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35
The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
, 2007
"... Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained ..."
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Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. This development has formed a major part of Gordon Plotkin’s mature work, and we study its history here, in particular asking why Lawvere theories were eclipsed by monads in the 1960’s, and how the renewed interest in them in a computer science setting might develop in future.
STACKS OF GROUP REPRESENTATIONS
"... Abstract. We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extensionofscalars. We deduce that, given a group G, the derived and the stable categories of representations of a subgroup H can be constructed ..."
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Abstract. We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extensionofscalars. We deduce that, given a group G, the derived and the stable categories of representations of a subgroup H can be constructed out of the corresponding category for G by a purely triangulatedcategorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods to investigate when modular representations of the subgroup H can be extended to G. We show that the presheaves of plain, derived and stable representations all form
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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Cited by 6 (0 self)
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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.
Syntactic Composition of TopDown Tree Transducers is Short Cut Fusion
, 2001
"... We compare two deforestation techniques: short cut fusion formalized in category theory and the syntactic composition of tree transducers. The former strongly depends on types and uses the parametricity property or free theorem whereas the latter makes no use of types at all and allows more general ..."
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Cited by 5 (1 self)
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We compare two deforestation techniques: short cut fusion formalized in category theory and the syntactic composition of tree transducers. The former strongly depends on types and uses the parametricity property or free theorem whereas the latter makes no use of types at all and allows more general compositions. We introduce the notion of a categorical transducer which is a generalization of a catamorphism and show a respective fusion result which is a generalization of the `acid rain theorem'. We prove the following main theorems: (i) The class of all categorical transducers builds a category where composition is fusion. (ii) The semantics of categorical transducers is a functor. (iii) The subclass of topdown categorical transducers is a subcategory. (iv) Syntactic composition of topdown tree transducers is equivalent to the fusion of topdown categorical transducers.
The odd origin of Gerstenhaber, BV and the master equation
, 2012
"... Abstract. In this paper we show that Gerstenhaber brackets, BV operators and related master equations arise in a very natural way when considering odd operads and their generalizations. We show that many known examples such as BV operators in the Calabi– Yau setting, brackets in string field theor ..."
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Abstract. In this paper we show that Gerstenhaber brackets, BV operators and related master equations arise in a very natural way when considering odd operads and their generalizations. We show that many known examples such as BV operators in the Calabi– Yau setting, brackets in string field theory, the master equation in that setting, the master equation for Feynman transforms come from this type of setup. We give a systematic and comprehensive treatment of all the usual setups involving (cyclic/modular) operads and PROP(erad)s including new results. Further generalizations and categorical constructions will be presented in a sequel.
Monads on Composition Graphs
 UNIVERSITY OF BREMEN
, 2000
"... Collections of objects and morphisms that fail to form categories inasmuch as the expected composites of two morphisms need not always be defined have been introduced in [14, 15] under the name composition graphs. In [14, 16], notions of adjunction and weak adjunction for composition graphs have bee ..."
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Collections of objects and morphisms that fail to form categories inasmuch as the expected composites of two morphisms need not always be defined have been introduced in [14, 15] under the name composition graphs. In [14, 16], notions of adjunction and weak adjunction for composition graphs have been proposed. Building on these definitions, we now introduce a concept of monads for composition graphs and show that the usual correspondence between adjunctions and monads remains correct, i.e. that (weak) adjunctions give rise to monads and that all monads are induced by adjunctions. Monads are described in terms of natural transforms as well as in terms of Kleisli triples, which seem to be better suited in the absence of associativity. The realization of a monad by an adjunction relies on a generalization of the Kleisli construction to composition graphs; on the other hand, the EilenbergMoore construction produces only a weak adjunction and admits comparison functors from weak adjuncti...
Unifying structured recursion schemes
 in International Conference on Functional Programming. ACM
"... Folds over inductive datatypes are well understood and widely used. In their plain form, they are quite restricted; but many disparate generalisations have been proposed that enjoy similar calculational benefits. There have also been attempts to unify the various generalisations: two prominent su ..."
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Folds over inductive datatypes are well understood and widely used. In their plain form, they are quite restricted; but many disparate generalisations have been proposed that enjoy similar calculational benefits. There have also been attempts to unify the various generalisations: two prominent such unifications are the ‘recursion schemes from comonads ’ of Uustalu, Vene and Pardo, and our own ‘adjoint folds’. Until now, these two unified schemes have appeared incompatible. We show that this appearance is illusory: in fact, adjoint folds subsume recursion schemes from comonads. The proof of this claim involves standard constructions in category theory that are nevertheless not well known in functional programming: EilenbergMoore categories and bialgebras.
Maps and Monads for Modal Frames
, 2004
"... The categorytheoretic nature of general frames for modal logic is explored. A new notion of "modal map" between frames is defined, generalizing the usual notion of bounded morphism/pmorphism. The category Fm of all frames and modal maps has reflective subcategories CHFm of compact Hausdo ..."
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The categorytheoretic nature of general frames for modal logic is explored. A new notion of "modal map" between frames is defined, generalizing the usual notion of bounded morphism/pmorphism. The category Fm of all frames and modal maps has reflective subcategories CHFm of compact Hausdorff frames, DFm of descriptive frames, and UEFm of ultrafilter enlargements of frames. All three subcategories are equivalent, and are dual to the category of modal algebras and their homomorphisms. The ultrafilter