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19
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.
A Study of Categories of Algebras and Coalgebras
, 2001
"... This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and ..."
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This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and the dual notion of coalgebras. Following this, we discuss (co)algebras for a (co)monad and develop a theory of regular subcoalgebras which will be used in the internal logic. We also prove that categories of coalgebras are complete, under reasonably weak conditions, and simultaneously prove the wellknown dual result for categories of algebras. We close the second chapter with a discussion of bisimulations in which we introduce a weaker notion of bisimulation than is current in the literature, but which is wellbehaved and reduces to the standard definition under the assumption of choice. The third chapter is a detailed look at three theorem's of G. Birkho# [Bir35, Bir44], presenting categorical proofs of the theorems which generalize the classical results and which can be easily dualized to apply to categories of coalgebras. The theorems of interest are the variety theorem, the equational completeness theorem and the subdirect product representation theorem. The duals of each of these theorems is discussed in detail, and the dual notion of "coequation" is introduced and several examples given. In the final chapter, we show that first order logic can be interpreted in categories of coalgebras and introduce two modal operators to first order logic to allow reasoning about "endomorphisminvariant" coequations and bisimulations internally. We also develop a translation of terms and formulas into the internal language of the base category, which preserves and reflects truth. La...
Corings over operads characterize morphisms, math.AT/0505559
"... objects in M, with its composition monoidal structure. Let R be a Pcoring, ..."
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objects in M, with its composition monoidal structure. Let R be a Pcoring,
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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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.
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...
Bimodules over operads characterize morphisms, preprint at arXiv:math.AT/0505559
, 2005
"... Abstract. Let P be any operad. A Pbimodule R that is a Pcooperad ..."
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Abstract. Let P be any operad. A Pbimodule R that is a Pcooperad
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.
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 Hausdorff frames ..."
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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
Abstraction and Performance from Explicit Monadic Reflection
, 1999
"... Much of the monadic programming literature gets the types right but the abstraction wrong. Using monadic parsing as the motivating example, we demonstrate standard monadic programs in Scheme, recognize how they violate abstraction boundaries, and recover clean abstraction crossings through monadic r ..."
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Much of the monadic programming literature gets the types right but the abstraction wrong. Using monadic parsing as the motivating example, we demonstrate standard monadic programs in Scheme, recognize how they violate abstraction boundaries, and recover clean abstraction crossings through monadic reflection. Once monadic reflection is made explicit, it is possible to construct a grammar for monadic programming. This grammar, in turn, enables the redefinition of the monadic operators as macros that eliminate at expansion time the overhead imposed by functional representations. The result is very efficient monadic programs; for parsing, the output code is competitive with good handcrafted parsers.