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Algebras versus coalgebras
 Appl. Categorical Structures, DOI
, 2007
"... Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of the ..."
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Cited by 12 (10 self)
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Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 70’s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between
MONADS AND COMONADS ON MODULE CATEGORIES
"... known in module theory that any Abimodule B is an Aring if and only if the functor − ⊗A B: MA → MA is a monad (or triple). Similarly, an Abimodule C is an Acoring provided the functor − ⊗A C: MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of − ⊗A B and comodu ..."
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known in module theory that any Abimodule B is an Aring if and only if the functor − ⊗A B: MA → MA is a monad (or triple). Similarly, an Abimodule C is an Acoring 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
Generic commutative separable algebras and cospans of graphs. Theory and Applications of Categories
 Applications of Categories
, 2005
"... Abstract. We show that the generic symmetric monoidal category with a commutative separable algebra which has a Σfamily of actions is the category of cospans of finite Σlabelled graphs restricted to finite sets as objects, thus providing a syntax for automata on the alphabet Σ. We use this result ..."
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Cited by 11 (6 self)
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Abstract. We show that the generic symmetric monoidal category with a commutative separable algebra which has a Σfamily of actions is the category of cospans of finite Σlabelled graphs restricted to finite sets as objects, thus providing a syntax for automata on the alphabet Σ. We use this result to produce semantic functors for Σautomata. 1. Introduction. A variety of authors have considered (bi)categories of cospans (and spans) of graphs in the study of algebras of processes. The present authors have concentrated attention on algebras of automata (in [11] , [12], [13], [20], [23]), cospan operations providing the sequential operations, and span operations corresponding parallel operations. In another
Resource modalities in tensor logic
"... The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more ..."
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The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more primitive than linear logic. This revised point of view leads us to introduce tensor logic, a primitive variant of linear logic where negation is not involutive. After formulating its categorical semantics, we interpret tensor logic in a model based on Conway games equipped with a notion of payoff, in order to reflect the various resource policies of the logic: linear, affine, relevant or exponential.
COPRODUCTS OF IDEAL MONADS
, 2004
"... The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by ..."
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Cited by 4 (1 self)
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The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by
COHERENCE FOR PSEUDODISTRIBUTIVE LAWS REVISITED
"... Abstract. In this paper we show that eight coherence conditions suffice for the definition of a pseudodistributive law between pseudomonads. 1. ..."
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Cited by 3 (2 self)
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Abstract. In this paper we show that eight coherence conditions suffice for the definition of a pseudodistributive law between pseudomonads. 1.
A Monadic Approach to PolyCategories
 Theory Appl. Categ
, 2002
"... Polycategories form a rather natural generalization of multicategories. Besides the domains also the codomains of morphisms are allowed to be strings of objects. Multicategories are known to have an elegant global characterization as monads in a suitable bicategory of special spans with free m ..."
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Polycategories form a rather natural generalization of multicategories. Besides the domains also the codomains of morphisms are allowed to be strings of objects. Multicategories are known to have an elegant global characterization as monads in a suitable bicategory of special spans with free monoid as domains. To describe polycategories in similar terms, we investigate distributive laws in the sense of Beck between cartesian monads as tools for constructing new bicategories of modi ed spans. Three very simple such laws produce a bicategory in which the monads are precisely the planar polycategories (where composition only is de ned if the corresponding circuit diagram is planar). General polycategories, which only satisfy a local planarity condition, require a slightly more complicated construction.
The weak theory of monads
 Adv. in Math
"... ABSTRACT. We construct a ‘weak ’ version EM w (K) of Lack & Street’s 2category of monads in a 2category K, by replacing their compatibility constraint of 1cells with the units of monads by an additional condition on the 2cells. A relation between monads in EM w (K) and composite premonads in K ..."
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ABSTRACT. We construct a ‘weak ’ version EM w (K) of Lack & Street’s 2category of monads in a 2category K, by replacing their compatibility constraint of 1cells with the units of monads by an additional condition on the 2cells. A relation between monads in EM w (K) and composite premonads in K is discussed. If K admits EilenbergMoore constructions for monads, we define two symmetrical notions of ‘weak liftings ’ for monads in K. If moreover idempotent 2cells in K split, we describe both kinds of a weak lifting via an appropriate 2functor EM w (K) → K. Weak entwining structures and partial entwining structures are shown to realize weak liftings of a comonad for a monad in these respective senses. Weak bialgebras are characterized as algebras and coalgebras, such that the corresponding monads weakly lift for the corresponding comonads and also the comonads weakly lift for the monads.
Lifting theorems for tensor functors on module categories
 Department of Mathematics, Heinrich Heine University Düsseldorf, Germany
"... groups. These notions were generalised to monads and comonads on arbitrary categories. Starting around 1970 with papers by Beck, Barr and others a rich theory of the interplay between such endofunctors was elaborated based on distributive laws between them and Applegate’s lifting theorem of functors ..."
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Cited by 2 (2 self)
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groups. These notions were generalised to monads and comonads on arbitrary categories. Starting around 1970 with papers by Beck, Barr and others a rich theory of the interplay between such endofunctors was elaborated based on distributive laws between them and Applegate’s lifting theorem of functors between categories to related (co)module categories. Curiously enough some of these results were not noticed by researchers in module theory and thus notions like entwining structures and smash products between algebras and coalgebras were introduced (in the nineties) without being aware that these are special cases of the more general theory. The purpose of this survey is to explain several of these notions and recent results from general category theory in the language of elementary module theory focussing on functors between module categories given by tensoring with a bimodule. This provides a simple and systematic approach to smash products, wreath products, corings and rings over corings (Crings). We also highlight the relevance of the YangBaxter equation for the structures on the threefold tensor product of algebras or coalgebras (see 3.6).
The lattice path operad and Hochschild cochains
 CONTEMPORARY MATHEMATICS
"... We introduce two coloured operads in sets – the lattice path operad and a cyclic extension of it – closely related to iterated loop spaces and to universal operations on cochains. As main application we present a formal construction of an E2action (resp. framed E2action) on the Hochschild cochai ..."
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We introduce two coloured operads in sets – the lattice path operad and a cyclic extension of it – closely related to iterated loop spaces and to universal operations on cochains. As main application we present a formal construction of an E2action (resp. framed E2action) on the Hochschild cochain complex of an associative (resp. symmetric Frobenius) algebra.