagrams of canonical arrows commute. This is the sense in which the notions of monoidal category and bicategory are coherent. (The coherence of monoidal categories, which is due to Mac Lane, is the best known coherence result.) The coherence proof in the paper is much simplified by indexing the insertion operations in a context-independent way, and not in the usual manner. This proof, which is in the style of term rewriting, involves an argument with normal forms that generalizes what is established with the completeness proof for the standard presentation of symmetric groups. This generalization may be of an independent interest, and related to other matters than those studied in this paper. Some of the coherence conditions for weak Cat-operads lead to the hemiassociahedron, which is a polyhedron related to, but different from, the three-dimensional associahedron and permutohedron.