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 $K$ to $DG$-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 $DG$ 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 $K$, 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-Konstant-Rosenberg and Loday-Quillen theorems. Our version applies to the cyclic module that arises from a homomorphism $R\to S$ of not necessarily commutative rings when the coproduct $\coprod_R$ of associative $R$-algebras is substituted for $\otimes_R$. As another application of the properties of $Q$, we obtain a simple, braid-free description of a product on the tensor power $S^{\otimes_R^n}$ 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

Peiffer elements in simplicial groups and algebras

The main objective of this paper is to prove in full generality the following two facts: A. For an operad O in Ab, let A be a simplicial O-algebra such that Am is generated as an O-ideal by (∑i = 0m-1 si (Am-1)), for m > 1, and let NA be the Moore complex of A. Then d(NmA) = ∑Iγ (Op⊗ ∩ i∈I1 ker di ⊗ ⋯ ⊗ ∩ i∈Ip ker di) where the sum runs over those partitions of [m - 1], I = (I1, ..., Ip), p ≥ 1, and γ is the action of O on A. B. Let G be a simplicial group with Moore complex NG in which Gn is generated as a normal subgroup by the degenerate elements in dimensionn > 1, then d (NnG) = ∏I, J [∩i∈I ker di, ∩i∈J ker dj], for I, J ⊆ [n - 1] with I ∪ J = [n - 1]. In both cases, di is the i-th face of the corresponding simplicial object. The former result completes and generalizes results from Akça and Arvasi [I. Akça, Z. Arvasi, Simplicial and crossed Lie algebras, Homology Homotopy Appl. 4 (1) (2002) 43-57], and Arvasi and Porter [Z. Arvasi, T. Porter, Higher dimensional Peiffer elements in simplicial commutative algebras, Theory Appl. Categ. 3 (1) (1997) 1-23]; the latter completes a result from Mutlu and Porter [A. Mutlu, T. Porter, Applications of Peiffer pairings in the Moore complex of a simplicial group, Theory Appl. Categ. 4 (7) (1998) 148-173]. Our approach to the problem is different from that of the cited works. We have first succeeded with a proof for the case of algebras over an operad by introducing a different description of the inverse of the normalization functor N:AbΔop → Ch≥ 0. For the case of simplicial groups, we have then adapted the construction for the inverse equivalence used for algebras to get a simplicial group NG ⊠ Λ from the Moore complex N G of a simplicial group G. This construction could be of interest in itself.Facultad de Ciencias Exacta

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