The Gray monoidal product of double categories

The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $\mathbb A$, the corresponding internal hom functor $|[ \mathbb A,-]|$ sends a double category $\mathbb B$ to the double category whose 0-cells are the double functors $\mathbb A \to \mathbb B$, whose horizontal and vertical 1-cells are the horizontal and vertical pseudotransformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.Comment: 35 pages, 5 large and many small diagram

Galois extensions over commutative and non-commutative base

This paper is a written form of a talk. It gives a review of various notions of Galois (and in particular cleft) extensions. Extensions by coalgebras,bialgebras and Hopf algebras (over a commutative base ring) and by corings,bialgebroids and Hopf algebroids (over a non-commutative base algebra) are systematically recalled and compared. In the first version of this paper, the journal version of [15, Theorem 2.6] was heavily used, in two respects. First, it was applied to establish an isomorphism between the comodule categories of two constituent bialgebroids in a Hopf algebroid. Second, it was used to construct a Morita context for any bicomodule for a coring extension. Regrettably, it turned out that the proof of [15, Theorem 2.6] contains an unjustified step. Therefore, our derived results are not expected to hold at the stated level of generality either. In the revised version we make the necessary corrections in both respects. In doing so, we obtain a corrected version of \cite[5, Theorem 4.2] as well, whose original proof contains a very similar error to [15, Theorem 2.6].Comment: written form of a talk, LaTeX file, 27 pages. v2: Substantial revision, distinguishing between comodules of both constituent bialgebroids in a Hopf algebroi

Integral theory for Hopf Algebroids

The theory of integrals is used to analyse the structure of Hopf algebroids, introduced in math.QA/0302325. We prove that the total algebra of the Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. The total algebra of a finitely generated and projective Hopf algebroid is a Frobenius extension of the base algebrahe Hopf algebroid possesses anormalized integral. We give also a sufficient and necessary condition in terms of integrals under which it is a quasi-Frobenius extension and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras.Comment: LaTeX2e file 30 pages, no figures. v4:missing assumptions added in Thms 4.2, 4.7 and 5.2, new Corollary 4.

Comodules over weak multiplier bialgebras

This is a sequel paper of arXiv:1306.1466 in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.Comment: LaTeX source, 47 page

Galois theory for Hopf algebroids

An extension B\subset A of algebras over a commutative ring k is an H-extension for an L-bialgebroid H if A is an H-comodule algebra and B is the subalgebra of its coinvariants. It is H-Galois if the canonical map A\otimes_B A\to A\otimes_L H is an isomorphism or, equivalently, if the canonical coring (A\otimes_L H:A) is a Galois coring. In the case of a Hopf algebroid H=(H_L,H_R,S) any H_R-extension is shown to be also an H_L-extension. If the antipode is bijective then also the notions of H_R-Galois extensions and of H_L-Galois extensions are proven to coincide. Results about bijective entwining structures are extended to entwining structures over non-commutative algebras in order to prove a Kreimer-Takeuchi type theorem for a finitely generated projective Hopf algebroid H with bijective antipode. It states that any H-Galois extension B\subset A is projective, and if A is k-flat then already the surjectivity of the canonical map implies the Galois property. The Morita theory, developed for corings by Caenepeel, Vercruysse and Wang, is applied to obtain equivalent criteria for the Galois property of Hopf algebroid extensions. This leads to Hopf algebroid analogues of results for Hopf algebra extensions by Doi and, in the case of Frobenius Hopf algebroids, by Cohen, Fishman and Montgomery.Comment: 19 pages, no figure v2: Prop 3.1 added. v3: Prop 3.1 is withdrawn, without affecting any other result in the pape

Crossed modules of monoids I. Relative categories

This is the first part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans - crossed modules of monoids relative to this class of spans - simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this first part the theory of relative pullbacks is worked out leading to the definition of a relative category.Comment: 22 pages, first item of a three parts serie

Doi-Hopf Modules over Weak Hopf Algebras

The theory of Doi-Hopf modules is generalized to Weak Hopf algebras.Comment: 11 pages, LaTeX fil

Hopf comonads on naturally Frobenius map-monoidales

We study monoidal comonads on a naturally Frobenius map-monoidale $M$ in a monoidal bicategory $\mathcal M$. We regard them as bimonoids in the duoidal hom-category $\mathcal M(M,M)$, and generalize to that setting various conditions distinguishing classical Hopf algebras among bialgebras; in particular, we define a notion of antipode in that context. Assuming the existence of certain conservative functors and the splitting of idempotent 2-cells in $\mathcal M$, we show all these Hopf-like conditions to be equivalent. Our results imply in particular several equivalent characterizations of Hopf algebras in braided monoidal categories, of small groupoids, of Hopf algebroids over commutative base algebras, of weak Hopf algebras, and of Hopf monads in the sense of Brugui\`eres and Virelizier.Comment: 37 pages, a number of commutative diagram

A simplicial approach to multiplier bimonoids

Although multiplier bimonoids in general are not known to correspond to comonoids in any monoidal category, we classify them in terms of maps from the Catalan simplicial set to another suitable simplicial set; thus they can be regarded as (co)monoids in something more general than a monoidal category (namely, the simplicial set itself). We analyze the particular simplicial maps corresponding to that class of multiplier bimonoids which can be regarded as comonoids.Comment: 14 pages; v2 minor changes only, to appear in Bulletin of the Belgian Mathematical Society -- Simon Stevi

A categorical approach to cyclic duality

The aim of this paper is to provide a unifying categorical framework for the many examples of para-(co)cyclic modules arising from Hopf cyclic theory. Functoriality of the coefficients is immediate in this approach. A functor corresponding to Connes's cyclic duality is constructed. Our methods allow, in particular, to extend Hopf cyclic theory to (Hopf) bialgebroids.Comment: LaTeX 2e, 47 pages, lots of figures. v2: the same mathematical content as in v1 is presented via string diagrams. Final version to appear in J. Noncommutative Geometr