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Notes on enriched categories with colimits of some class
 Theory Appl. Categ
"... The paper is in essence a survey of categories having φweighted colimits for all the weights φ in some class Φ. We introduce the class Φ + of Φflat weights which are those ψ for which ψcolimits commute in the base V with limits having weights in Φ; and the class Φ − of Φatomic weights, which are ..."
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The paper is in essence a survey of categories having φweighted colimits for all the weights φ in some class Φ. We introduce the class Φ + of Φflat weights which are those ψ for which ψcolimits commute in the base V with limits having weights in Φ; and the class Φ − of Φatomic weights, which are those ψ for which ψlimits commute in the base V with colimits having weights in Φ. We show that both these classes are saturated (that is, what was called closed in the terminology of [AK88]). We prove that for the class P of all weights, the classes P + and P − both coincide with the class Q of absolute weights. For any class Φ and any category A, we have the free Φcocompletion Φ(A) of A; and we recognize Q(A) as the Cauchycompletion of A. We study the equivalence between (Q(A op)) op and Q(A), which we exhibit as the restriction of the Isbell adjunction between [A, V] op and [A op, V] when A is small; and we give a new Morita theorem for any class Φ containing Q. We end with the study of Φcontinuous weights and their relation to the Φflat weights. 1
Algebras of higher operads as enriched categories II
 In preparation
"... Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the ..."
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Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the category of nglobular sets from any normalised (n + 1)operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)category is something like a category enriched in weak ncategories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.
Formal proof, computation, and the construction problem in algebraic geometry
"... It has become a classical technique to turn to theoretical computer science to provide computational tools for algebraic geometry. A more recent transformation is that now we also get logical tools, and these too should be useful in the study of algebraic varieties. The purpose of this note is to co ..."
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It has become a classical technique to turn to theoretical computer science to provide computational tools for algebraic geometry. A more recent transformation is that now we also get logical tools, and these too should be useful in the study of algebraic varieties. The purpose of this note is to consider a very small part of this picture, and try to motivate the study of computer theoremproving techniques by looking at how they might be relevant to a particular class of problems in algebraic geometry. This is only an informal discussion, based more on questions and possible research directions than on actual results. This note amplifies the themes discussed in my talk at the “Arithmetic and Differential Galois Groups ” conference (March 2004, Luminy), although many specific points in the discussion were only finished more recently. I would like to thank: André Hirschowitz and Marco Maggesi, for their invaluable insights about computerformalized mathematics as it relates
A Duality Relative To A Limit Doctrine
, 2002
"... We give a unified proof of GabrielUlmer duality for locally finitely presentable categories, AdamekLawvereRosicky duality for varieties and Morita duality for presheaf categories. As an application, we compare presheaf categories and varieties. ..."
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We give a unified proof of GabrielUlmer duality for locally finitely presentable categories, AdamekLawvereRosicky duality for varieties and Morita duality for presheaf categories. As an application, we compare presheaf categories and varieties.
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.
FINAL COALGEBRAS IN ACCESSIBLE CATEGORIES
, 905
"... Abstract. We give conditions on a finitary endofunctor of a finitely accessible category to admit a final coalgebra. Our conditions always apply to the case of a finitary endofunctor of a locally finitely presentable (l.f.p.) category and they bring an explicit construction of the final coalgebra in ..."
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Abstract. We give conditions on a finitary endofunctor of a finitely accessible category to admit a final coalgebra. Our conditions always apply to the case of a finitary endofunctor of a locally finitely presentable (l.f.p.) category and they bring an explicit construction of the final coalgebra in this case. On the other hand, there are interesting examples of final coalgebras beyond the realm of l.f.p. categories to which our results apply. We rely on ideas developed by Tom Leinster for the study of selfsimilar objects in topology. 1.
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