Weak Cat-Operads

TL;DR

This paper introduces the concept of weak Cat-operads, extending operads with isomorphisms replacing equations, and provides a coherence theorem with a simplified, index-free proof involving normal forms and polyhedral structures.

Contribution

It formulates coherence conditions for weak Cat-operads, introduces a novel, simplified proof method, and relates the structure to the hemiassociahedron, advancing understanding of higher categorical operads.

Findings

Coherence conditions ensure all diagrams of canonical arrows commute.

A simplified, index-independent proof method is developed.

The hemiassociahedron polyhedron arises from certain coherence conditions.

Abstract

An operad (this paper deals with non-symmetric operads)may be conceived as a partial algebra with a family of insertion operations, Gerstenhaber's circle-i products, which 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. 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. 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 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 matters other 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.