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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.
Formalized proof, computation, and the construction problem in algebraic geometry
, 2004
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A characterization of locally Dpresentable categories. Cahiers Topologie et Géométrie Différentielle
, 2004
"... ABSTRACT. Locally finitely presentable categories have been generalized in [1], under the name of locally Dpresentable categories, replacing filtered colimits by colimits commuting in Set with limits indexed by an arbitrary doctrine D. In this note, we characterize locally Dpresentable categories ..."
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ABSTRACT. Locally finitely presentable categories have been generalized in [1], under the name of locally Dpresentable categories, replacing filtered colimits by colimits commuting in Set with limits indexed by an arbitrary doctrine D. In this note, we characterize locally Dpresentable categories as cocomplete categories with a strong generator consisting of Dpresentable objects. This extends known results on locally finitely presentable categories, varieties and presheaf categories. 1.
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.
Foundation.
, 2011
"... and the category Alg of “algebras ” that arise from a schizophrenic object , which is both an “algebra ” and a “space”. We call such adjunctions logical connections. We prove that the exact nature of is that of a module that allows to lift optimally the structure of a “space ” and an “algebra ” to ..."
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and the category Alg of “algebras ” that arise from a schizophrenic object , which is both an “algebra ” and a “space”. We call such adjunctions logical connections. We prove that the exact nature of is that of a module that allows to lift optimally the structure of a “space ” and an “algebra ” to certain diagrams. Our approach allows to give a unified framework known from logical connections over the category of sets and analyzed, e.g., by Hans Porst and Walter Tholen, with future applications of logical connections in coalgebraic logic and elsewhere, where typically, both
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.
4 FORMALIZED PROOF, COMPUTATION, AND THE CONSTRUCTION PROBLEM IN ALGEBRAIC GEOMETRY
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DENSE MORPHISMS OF MONADS
"... Abstract. Given an arbitrary locally finitely presentable category K and finitary monads T and S on K, we characterize monad morphisms α: S − → T with the property that the induced functor α ∗ : KT − → KS between the categories of EilenbergMoore algebras is fully faithful. We call such monad morphi ..."
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Abstract. Given an arbitrary locally finitely presentable category K and finitary monads T and S on K, we characterize monad morphisms α: S − → T with the property that the induced functor α ∗ : KT − → KS between the categories of EilenbergMoore algebras is fully faithful. We call such monad morphisms dense and give a characterization of them in the spirit of Beth’s definability theorem: α is a dense monad morphism if and only if every Toperation is explicitly defined using Soperations. We also give a characterization in terms of epimorphic property of α and clarify the connection between various notions of epimorphisms between monads.