Weak Cat-Operads
Kosta DOSEN (Mathematical Institute SANU), Zoran Petric (Mathematical, Institute SANU)

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…
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