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Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to &quot;strongly separable &quot; Frobenius algebras and &quot;weak monoidal Morita equivalence&quot;. Wreath products of Frobenius algebras are discussed.
Quantum categories, star autonomy, and quantum groupoids
 in &quot;Galois Theory, Hopf Algebras, and Semiabelian Categories&quot;, Fields Institute Communications 43 (American Math. Soc
, 2004
"... Abstract A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. Generalizing this concept, we define the term "quantum category"in a braided monoidal category with equalizers dist ..."
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Cited by 19 (9 self)
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Abstract A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. Generalizing this concept, we define the term &quot;quantum category&quot;in a braided monoidal category with equalizers distributed over by tensoring with an object. The definition of antipode for a bialgebroid is less resolved in the literature. Our suggestion is that the kind of dualization occurring in Barr's starautonomous categories is more suitable than autonomy ( = compactness = rigidity). This leads to our definition of quantum groupoid intended as a &quot;Hopf algebra with several objects&quot;. 1.
A Formal Foundation for OntologyAlignment Interaction Models
"... Abstract. Ontology alignment foundations are hard to find in the literature. The abstract nature of the topic and the diverse means of practice make it difficult to capture it in a universal formal foundation. We argue that such a lack of formality hinders further development and convergence of prac ..."
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Cited by 9 (7 self)
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Abstract. Ontology alignment foundations are hard to find in the literature. The abstract nature of the topic and the diverse means of practice make it difficult to capture it in a universal formal foundation. We argue that such a lack of formality hinders further development and convergence of practices, and in particular, prevents us from achieving greater levels of automation. In this article we present a formal foundation for ontology alignment that is based on interaction models between heterogeneous agents on the Semantic Web. We use the mathematical notion of information flow in a distributed system to ground our three hypotheses of enabling semantic interoperability and we use a motivating example throughout the article: how to progressively align two ontologies of research quality assessment through meaning coordination. We conclude the article with the presentation—in an executable specification language—of such an ontologyalignment interaction model. 1.
Entropic hopf algebras and models of noncommutative linear logic
 THEORY AND APPLICATIONS OF CATEGORIES 10
, 2002
"... We give a definition of categorical model for the multiplicative fragment of noncommutative logic. We call such structures entropic categories. We demonstrate the soundness and completeness of our axiomatization with respect to cutelimination. We then focus on several methods of building entropic ..."
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We give a definition of categorical model for the multiplicative fragment of noncommutative logic. We call such structures entropic categories. We demonstrate the soundness and completeness of our axiomatization with respect to cutelimination. We then focus on several methods of building entropic categories. Our first models are constructed via the notion of a partial bimonoid acting on a cocomplete category. We also explore an entropic version of the Chu construction, and apply it in this setting. It has recently been demonstrated that Hopf algebras provide an excellent framework for modeling a number of variants of multiplicative linear logic, such as commutative, braided and cyclic. We extend these ideas to the entropic setting by developing a new type of Hopf algebra, which we call entropic Hopf algebras. We show that the category of modules over an entropic Hopf algebra is an entropic category, (possibly after application of the Chu construction). Several examples are discussed, based first on the notion of a bigroup. Finally the TannakaKrein reconstruction theorem is extended to the entropic setting.
A 2dimensional view of the Chuconstruction
, 2000
"... The cyclic Chuconstruction for closed bicategories, generalizing the original Chuconstruction for symmetric monoidal closed categories, turns out to have a noncyclic counterpart. Both constructions are based on socalled Chucells and can be generalized to chains of composable 1cells. This leads ..."
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The cyclic Chuconstruction for closed bicategories, generalizing the original Chuconstruction for symmetric monoidal closed categories, turns out to have a noncyclic counterpart. Both constructions are based on socalled Chucells and can be generalized to chains of composable 1cells. This leads to two hierarchies of closed bicategories for any closed bicategory with local pullbacks. Chucells in rel correspond to bipartite state transition systems. Even though their vertical composition may fail here due to the lack of pullbacks, basic gametheoretic constructions can be performed on cyclic Chucells. These generalize to all symmetric monoidal closed categories. If finite limits exist, the cyclic Chucells form the objects of a *autonomous category.
AN EXTENDED VIEW OF THE CHUCONSTRUCTION
"... Abstract. The cyclic Chuconstruction for closed bicategories with pullbacks, which generalizes the original Chuconstruction for symmetric monoidal closed categories, turns out to have a noncyclic counterpart. Both use socalled Chuspans as new 1cells between 1cells of the underlying bicategory ..."
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Abstract. The cyclic Chuconstruction for closed bicategories with pullbacks, which generalizes the original Chuconstruction for symmetric monoidal closed categories, turns out to have a noncyclic counterpart. Both use socalled Chuspans as new 1cells between 1cells of the underlying bicategory, which form the new objects. Chuspans may be seen as a natural generalization of 2cellspans in the base bicategory that no longer are confined to a single homcategory. This view helps to clarify the composition of Chuspans. We consider various approaches of linking the underlying bicategory with the newly constructed ones, for example, by means of twodimensional generalizations of bifibrations. In the quest for
THE CHU CONSTRUCTION: HISTORY OF AN IDEA
"... Abstract. This paper describes the historical background and motivation involved in the discovery (or invention) of Chu categories. In 1975, I began a sabbatical leave at the ETH in Zürich, with the idea of studying duality in categories in some depth. By this, I meant not such things as the duality ..."
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Abstract. This paper describes the historical background and motivation involved in the discovery (or invention) of Chu categories. In 1975, I began a sabbatical leave at the ETH in Zürich, with the idea of studying duality in categories in some depth. By this, I meant not such things as the duality between Boolean algebras and Stone spaces, nor between compact and discrete abelian groups, but rather selfdual categories such as complete semilattices, finite abelian groups, and locally compact abelian groups. Moreover, I was interested in the possibilities of having a category that was not only self dual but one that had an internal hom and for which the duality was implemented as the internal hom into a “dualizing object”. This was already true for the complete semilattices, but not for finite abelian groups or locally compact abelian groups. The category of finite abelian groups has an internal hom, but lacks a dualizing object, while locally compact groups have a dualizing object, but not an internal hom that is defined everywhere. Although you could define an abelian group of continuous homomorphisms between locally compact abelian groups, there was no way of systematically putting a locally compact topology on the hom set that would lead to the
Morphisms and Modules for Linear Bicategories
, 2000
"... Linear bicategories are a generalization of bicategories, in which the horizontal composition of 1cells is replaced by two (coherently linked) horizontal compositions. This notion combines and integrates compositional features typical of bicategories with those arising from linear logic. Linear bic ..."
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Linear bicategories are a generalization of bicategories, in which the horizontal composition of 1cells is replaced by two (coherently linked) horizontal compositions. This notion combines and integrates compositional features typical of bicategories with those arising from linear logic. Linear bicategories, therefore, provide a natural categorical semantics and interpretation for (relational) noncommutative linear logic. In particular, the logical notion of negation (or complementation) turns into a linear notion of adjunction, with involutive negations corresponding to cyclic linear adjunctions. The latter are crucial for the construction of a tricategory of linear bicategories, linear functors, linear transformations and linear modications. This paper rst develops the structure of the aforementioned tricategory and describes how the various components have a natural interpretation using the diagrammatic calculus of circuits. Then we transfer the module construction to the linea...
MIQIS: Modular Integration of Queryable Information Systems
, 2004
"... Information integration is not a new problem. ..."