60 research outputs found
Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence
The (dual) Dold-Kan correspondence says that there is an equivalence of
categories K:\cha\to \Ab^\Delta between nonnegatively graded cochain
complexes and cosimplicial abelian groups, which is inverse to the
normalization functor. We show that the restriction of to -rings can be
equipped with an associative product and that the resulting functor
DGR^*\to\ass^\Delta, although not itself an equivalence, does induce one at
the level of homotopy categories. The dual of this result for chain and
simplicial rings was obtained independently by S. Schwede and B. Shipley
through different methods ({\it Equivalences of monoidal model categories}.
Algebraic and Geometric Topology 3 (2003), 287-334). Our proof is based on a
functor Q:DGR^*\to \ass^\Delta, naturally homotopy equivalent to , which
preserves the closed model structure. It also has other interesting
applications. For example, we use to prove a noncommutative version of the
Hochschild-Konstant-Rosenberg and Loday-Quillen theorems. Our version applies
to the cyclic module that arises from a homomorphism of not
necessarily commutative rings when the coproduct of associative
-algebras is substituted for . As another application of the
properties of , we obtain a simple, braid-free description of a product on
the tensor power originally defined by P. Nuss using braids
({\it Noncommutative descent and nonabelian cohomology,} K-theory {\bf 12}
(1997) 23-74.).Comment: Final version to appear in JPAA. Large parts rewritten, especially in
the last section.Proof of main theorem simplifie
Cosimplicial versus DG-rings: A version of the Dold-Kan correspondence
The (dual) Dold-Kan correspondence says that there is an equivalence of categories K: Ch≥0→AbΔ between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show that the restriction of K to DG-rings can be equipped with an associative product and that the resulting functor DGR*→RingsΔ, although not itself an equivalence, does induce one at the level of homotopy categories. In other words both DGR* and RingsΔ are Quillen closed model categories and the total left derived functor of K is an equivalence: LK: Ho DGR* Ho RingsΔ. The dual of this result for chain DG and simplicial rings was obtained independently by Schwede and Shipley, Algebraic and Geometric Topology 3 (2003) 287, through different methods. Our proof is based on a functor Q:DGR*→RingsΔ, naturally homotopy equivalent to K, and which preserves the closed model structure. It also has other interesting applications. For example, we use Q to prove a noncommutative version of the Hochschild-Kostant-Rosenberg and Loday-Quillen theorems. Our version applies to the cyclic module [n] ∐nRS that arises from a homomorphism R→S of not necessarily commutative rings, using the coproduct ∐R of associative R-algebras. As another application of the properties of Q, we obtain a simple, braid-free description of a product on the tensor power S⊗Rn originally defined by Nuss K-theory 12 (1997) 23, using braids.Facultad de Ciencias Exacta
The finite model property for the variety of Heyting algebras with successor
The finite model property of the variety of S-algebras was proved by X. Caicedo using Kripke model techniques of the associated calculus. A more algebraic proof, but still strongly based on Kripke model ideas, was given by Muravitskii. In this article we give a purely algebraic proof for the finite model property which is strongly based on the fact that for every element x in a S-algebra the interval [x, S(x)] is a Boolean lattice.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica
Variations of the free implicative semilattice extension of a Hilbert algebra
Celani and Jansana (Math Log Q 58(3):188–207, 2012) give an explicit description of the free implicative semilattice extension of a Hilbert algebra. In this paper, we give an alternative path conducing to this construction. Furthermore, following our procedure, we show that an adjunction can be obtained between the algebraic categories of Hilbert algebras with supremum and that of generalized Heyting algebras. Finally, in the last section, we describe a functor from the algebraic category of Hilbert algebras to that of generalized Heyting algebras, of possible independent interest.Fil: Castiglioni, José Luis. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: San Martín, Hernán Javier. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentin
Modal operators for meet-complemented lattices
We investigate some modal operators of necessity and possibility in the context of meet-complemented (not necessarily distributive) lattices. We proceed in stages. We compare our operators with others.Facultad de Ciencias Exacta
On the variety of Heyting algebras with successor generated by all finite chains
Contrary to the variety of Heyting algebras, finite Heyting algebras with successor only generate a proper subvariety of that of all Heyting algebras with successor. In particular, all finite chains generate a proper subvariety, SLHω, of the latter. There is a categorical duality between Heyting algebras with successor and certain Priestley spaces. Let X be the Heyting space associated by this duality to the Heyting algebra with successor H. If there is an ordinal κ and a filtration on X such that X = S λ≤κ Xλ, the height of X is the minimun ordinal ξ ≤ κ such that Xc ξ = ∅. In this case, we also say that H has height ξ. This filtration allows us to write the space X as a disjoint union of antichains. We may think that these antichains define levels on this space. We study the way of characterize subalgebras and homomorphic images in finite Heyting algebras with successor by means of their Priestley spaces. We also depict the spaces associated to the free algebras in various subcategories of SLH.Fil: Castiglioni, José Luis. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaFil: San Martín, Hernán Javier. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentin
The finite model property for the variety of Heyting algebras with successor
The finite model property of the variety of S-algebras was proved by X. Caicedo using Kripke model techniques of the associated calculus. A more algebraic proof, but still strongly based on Kripke model ideas, was given by Muravitskii. In this article we give a purely algebraic proof for the finite model property which is strongly based on the fact that for every element x in a S-algebra the interval [x, S(x)] is a Boolean lattice.Fil: Castiglioni, José Luis. Universidad Nacional de la Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: San Martin, Hernan Javier. Universidad Nacional de la Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
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