Coherence constraints for operads, categories and algebras (Q2722339)

The authors investigate coherence phenomena in the general context of the theory of operads. Coherence constraints in particular appear in two ways. In category theory they consist of a set of commutative diagrams implying the commutativity of (any) larger class of diagrams. For algebras coherence constraints are given by the minimal set of generators for the second syzygy. It will be shown that both types of coherence are described by the coherence constraints of the underlying operad of the respective algebraic structure. Coherence constraints form a homological invariant of the operad. They can be informally described as ``generating relations'' among the ``defining relations'' which generate the ideal through which the given operad is a quotient of a certain free operad. The coherence constraints can be obtained from the bigraded model of the operad. In the case of Koszul quadratic operads the coherence constraints are isomorphic to \(sgn\otimes {\mathcal P}^!(4)\) where \(sgn\) is the one-dimensional signum representation and \({\mathcal P}^!\) is the quadratic dual of the operad. NEWLINENEWLINENEWLINECategorical coherence will be described by operads given by commutative diagrams. It will be shown that the diagrams corresponding to the set of coherence constraints implies commutativity of all diagrams. For algebras the coherence of a deformation is given by linear equations. Any solution of a linear equation in the undeformed system deforms to a solution of the quantization. In terms of operads this will be formalized through so-called \(V\)-relative operads \({\mathcal P}_V\) of a given operad \({\mathcal P}\), where \(V\) is a unital algebra with augmentation. The \(V\)-relative operad \({\mathcal P}_V\) is called \(V\)-quantization if the operadic ideal will be generated by a collection whose \(n\)th components are free \(V^{\otimes_n}\)-modules. Coherence then means that the defining relations of the quantization have the same rank as the original structure, i.e., for each degree \(n\), \({\mathcal P}_V(n)\) is isomorphic to \({\mathcal P}(n)\otimes V^{\otimes_n}\) as right \(V^{\otimes_n}\)-module. The theory will be illustrated by proving Mac Lane's coherence theorem for monoidal categories using that the operad \(Ass\) is Koszul. For the algebraic type of coherence Drinfel'd's quasi-Hopf algebras and generalized Lie algebras have been discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00020].