  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.

Category Theory;Algebraic Topology