Confluence laws and Hopf-Borel type theorem for operads (original) (raw)
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Generalised bialgebras and entwined monads and comonads
Journal of Pure and Applied Algebra, 2015
Jean-Louis Loday has defined generalised bialgebras and proved structure theorems in this setting which can be seen as general forms of the Poincaré-Birkhoff-Witt and the Cartier-Milnor-Moore theorems. It was observed by the present authors that parts of the theory of generalised bialgebras are special cases of results on entwined monads and comonads and the corresponding mixed bimodules. In this article the Rigidity Theorem of Loday is extended to this more general categorical framework.
Bimonads and Hopf monads on categories
Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2011
The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal structure of the category of endofunctors on any category A and by this we retain some of the combinatorial complexity which makes the theory so interesting. As a basic tool we use distributive laws between monads and comonads (entwinings) on A: we define a bimonad on A as an endofunctor B which is a monad and a comonad with an entwining W BB ! BB satisfying certain conditions. This is also employed to define the category A B B of (mixed) B-bimodules. In the classical situation, an entwining is derived from the twist map for vector spaces. Here this need not be the case but there may exist special distributive laws W BB ! BB satisfying the Yang-Baxter equation (local prebraidings) which induce an entwining and lead to an extension of the theory of braided Hopf algebras.
Equivariant monads and equivariant lifts versus a 2-category of distributive laws
2007
Fix a monoidal category C. The 2-category of monads in the 2-category of C-actegories, colax C-equivarant functors, and C-equivariant natural transformations of colax functors, may be recast in terms of pairs consisting of a usual monad and a distributive law between the monad and the action of C, morphisms of monads respecting the distributive law, and transformations of monads satisfying some compatibility with the actions and distributive laws involved. The monads in this picture may be generalized to actions of monoidal categories, and actions of PRO-s in particular. If C is a PRO as well, then in special cases one gets various distributive laws of a given classical type, for example between a comonad and an endofunctor or between a monad and a comonad. The usual pentagons are in general replaced by multigons, and there are also ``mixed'' multigons involving two distinct distributive laws. Beck's bijection between the distributive laws and lifts of one monad to the Eilenberg-Moore category of another monad is here extended to an isomorphism of 2-categories. The lifts of maps of above mentioned pairs are colax C-equivariant. We finish with a short treatment of relative distributive laws between two pseudoalgebra structures which are relative with respect to the distributivity of two pseudomonads involved, what gives a hint toward the generalizations.
On Hopf algebra structures over free operads
Advances in Mathematics, 2006
The operad Lie can be constructed as the operad of primitives PrimAs from the operad As of associative algebras. This is reflected by the theorems of Friedrichs, Poincaré-Birkhoff-Witt and Cartier-Milnor-Moore. We replace the operad As by families of free operads P, which include the operad Mag freely generated by a noncommutative non-associative binary operation and the operad of Stasheff polytopes. We obtain Poincaré-Birkhoff-Witt type theorems and collect information about the operads PrimP, e.g. in terms of characteristic functions.
Notes on bimonads and Hopf monads
Theory and Applications of Categories, 2012
For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguières and A. Virelizier study opmonoidal monads on monoidal categories (which they called bimonads). In a recent joint paper with S. Lack the same authors define the notion of a pre-Hopf monad by requiring only a special form of the fusion operator to be invertible. In previous papers it was observed by the present authors that bimonads yield a special case of an entwining of a pair of functors (on arbitrary categories). The purpose of this note is to show that in this setting the pre-Hopf monads are a special case of Galois entwinings. As a byproduct some new properties are detected which make a (general) bimonad on a Cauchy complete category to a Hopf monad. In the final section applications to cartesian monoidal categories are considered.
2012
For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguières and A. Virelizier study opmonoidal monads on monoidal categories (which they called bimonads). In a recent joint paper with S. Lack the same authors define the notion of a pre-Hopf monad by requiring only a special form of the fusion operator to be invertible. In previous papers it was observed by the present authors that bimonads yield a special case of an entwining of a pair of functors (on arbitrary categories). The purpose of this note is to show that in this setting the pre-Hopf monads are a special case of Galois entwinings. As a byproduct some new properties are detected which make a (general) bimonad on a Cauchy complete category to a Hopf monad. In the final section applications to cartesian monoidal categories are considered.
Coherence Constraints for Operads, Categories and Algebras. ArXiv: q-alg/9712027
2016
ABSTRACT. Coherence phenomena appear in two different situations. In the context of category theory the term 'coherence constraints ' refers to a set of diagrams whose commutativity implies the commutativity of a larger class of diagrams. In the context of algebra coherence constrains are a minimal set of generators for the second syzygy, that is, a set of equations which generate the full set of identities among the defining relations of an algebraic theory. A typical example of the first type is Mac Lane's coherence theorem for monoidal categories [9, Theorem 3.1], an example of the second type is the result of [2] saying that pentagon identity for the 'associator ' $ of a quasi-Hopf algebra implies the validity of a set of identities with higher instances of $. We show that both types of coherence are governed by a homological invariant of the operad for the underlying algebraic structure. We call this invariant the (space of) coherence constraints. In ma...
Operads as abstract algebras, and the Koszul property
Journal of Pure and Applied Algebra, 1999
The purpose of this work is to clarify some aspects of the theory of operads – especially of Koszul operads. We describe the free operad and give the categorical bar construction in terms of it. After giving a description of operads as abstract “algebras”, we give a self-contained proof of Theorem 4.1.13 of Ginzburg and Kapranov (1994). Our approach uses two spectral sequences associated to the categorical cobar complex of a Koszul operad.