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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 "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.
Quantum categories, star autonomy, and quantum groupoids
 in "Galois Theory, Hopf Algebras, and Semiabelian Categories", 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 distributed ..."
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Cited by 19 (8 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 "quantum category"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 "Hopf algebra with several objects". 1.
G.: Approximable concepts, Chu spaces, and information systems. Theory and Applications of Categories (200x
"... ABSTRACT. This paper serves to bring three independent but important areas of computer science to a common meeting point: Formal Concept Analysis (FCA), Chu Spaces, and Domain Theory (DT). Each area is given a perspective or reformulation that is conducive to the flow of ideas and to the exploration ..."
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Cited by 12 (8 self)
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ABSTRACT. This paper serves to bring three independent but important areas of computer science to a common meeting point: Formal Concept Analysis (FCA), Chu Spaces, and Domain Theory (DT). Each area is given a perspective or reformulation that is conducive to the flow of ideas and to the exploration of crossdisciplinary connections. Among other results, we show that the notion of state in Scott’s information system corresponds precisely to that of formal concepts in FCA with respect to all finite Chu spaces, and the entailment relation corresponds to “association rules”. We introduce, moreover, the notion of approximable concept and show that approximable concepts represent algebraic lattices which are identical to Scott domains except the inclusion of a top element. This notion serves as a stepping stone in the recent work [Hitzler and Zhang, 2004] in which a new notion of morphism on formal contexts results in a category equivalent to (a) the category of complete algebraic lattices and Scott continuous functions, and (b) a category of information systems and approximable mappings. 1.
Monoidal functor categories and graphic Fourier transforms, arXiv: math/0612496v1 [math. QA
, 2006
"... Abstract. This article represents a preliminary attempt to link Kan extensions, and some of their further developments, to Fourier theory and quantum algebra through ∗autonomous monoidal categories and related structures. There is a close resemblance to convolution products and the Wiener algebra ( ..."
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Cited by 3 (1 self)
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Abstract. This article represents a preliminary attempt to link Kan extensions, and some of their further developments, to Fourier theory and quantum algebra through ∗autonomous monoidal categories and related structures. There is a close resemblance to convolution products and the Wiener algebra (of transforms) in functional analysis. The analysis term “kernel ” (of a distribution) has also been adapted below in connection with certain special types of “distributors ” (in the terminology of J. Bénabou) or “modules” (in the terminology of R. Street) in category theory. In using the term “graphic”, in a very broad sense, we are clearly distinguishing the categorical methods employed in this article from standard Fourier and wavelet mathematics. The term “graphic ” also applies to promultiplicative graphs, and related concepts, which can feature prominently in the theory.
Bifinite Chu spaces
"... ABSTRACT. This paper studies colimits of sequences of finite Chu spaces and their ramifications. We consider three base categories of Chu spaces: the generic Chu spaces (C), the extensional Chu spaces (E), and the biextensional Chu spaces (B). The main results are: (1) a characterization of monics i ..."
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Cited by 2 (1 self)
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ABSTRACT. This paper studies colimits of sequences of finite Chu spaces and their ramifications. We consider three base categories of Chu spaces: the generic Chu spaces (C), the extensional Chu spaces (E), and the biextensional Chu spaces (B). The main results are: (1) a characterization of monics in each of the three categories; (2) existence (or the lack thereof) of colimits and a characterization of finite objects in each of the corresponding categories using monomorphisms/injections (denoted as iC, iE, and iB, respectively); (3) a formulation of bifinite Chu spaces with respect to iC; (4) the existence
Proof of a Conjecture of S. Mac Lane
, 1996
"... Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these ..."
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Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions (only this case was considered in the original Mac Lane conjecture). Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered (together with categorical equivalence). Two derivations of the same sequent are equivalent if and only if all their interpretations in K are equal. In fact, the assignment of values (objects of K) to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the "abstract coherence theorem" found by the author, it was necessary to modify her description of the class of noncommutative diagrams in SMC categories; our proof of S. Mac Lane conjecture proves also the correctness of the modified description. 1 Preface Since the notion of Symmetric Monoidal Closed (SMC) Category, in its axiomatic formulation, was introduced, the category of vector spaces over a field was considered as one of its principal models (see, e.g., [4]). The structure of an SMC category includes tensor product and internal homfunctor, and corresponding natural transformations. A diagram commutes in the free SMC category, iff it commutes in all its models, including vector spaces. But "how faithfully" does the notion of SMC category capture the categorical properties of a model? For example, Does a diagram commute in a free SMC category (and hence in all SMC categories) iff all instantiations by vector spaces give a commutative diagram? The positive answer would...
Proof of a conjecture of S.Mac Lane and some its algorithmic consequences. (extended )
"... coherence theorem. In [20] R.Voreadou presented a description all nonequivalent pairs of canonical maps (noncommutative diagrams in F(A) ) in terms of two classes W 0 and W . The class W 0 was the class of "generating" pairs, in the sense, that all the pairs of W were obtained from the pairs belo ..."
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coherence theorem. In [20] R.Voreadou presented a description all nonequivalent pairs of canonical maps (noncommutative diagrams in F(A) ) in terms of two classes W 0 and W . The class W 0 was the class of "generating" pairs, in the sense, that all the pairs of W were obtained from the pairs belonging to W 0 by certain rules. According to R.Voreadou, two canonical maps are not equivalent iff they belong to W . R.Voreadou's method was 1) to prove an "abstract coherence" theorem (in our terms, that if two derivations /; ' with the same final sequent do not belong to W , then they are equivalent) and 2) to build a model (also a kind of calculus in Voreadou's case) where the pairs /; ' belonging to W are interpreted as noncommutative diagrams. Since recently the author found a mistake in R.Voreadou's proof of her "abstract coherence theorem" (there is a counterexample to the proposition 2 [20], p.3), in this work the definition of R.Voreadou's classes is modified in such a way, that ge...
Example
, 2004
"... Recall the ordinary notion of Frobenius algebra over a field k. Step 2 Lift the concept from linear algebra to a general monoidal category and justify this with examples and theorems. Step 3 Lift the concept up a dimension so that monoidal categories themselves can be examples. 1 Frobenius algebras ..."
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Recall the ordinary notion of Frobenius algebra over a field k. Step 2 Lift the concept from linear algebra to a general monoidal category and justify this with examples and theorems. Step 3 Lift the concept up a dimension so that monoidal categories themselves can be examples. 1 Frobenius algebras An algebra A over a field k is called Frobenius when it is finite dimensional and equipped with a linear function e:A æÆ æ k such that: e ( ab) = 0 for all a ŒA implies b = 0.