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52
(CO)CYCLIC (CO)HOMOLOGY OF BIALGEBROIDS: AN APPROACH VIA (CO)MONADS
, 2008
"... ... there is a (co)simplex Z ∗: = ΠTl ∗+1 X in C. The aim of this paper is to find criteria for para(co)cyclicity of Z ∗. Our construction is built on a distributive law of Tl with a second (co)monad Tr on M, a natural transformation i: ΠTl → ΠTr, and a morphism w: TrX → TlX in M. The (symmetrical) ..."
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... there is a (co)simplex Z ∗: = ΠTl ∗+1 X in C. The aim of this paper is to find criteria for para(co)cyclicity of Z ∗. Our construction is built on a distributive law of Tl with a second (co)monad Tr on M, a natural transformation i: ΠTl → ΠTr, and a morphism w: TrX → TlX in M. The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads Tl = T ⊗R (−) and Tr = (−) ⊗R T on the category of Rbimodules. The functor Π can be chosen such that Z n = T b⊗R... b⊗RT b⊗RX is the cyclic Rmodule tensor product. A natural transformation i: T b⊗R(−) → (−)b⊗RT is given by the flip map and a morphism w: X ⊗R T → T ⊗R X is constructed whenever T is a (co)module algebra or coring of an Rbialgebroid. The notion of a stable anti YetterDrinfel’d module over certain bialgebroids, so called ×RHopf algebras, is introduced. In the particular example when T is a module coring of a ×RHopf algebra B and X is a stable anti YetterDrinfel’d Bmodule, the paracyclic object Z ∗ is shown to project to a cyclic structure on T ⊗ R ∗+1 ⊗B X. For a BGalois extension S ⊆ T, a stable anti YetterDrinfel’d Bmodule TS is constructed, such that
Complete Lax Logical Relations for Cryptographic LambdaCalculi
 In Proceedings of CSL’2004, volume 3210 of LNCS
, 2004
"... Security properties are profitably expressed using notions of contextual equivalence, and logical relations are a powerful proof technique to establish contextual equivalence in typed lambda calculi, see e.g. Sumii and Pierce's logical relation for a cryptographic lambdacalculus. We clarify Sumii a ..."
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Security properties are profitably expressed using notions of contextual equivalence, and logical relations are a powerful proof technique to establish contextual equivalence in typed lambda calculi, see e.g. Sumii and Pierce's logical relation for a cryptographic lambdacalculus. We clarify Sumii and Pierce's approach, showing that the right tool is prelogical relations, or lax logical relations in general: relations should be lax at encryption types, notably. To explore the difficult aspect of fresh name creation, we use Moggi's monadic lambdacalculus with constants for cryptographic primitives, and Stark's name creation monad. We define logical relations which are lax at encryption and function types but strict (nonlax) at various other types, and show that they are sound and complete for contextual equivalence at all types.
GENERAL TWISTING OF ALGEBRAS
, 2006
"... ABSTRACT. We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra (A, µ, u) in a monoidal category, as a morphism T: A ⊗ A → A ⊗ A satisfying a list of axioms ensuring that (A,µ ◦ T, u) is also an algebra in the category. This concept provi ..."
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ABSTRACT. We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra (A, µ, u) in a monoidal category, as a morphism T: A ⊗ A → A ⊗ A satisfying a list of axioms ensuring that (A,µ ◦ T, u) is also an algebra in the category. This concept provides a unifying framework for various deformed (or twisted) algebras from the literature, such as twisted tensor products of algebras, twisted bialgebras and algebras endowed with Fedosov products. Pseudotwistors appear also in other topics from the literature, e.g. Durdevich’s braided quantum groups and ribbon algebras. We also focus on the effect of twistors on the universal first order differential calculus, as well as on lifting twistors to braided twistors on the algebra of universal differential forms. 1.
MONAD COMPOSITIONS I: GENERAL CONSTRUCTIONS AND RECURSIVE DISTRIBUTIVE LAWS
"... ABSTRACT. New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product monads, recursivelydefined distributive laws, and linear equations. Sequel papers will consider distributive laws in closed categories and will construct monad ..."
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ABSTRACT. New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product monads, recursivelydefined distributive laws, and linear equations. Sequel papers will consider distributive laws in closed categories and will construct monad approximations for compositions which fail to be a monad. 1.
Multitensors and monads on categories of enriched graphs
"... Abstract. In this paper we unify the developments of [Batanin, 1998], [BataninWeber, 2011] and [Cheng, 2011] into a single framework in which the interplay between multitensors on a category V, and monads on the category GV of graphs enriched in V, is taken as fundamental. The material presented he ..."
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Abstract. In this paper we unify the developments of [Batanin, 1998], [BataninWeber, 2011] and [Cheng, 2011] into a single framework in which the interplay between multitensors on a category V, and monads on the category GV of graphs enriched in V, is taken as fundamental. The material presented here is the conceptual background for subsequent work: in [BataninCisinskiWeber, 2012] the Gray tensor product of 2categories and the Crans tensor product [Crans, 1999] of Gray categories are exhibited as existing within our framework, and in [Weber, 2013] the explicit construction of the funny tensor product of categories is generalised to a large class of Batanin operads. 1.
An algebraic view of program composition
 Algebraic Methodology and Software Technology, number 1548 in Lect. Notes Comp. Sci
, 1998
"... Abstract. We propose a general categorical setting for modeling program composition in which the callbyvalue and callbyname disciplines fit as special cases. Other notions of composition arising in denotational semantics are captured in the same framework: our leading examples are nondeterminist ..."
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Abstract. We propose a general categorical setting for modeling program composition in which the callbyvalue and callbyname disciplines fit as special cases. Other notions of composition arising in denotational semantics are captured in the same framework: our leading examples are nondeterministic callbyneed programs and nonstrict functions with side effects. Composition of such functions is treated in our framework with the same degree of abstraction that Moggi’s categorical approach based on monads allows in the treatment of callbyvalue programs. By virtue of such abstraction, interesting program equivalences can be validated axiomatically in mathematical models obtained by means of modular constructions. 1
The expression lemma ⋆
"... Abstract. Algebraic data types and catamorphisms (folds) play a central role in functional programming as they allow programmers to define recursive data structures and operations on them uniformly by structural recursion. Likewise, in objectoriented (OO) programming, recursive hierarchies of objec ..."
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Abstract. Algebraic data types and catamorphisms (folds) play a central role in functional programming as they allow programmers to define recursive data structures and operations on them uniformly by structural recursion. Likewise, in objectoriented (OO) programming, recursive hierarchies of object types with virtual methods play a central role for the same reason. There is a semantical correspondence between these two situations which we reveal and formalize categorically. To this end, we assume a coalgebraic model of OO programming with functional objects. The development may be helpful in deriving refactorings that turn sufficiently disciplined functional programs into OO programs of a designated shape and vice versa. Key words: expression lemma, expression problem, functional object, catamorphism, fold, the composite design pattern, program calculation, distributive law, free monad, cofree comonad. 1
MONADS AS EXTENSION SYSTEMS —NO ITERATION IS NECESSARY
"... Abstract. We introduce a description of the algebras for a monad in terms of extension systems, similar to the one for monads given in [Manes, 1976]. We rewrite distributive laws for monads and wreaths in terms of this description, avoiding the iteration of the functors involved. We give a profuncto ..."
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Abstract. We introduce a description of the algebras for a monad in terms of extension systems, similar to the one for monads given in [Manes, 1976]. We rewrite distributive laws for monads and wreaths in terms of this description, avoiding the iteration of the functors involved. We give a profunctorial explanation of why Manes’ description of monads in terms of extension systems works. 1.
The Powerdomain of Indexed Valuations
 In Proceedings of the 17th IEEE Symposium on Logic in Computer Science (LICS
, 2002
"... This paper is about combining nondeterminism and probabilities. We study this phenomenon from a domain theoretic point of view. In domain theory, nondeterminism is modeled using the notion of powerdomain, while probability is modeled using the powerdomain of valuations. Those two functors do not com ..."
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This paper is about combining nondeterminism and probabilities. We study this phenomenon from a domain theoretic point of view. In domain theory, nondeterminism is modeled using the notion of powerdomain, while probability is modeled using the powerdomain of valuations. Those two functors do not combine well, as they are. We define the notion of powerdomain of indexed valuations, which can be combined nicely with the usual nondeterministic powerdomain. We show an equational characterization of our construction. Finally we discuss the computational meaning of indexed valuations, and we show how they can be used, by giving a denotational semantics of a simple imperative language.