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Enrico Vitale - One of the best experts on this subject based on the ideXlab platform.
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The snail lemma for internal groupoids
Journal of Algebra, 2019Co-Authors: Sandra Mantovani, Giuseppe Metere, Enrico VitaleAbstract:Abstract We establish a generalized form both of the Gabriel-Zisman exact sequence associated with a pointed functor between pointed groupoids, and of the Brown exact sequence associated with a fibration of pointed groupoids. Our generalization consists in replacing pointed groupoids with groupoids internal to a pointed regular category with reflexive Coequalizers.
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What are sifted colimits
Theory and Applications of Categories, 2010Co-Authors: Jiří Adámek, Jiri Rosicky, Enrico VitaleAbstract:Sifted colimits, important for algebraic theories, are "almost" just the combination of filtered colimits and reflexive Coequalizers. For example, given a finitely cocomplete category $cal A$, then a functor with domain $cal A$ preserves sifted colimits iff it preserves filtered colimits and reflexive Coequalizers. But for general categories $cal A$ that statement is not true: we provide a counter-example.
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A Picard–Brauer exact sequence of categorical groups
Journal of Pure and Applied Algebra, 2002Co-Authors: Enrico VitaleAbstract:A categorical group is a monoidal groupoid in which each object has a tensorial inverse. Two main examples are the Picard categorical group of a monoidal category and the Brauer categorical group of a braided monoidal category with stable Coequalizers. After discussing the notions of kernel, cokemel and exact sequence for categorical groups, we show that, given a suitable monoidal functor between two symmetric monoidal categories with stable Coequalizers, it is possible to build up a five-term Picard-Brauer exact sequence of categorical groups. The usual Units-Picard and Picard-Brauer exact sequences of abelian groups follow from this exact sequence of categorical groups. We also discuss the direct sum decomposition of the Brauer-Long group. (C) 2002 Elsevier Science B.V. All rights reserved.
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Beck's theorem for pseudo-monads
Journal of Pure and Applied Algebra, 2002Co-Authors: I. J. Le Creurer, F. Marmolejo, Enrico VitaleAbstract:In this work we establish a 2-categorical analogue of Beck's theorem characterizing monadic functors. We show that a 2-functor (a pseudo-functor) U is monadic iff it is a right pseudo-adjoint, it reflects adjoint equivalences and it creates U-absolute pseudo-Coequalizers of codescent objects. (C) 2002 Elsevier Science B.V. All rights reserved.
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Exact completion and representations in abelian categories
Homology Homotopy and Applications, 2001Co-Authors: J. Rosicky, Enrico VitaleAbstract:When the exact completion of a category with weak finite limits is a Mal’cev category, it is possible to combine the universal property of the exact completion and the universal property of the Coequalizer completion. We use this fact to explain Freyd’s representation theorems in abelian and Frobenius categories.
Javier Paris - One of the best experts on this subject based on the ideXlab platform.
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the foldl operator as a Coequalizer using coq
Computer Aided Systems Theory, 2009Co-Authors: Antonio Blanco, Enrique Freire, José Luis Freire, Javier ParisAbstract:In the present work a Coq based approach is taken to characterize the foldl using a functorial structure from which an inductive type is determined. With μF being an initial F---algebra and (B,?) another F---algebra, two F---algebras with support B×μF are constructed and then coequalized. This coequalization morphism allows the definition of foldl structurally. After examining some significant examples we propose the following methodology to define a foldl operator. Let F be a polynomial endofunctor and (μF,inF) its initial algebra. We define two F---algebras with support B×μF, and h1,h2:F(B×μF) ?B ×μF constructed such that in one of them the argument of the initial type is syntactically (structurally) lower than that in the other. Then, ${\mathit{foldl}}:B\times\mu_F \rightarrow B$ can be defined as a specific morphism that coequalizes them $(h_1;{\mathit{foldl}}=h_2;{\mathit{foldl}}).$ For an initial F---algebra with distinguished element (as in the case of lists), foldl is a Coequalizer of h1 and h2. The proofs are performed using the Coq proof system. In this context, constructive approach stress that existence is constructive, therefore with a computational content. Detailed structure for proofs in Coq is included but interactive code is omited. See the section 4 for a repository with the complete source code.
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EUROCAST - The Foldl Operator as a Coequalizer Using Coq
Computer Aided Systems Theory - EUROCAST 2009, 2009Co-Authors: Antonio Blanco, Enrique Freire, José Luis Freire, Javier ParisAbstract:In the present work a Coq based approach is taken to characterize the foldl using a functorial structure from which an inductive type is determined. With μF being an initial F---algebra and (B,?) another F---algebra, two F---algebras with support B×μF are constructed and then coequalized. This coequalization morphism allows the definition of foldl structurally. After examining some significant examples we propose the following methodology to define a foldl operator. Let F be a polynomial endofunctor and (μF,inF) its initial algebra. We define two F---algebras with support B×μF, and h1,h2:F(B×μF) ?B ×μF constructed such that in one of them the argument of the initial type is syntactically (structurally) lower than that in the other. Then, ${\mathit{foldl}}:B\times\mu_F \rightarrow B$ can be defined as a specific morphism that coequalizes them $(h_1;{\mathit{foldl}}=h_2;{\mathit{foldl}}).$ For an initial F---algebra with distinguished element (as in the case of lists), foldl is a Coequalizer of h1 and h2. The proofs are performed using the Coq proof system. In this context, constructive approach stress that existence is constructive, therefore with a computational content. Detailed structure for proofs in Coq is included but interactive code is omited. See the section 4 for a repository with the complete source code.
Steven Vickers - One of the best experts on this subject based on the ideXlab platform.
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The localic compact interval is an Escardo-Simpson interval object
Mathematical Logic Quarterly, 2017Co-Authors: Steven VickersAbstract:The locale corresponding to the real interval [-1,1] is an interval object, in the sense of Escard\'o and Simpson, in the category of locales. The map c from 2^\omega to [-1,1], mapping a stream s of signs +1 or -1 to \Sum_{i=1}^\infty s_i 2^{-i}, is a proper localic surjection; it is also expressed as a Coequalizer.
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the localic compact interval is an escardo simpson interval object
Mathematical Logic Quarterly, 2017Co-Authors: Steven VickersAbstract:The locale corresponding to the real interval [−1, 1] is an interval object, in the sense of Escardo and Simpson, in the category of locales. The map c : 2 → [−1, 1], mapping a stream s of signs ±1 to Σi=1si2, is a proper localic surjection; it is also expressed as a Coequalizer.
J. N. Alonso Álvarez - One of the best experts on this subject based on the ideXlab platform.
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Faithfully flat descent for magmas
Journal of Pure and Applied Algebra, 2020Co-Authors: J. N. Alonso Álvarez, J. M. Fernández Vilaboa, R. González Rodríguez, M.p. López LópezAbstract:Abstract In this paper we develop a descent theory for morphisms α between a monoid B and a unital magma A in a monoidal category with equalizers and Coequalizers. We introduce the category of strong descent data for α and we prove that under faithfully flat conditions this category is equivalent to the one of right B-modules. As an application we prove that the category of strong Hopf modules, introduced by us for Hopf quasigroups and weak Hopf quasigroups, is equivalent to a suitable category of strong descent data.
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Weak Galois and Weak Cocleft Coextensions
Algebra Colloquium, 2007Co-Authors: J. N. Alonso Álvarez, J. M. Fernández Vilaboa, R. González Rodríguez, A. B. Rodríguez RaposoAbstract:For a weak entwining structure (A, C, ψ) living in a braided monoidal category with equalizers and Coequalizers, we formulate the notion of weak A-Galois coextension with normal basis and we show that these Galois coextensions are equivalent to the weak A-cocleft coextensions introduced by the authors.
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THE GROUP OF HOPF-GALOIS EXTENSIONS WITH CENTRAL INVARIANTS
Communications in Algebra, 2001Co-Authors: J. N. Alonso Álvarez, J. M. Fernández Vilaboa, Ramón González RodríguezAbstract:In this paper, for an algebra A and a flat Hopf algebra H, in a symmetric closed category C with equalizers and Coequalizers, we define the product of H-extensions of A with central invariants. This product is a generalization of the product of H-Galois objects. Finally, we define the Hopf-Galois H-extensions of A with central invariants and we prove that, if A is a commutative faithfully flat algebra and H is cocommutative and faithfully flat, the set of isomorphism classes of Hopf-Galois H-extensions of A with central invariants, with the product of H-extensions, is an abelian group denoted by Gal C (A, H).
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Yetter—Drinfel'd H-Azumaya Monoids in Closed Categories
Applied Categorical Structures, 1998Co-Authors: J. N. Alonso Álvarez, J. M. Fernández Vilaboa, E. Villanueva NovoaAbstract:When C is a symmetric closed category with equalizers and Coequalizers and H is a Hopf algebra in C, the category of Yetter—Drinfel’d H-modules is a braided monoidal category.
J. Rosicky - One of the best experts on this subject based on the ideXlab platform.
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Generalized Brown representability in homotopy categories: erratum.
Theory and Applications of Categories, 2008Co-Authors: J. RosickyAbstract:Propositions 4.2 and 4.3 of the author's article (Theory Appl. Categ. 14 (2005), 451-479) are not correct. We show that their use can be avoided and all remaining results remain correct 1 . Propositions 4.2 and 4.3 of the author's (3) are not correct and I am grateful to J. F. Jardine for pointing it out. In fact, consider the diagram D sending the one morphism category to the point 0 in the homotopy category Ho(SSet) of simplicial sets. The standard weak colimit of D is the standard weak Coequalizer
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Comparing Coequalizer And Exact Completions
2007Co-Authors: M. C. Pedicchio, J. RosickyAbstract:. We characterize when the Coequalizer and the exact completion of a category C with finite sums and weak finite limits coincide. Introduction Our aim is to compare two well known completions: the Coequalizer completion C coeq of a small category C with finite sums (see [P]) and the exact completion of a small category C with weak finite limits (see [CV]). For a category C with finite sums and weak finite limits, C ex is always a full subcategory of C coeq . We characterize when the two completions are equivalent - it turns out that this corresponds to a finiteness condition expressed in terms of reflexive and symmetric graphs in C. 1. Two completions For a small category C with finite sums, the Coequalizer completion of C is a category C coeq with finite colimits together with a finite sums preserving functor G C : C ! C coeq such that, for any finite sums preserving functor F : C !X into a finitely cocomplete category, there is a unique finite colimits preserving functor F : C..
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Exact completion and representations in abelian categories
Homology Homotopy and Applications, 2001Co-Authors: J. Rosicky, Enrico VitaleAbstract:When the exact completion of a category with weak finite limits is a Mal’cev category, it is possible to combine the universal property of the exact completion and the universal property of the Coequalizer completion. We use this fact to explain Freyd’s representation theorems in abelian and Frobenius categories.
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comparing Coequalizer and exact completions
Theory and Applications of Categories, 1999Co-Authors: M. C. Pedicchio, J. RosickyAbstract:There is shown when the Coequalizer and the exact completions of a category with finite coproducts and weak finite limits coincide.
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COMPARING Coequalizer AND EXACT COMPLETIONS
1999Co-Authors: M. C. Pedicchio, J. RosickyAbstract:We characterize when the Coequalizer and the exact completion of a category C with nite sums and weak nite limits coincide