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The essence of dataflow programming
 In APLAS
, 2005
"... Abstract. We propose a novel, comonadic approach to dataflow (streambased) computation. This is based on the observation that both general and causal stream functions can be characterized as coKleisli arrows of comonads and on the intuition that comonads in general must be a good means to structure ..."
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Cited by 19 (3 self)
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Abstract. We propose a novel, comonadic approach to dataflow (streambased) computation. This is based on the observation that both general and causal stream functions can be characterized as coKleisli arrows of comonads and on the intuition that comonads in general must be a good means to structure contextdependent computation. In particular, we develop a generic comonadic interpreter of languages for contextdependent computation and instantiate it for streambased computation. We also discuss distributive laws of a comonad over a monad as a means to structure combinations of effectful and contextdependent computation. We apply the latter to analyse clocked dataflow (partial stream based) computation. 1
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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Cited by 14 (11 self)
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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
Algebraic model structures
"... Abstract. We define a new notion of an algebraic model structure, in which the cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads, and prove “algebraic ” analogs of classical results. Using a modified version of Quillen’s small object argument, we show that e ..."
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Cited by 13 (9 self)
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Abstract. We define a new notion of an algebraic model structure, in which the cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads, and prove “algebraic ” analogs of classical results. Using a modified version of Quillen’s small object argument, we show that every cofibrantly generated model structure in the usual sense underlies a cofibrantly generated algebraic model structure. We show how to pass a cofibrantly generated algebraic model structure across an adjunction, and we characterize the algebraic Quillen adjunction that results. We prove that pointwise algebraic weak factorization systems on diagram categories are cofibrantly generated if the original ones are, and we give an algebraic generalization of the projective model structure. Finally, we prove that certain fundamental comparison maps present in any cofibrantly generated model category are cofibrations when the cofibrations are monomorphisms, a conclusion that does not seem to be provable in the classical, nonalgebraic, theory. Contents
SYMMETRY AND CAUCHY COMPLETION OF QUANTALOIDENRICHED CATEGORIES
"... Abstract. We formulate an elementary condition on an involutive quantaloid Q under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Qenriched categories. For such quantaloids, which we call Cauchybilateral quantaloids, it follows ..."
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Abstract. We formulate an elementary condition on an involutive quantaloid Q under which there is a distributive law from the Cauchy completion monad over the symmetrisation comonad on the category of Qenriched categories. For such quantaloids, which we call Cauchybilateral quantaloids, it follows that the Cauchy completion of any symmetric Qenriched category is again symmetric. Examples include Lawvere’s quantale of nonnegative real numbers and Walters ’ small quantaloids of closed cribles.
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 i ..."
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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.
The essence of dataflow programming (short version
 Proc. of 3rd Asian Symp. on Programming Languages and Systems, APLAS 2005, v. 3780 of Lect. Notes in Comput. Sci
, 2005
"... Abstract. We propose a novel, comonadic approach to dataflow (streambased) computation. This is based on the observation that both general and causal stream functions can be characterized as coKleisli arrows of comonads and on the intuition that comonads in general must be a good means to structure ..."
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Cited by 2 (1 self)
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Abstract. We propose a novel, comonadic approach to dataflow (streambased) computation. This is based on the observation that both general and causal stream functions can be characterized as coKleisli arrows of comonads and on the intuition that comonads in general must be a good means to structure contextdependent computation. In particular, we develop a generic comonadic interpreter of languages for contextdependent computation and instantiate it for streambased computation. We also discuss distributive laws of a comonad over a monad as a means to structure combinations of effectful and contextdependent computation. We apply the latter to analyse clocked dataflow (partial stream based) computation. 1
THE 2CATEGORY OF WEAK ENTWINING STRUCTURES
, 902
"... ABSTRACT. A weak entwining structure in a 2category K consists of a monad t and a comonad c, together with a 2cell relating both structures in a way that generalizes a mixed distributive law. A weak entwining structure can be characterized as a compatible pair of a monad and a comonad, in 2catego ..."
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ABSTRACT. A weak entwining structure in a 2category K consists of a monad t and a comonad c, together with a 2cell relating both structures in a way that generalizes a mixed distributive law. A weak entwining structure can be characterized as a compatible pair of a monad and a comonad, in 2categories generalizing the 2category of comonads and the 2category of monads in K, respectively. This observation is used to define a 2category Entw w (K) of weak entwining structures in K. If the 2category K admits EilenbergMoore constructions for both monads and comonads and idempotent 2cells in K split, then there are 2functors from Entw w (K) to the 2category of monads and to the 2category of comonads in K, taking a weak entwining structure (t,c) to a ‘weak lifting ’ of t for c and a ‘weak lifting ’ of c for t, respectively. The EilenbergMoore objects of the lifted monad and the lifted comonad are shown to be isomorphic. If K is the 2category of functors induced by bimodules, then these isomorphic EilenbergMoore objects are isomorphic to
THE HOMOTOPY THEORY OF STRONG HOMOTOPY ALGEBRAS AND BIALGEBRAS
, 908
"... Abstract. Lada introduced strong homotopy algebras to describe the structures on a deformation retract of an algebra in topological spaces. However, there is no satisfactory general definition of a morphism of strong homotopy (s.h.) algebras. Given a monad ⊤ on a simplicial category C, we instead sh ..."
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Abstract. Lada introduced strong homotopy algebras to describe the structures on a deformation retract of an algebra in topological spaces. However, there is no satisfactory general definition of a morphism of strong homotopy (s.h.) algebras. Given a monad ⊤ on a simplicial category C, we instead show how s.h. ⊤algebras over C naturally form a Segal space. Given a distributive monadcomonad pair (⊤, ⊥), the same is true for s.h. (⊤, ⊥)bialgebras over C; in particular this yields the homotopy theory of s.h. sheaves of s.h. rings. There are similar statements for quasimonads and quasicomonads. We also show how the structures arising are related to derived connections on bundles.