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Pseudo algebras and pseudo double categories
 J. Homotopy Relat. Struct
"... Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, an ..."
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Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group.
Representations of matched pairs of groupoids and applications to weak Hopf algebras
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The double algebraic view of finite quantum groupoids
 Journal of Algebra
"... Abstract. Double algebra is the structure modelled by the properties of the ordinary and the convolution product in Hopf algebras, weak Hopf algebras and Hopf algebroids if a Frobenius integral is given. The Hopf algebroids possessing a Frobenius integral are precisely the Frobenius double algebras ..."
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Abstract. Double algebra is the structure modelled by the properties of the ordinary and the convolution product in Hopf algebras, weak Hopf algebras and Hopf algebroids if a Frobenius integral is given. The Hopf algebroids possessing a Frobenius integral are precisely the Frobenius double algebras in which the two multiplications satisfy distributivity. The double algebra approach makes it manifest that all comultiplications in such measured Hopf algebroids are of the AbramsKadison type, i.e., they come from a Frobenius algebra structure in some bimodule category. Antipodes for double algebras correspond to the ConnesMoscovici ‘deformed ’ antipode as we show by discussing Hopf and weak Hopf algebras from the double algebraic point of view. Frobenius algebra extensions provide further examples that need not be distributive. 1.
ON THE QUIVERTHEORETICAL QUANTUM YANGBAXTER EQUATION
, 2004
"... Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum YangBaxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a “solution ” for short. Results of Eting ..."
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Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum YangBaxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a “solution ” for short. Results of EtingofSchedlerSoloviev, LuYanZhu and Takeuchi on the settheoretical quantum YangBaxter equation are generalized to the context of quivers, with groupoids playing the rôle of groups. The notion of “braided groupoid ” is introduced. Braided groupoids are solutions and are characterized in terms of bijective 1cocycles. The structure groupoid of a nondegenerate solution is defined; it is shown that it is braided groupoid. The reduced structure groupoid of a nondegenerate solution is also defined. Nondegenerate solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct startriangular face models and realize them as modules over quasitriangular quantum groupoids introduced in recent papers by M. Aguiar, S. Natale and the author.
ON BRAIDED GROUPOIDS
, 2005
"... Abstract. We study and give examples of braided groupoids, and a fortiori, nondegenerate solutions of the quivertheoretical braid equation. ..."
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Abstract. We study and give examples of braided groupoids, and a fortiori, nondegenerate solutions of the quivertheoretical braid equation.
NTUPLE GROUPOIDS AND OPTIMALLY COUPLED FACTORIZATIONS
"... Abstract. In this paper, we prove that the category of vacant ntuple groupoids is equivalent to the category of factorizations of groupoids by n factors that satisfy some YangBaxter type equation. Moreover we extend this equivalence to the category of maximally exclusive ntuple groupoids, which w ..."
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Abstract. In this paper, we prove that the category of vacant ntuple groupoids is equivalent to the category of factorizations of groupoids by n factors that satisfy some YangBaxter type equation. Moreover we extend this equivalence to the category of maximally exclusive ntuple groupoids, which we define, by dropping one assumption. The paper concludes by a note on how these results could tell us more about some Lie groups of interest.
PSEUDO ALGEBRAS AND PSEUDO DOUBLE CATEGORIES
 JOURNAL OF HOMOTOPY AND RELATED STRUCTURES, VOL. 2(2), 2007, PP.119–170
, 2007
"... As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to ..."
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As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group.