### Contents

, 804

"... Abstract. Let A be a ring and MA the category of A-modules. It is well known in module theory that for any A-bimodule B, B is an A-ring if and only if the functor − ⊗A B: MA → MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor − ⊗A C: MA → MA is a comonad (or c ..."

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Abstract. Let A be a ring and MA the category of A-modules. It is well known in module theory that for any A-bimodule B, B is an A-ring if and only if the functor − ⊗A B: MA → MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor − ⊗A C: MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of − ⊗A B and comodules (or coalgebras) of − ⊗A C are well studied in the literature. On the other hand, the right adjoint endofunctors HomA(B, −) and HomA(C, −) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find

### Trichromatic Diagrams for Understanding

, 2011

"... The general context We build on the stream of work of categorical semantics for quantum in-formation processing, first initiated in the seminal paper of Abramsky and Coecke [1]. In this tradition, Coecke and Duncan developed and made ex-tensive use of a calculus of dichromatic diagrams to express qu ..."

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The general context We build on the stream of work of categorical semantics for quantum in-formation processing, first initiated in the seminal paper of Abramsky and Coecke [1]. In this tradition, Coecke and Duncan developed and made ex-tensive use of a calculus of dichromatic diagrams to express quantum pro-tocols and quantum states [2, 3]. This diagrammatic calculus turned out to be universal for quantum computing. The graphical calculus has been used to prove many statements useful to quantum computing, including some about graph states [9], measurement-based quantum computation [10], and a multitude of protocols [3]. Unfortunately, this calculus is known to be in-complete (with respect to stabilizer quantum mechanics) in a sense that is formalized in the main matter. The research problem The red-green calculus of [2, 3, 9, 10] talks of two complementary observ-ables in qubits. It is by now well known that it is possible to fit 3 comple-

### BACHUKI MESABLISHVILI, TBILISI AND

, 710

"... Abstract. 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 c ..."

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Abstract. 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

### Acknowledgements

, 2007

"... I hereby declare that this thesis is entirely my own creation, based on work done with my collaborator Michael Rios and under the supervision of Dr William Joyce (chapters 5 and 6). No part of the thesis will be used towards a qualification at any other institution. ..."

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I hereby declare that this thesis is entirely my own creation, based on work done with my collaborator Michael Rios and under the supervision of Dr William Joyce (chapters 5 and 6). No part of the thesis will be used towards a qualification at any other institution.

### 1 Physics, Topology, Logic and Computation:

"... Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objec ..."

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Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category has objects and morphisms, which represent things and ways to go between things. In physics, the objects are often physical systems, and the morphisms are processes turning

### ON FROBENIUS ALGEBRAS IN RIGID MONOIDAL CATEGORIES

, 2008

"... We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal catego ..."

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We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal categories, and for symmetric Frobenius algebras it is the one of sovereign monoidal categories. We also discuss some properties of Nakayama automorphisms.

### BIMONADS AND HOPF MONADS ON CATEGORIES BACHUKI MESABLISHVILI, TBILISI AND

, 710

"... Abstract. 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 c ..."

Abstract
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(Show Context)
Abstract. 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

### COALGEBRAIC STRUCTURES IN MODULE THEORY

"... Abstract. Although coalgebras and coalgebraic structures are well-known for a long time it is only in recent years that they are getting new attention from people working in algebra and module theory. The purpose of this survey is to explain the basic notions of the coalgebraic world and to show the ..."

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Abstract. Although coalgebras and coalgebraic structures are well-known for a long time it is only in recent years that they are getting new attention from people working in algebra and module theory. The purpose of this survey is to explain the basic notions of the coalgebraic world and to show their ubiquity in classical algebra. For this we recall the basic categorical notions and then apply them to linear algebra and module theory. It turns out that a number of results proven there were already contained in categorical papers from decades ago. Key Words: monads and comonads, algebras and coalgebras, module and

### Contents

"... Abstract. In the theory of coalgebras C over a ring R, the rational functor relates the ..."

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Abstract. In the theory of coalgebras C over a ring R, the rational functor relates the