Authors: Došen, Kosta 
Petrić, Zoran 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Medial commutativity
Journal: Annals of Pure and Applied Logic
Volume: 146
Issue: 2-3
First page: 237
Last page: 255
Issue Date: 1-May-2007
Rank: M22
ISSN: 0168-0072
DOI: 10.1016/j.apal.2007.03.002
Abstract: 
It is shown that all the assumptions for symmetric monoidal categories follow from a unifying principle involving natural isomorphisms of the type (A ∧ B) ∧ (C ∧ D) → (A ∧ C) ∧ (B ∧ D), called medial commutativity. Medial commutativity in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane's pentagonal and hexagonal coherence conditions for associativity and commutativity are derived from the preservation up to a natural isomorphism of medial commutativity by the biendofunctor ∧. This preservation boils down to an isomorphic representation of the Yang-Baxter equation of symmetric and braid groups. The assumptions of monoidal categories, and in particular Mac Lane's pentagonal coherence condition, are explained in the absence of commutativity, and also of the unit object, by a similar preservation of associativity by the biendofunctor ∧. In the final section one finds coherence conditions for medial commutativity in the absence of the unit object. These conditions are obtained by taking the direct product of the symmetric groups Sfenced(frac(n, i)) for 0 ≤ i ≤ n.
Keywords: Associativity | Binomial coefficients | Coherence | Commutativity | Mac Lane's hexagon | Mac Lane's pentagon | Monoidal categories | Symmetric groups | Symmetric monoidal categories | Yang-Baxter equation
Publisher: Elsevier
Project: Ministry of Science of Serbia, Grant no. 144013

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