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Notions of Lawvere theory
"... Categorical universal algebra can be developed either using Lawvere theories (singlesorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of ..."
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Categorical universal algebra can be developed either using Lawvere theories (singlesorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of
Functorial coalgebraic logic: The case of manysorted varieties
 Electron. Notes Theor. Comput. Sci
"... Following earlier work, a modal logic for Tcoalgebras is a functor L on a suitable variety. Syntax and proof system of the logic are given by presentations of the functor. This paper makes two contributions. First, a previous result characterizing those functors that have presentations is generaliz ..."
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Following earlier work, a modal logic for Tcoalgebras is a functor L on a suitable variety. Syntax and proof system of the logic are given by presentations of the functor. This paper makes two contributions. First, a previous result characterizing those functors that have presentations is generalized from endofunctors on onesorted varieties to functors between manysorted varieties. This yields an equational logic for the presheaf semantics of higherorder abstract syntax. As another application, we show how the move to functors between manysorted varieties allows to modularly combine syntax and proof systems of different logics. Second, we show how to associate to any setfunctor T a complete (finitary) logic L consisting of modal operators and Boolean connectives.
Instances of computational effects: an algebraic perspective
"... Abstract—We investigate the connections between computational effects, algebraic theories, and monads on functor categories. We develop a syntactic framework with variable binding that allows us to describe equations between programs while taking into account the idea that there may be different ins ..."
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Abstract—We investigate the connections between computational effects, algebraic theories, and monads on functor categories. We develop a syntactic framework with variable binding that allows us to describe equations between programs while taking into account the idea that there may be different instances of a particular computational effect. We use our framework to give a general account of several notions of computation that had previously been analyzed in terms of monads on presheaf categories: the analysis of local store by Plotkin and Power; the analysis of restriction by Pitts; and the analysis of the pi calculus by Stark. I.
WHAT ARE SIFTED COLIMITS?
"... Dedicated to Dominique Bourn on the occasion of his sixtieth birthday Abstract. Sifted colimits, important for algebraic theories, are “almost ” just the combination of filtered colimits and reflexive coequalizers. For example, given a finitely cocomplete category A, then a functor with domain A pre ..."
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Dedicated to Dominique Bourn on the occasion of his sixtieth birthday Abstract. Sifted colimits, important for algebraic theories, are “almost ” just the combination of filtered colimits and reflexive coequalizers. For example, given a finitely cocomplete category A, then a functor with domain A preserves sifted colimits iff it preserves filtered colimits and reflexive coequalizers. But for general categories A that statement is not true: we provide a counterexample.
ON HOMOTOPY VARIETIES
, 2005
"... Abstract. Given an algebraic theory T, a homotopy Talgebra is a simplicial set where all equations from T hold up to homotopy. All homotopy Talgebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties. We will also study ho ..."
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Abstract. Given an algebraic theory T, a homotopy Talgebra is a simplicial set where all equations from T hold up to homotopy. All homotopy Talgebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties. We will also study homotopy models of limit theories which leads to homotopy locally presentable categories. These were recently considered by Simpson, Lurie, Toën and Vezzosi. 1.
Note on the construction of free monoids
, 802
"... We construct free monoids in a monoidal category (C, ⊗, I) with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains. ..."
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We construct free monoids in a monoidal category (C, ⊗, I) with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains.
Accessible categories and . . .
, 2007
"... Accessible categories have recently turned out to be useful in homotopy theory. This text is prepared as notes for a series of lectures at ..."
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Accessible categories have recently turned out to be useful in homotopy theory. This text is prepared as notes for a series of lectures at
An algebraic presentation of predicate logic (extended abstract)
"... Abstract. We present an algebraic theory for a fragment of predicate logic. The fragment has disjunction, existential quantification and equality. It is not an algebraic theory in the classical sense, but rather within a new framework that we call ‘parameterized algebraic theories’. We demonstrate t ..."
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Abstract. We present an algebraic theory for a fragment of predicate logic. The fragment has disjunction, existential quantification and equality. It is not an algebraic theory in the classical sense, but rather within a new framework that we call ‘parameterized algebraic theories’. We demonstrate the relevance of this algebraic presentation to computer science by identifying a programming language in which every type carries a model of the algebraic theory. The result is a simple functional logic programming language. We provide a syntaxfree representation theorem which places terms in bijection with sieves, a concept from category theory. We study presentationinvariance for general parameterized algebraic theories by providing a theory of clones. We show that parameterized algebraic theories characterize a class of enriched monads. 1