Showing papers in "Applied Categorical Structures in 2010"

Freely Suspended Cellular “Backpacks” Lead to Cell Aggregate Self-Assembly

Abstract: Cellular “backpacks” are a new type of anisotropic, nanoscale thickness microparticle that may be attached to the surface of living cells creating a “bio-hybrid” material. Previous work has shown t...

Universal Strict General Actors and Actors in Categories of Interest

Abstract: For any category of interest ℂ we define a general category of groups with operations $\mathbb{C_G}, \mathbb{C}\hookrightarrow\mathbb{C_G}$ , and a universal strict general actor USGA(A) of an object A in ℂ, which is an object of $\mathbb{C_G}$ . The notion of actor is equivalent to the one of split extension classifier defined for an object in more general settings of semi-abelian categories. It is proved that there exists an actor of A in ℂ if and only if the semidirect product ${\text{USGA}}(A)\ltimes A$ is an object of ℂ and if it is the case, then USGA(A) is an actor of A. We give a construction of a universal strict general actor for any A ∈ ℂ, which helps to detect more properties of this object. The cases of groups, Lie, Leibniz, associative, commutative associative, alternative algebras, crossed and precrossed modules are considered. The examples of algebras are given, for which always exist actors.

Lawvere Completion and Separation Via Closure

Abstract: For a quantale \(\mathsf{V}\), first a closure-theoretic approach to completeness and separation in \(\mathsf{V}\text{-categories}\) is presented. This approach is then generalized to Open image in new window, where Open image in new window is a topological theory that entails a set monad \( \mathbb{T}\) and a compatible \( \mathbb{T}\text{-algebra}\) structure on \(\mathsf{V}\).

The Universal Covering of an Inverse Semigroup

Abstract: We examine an inverse semigroup T in terms of the universal locally constant covering of its classifying topos Open image in new window. In particular, we prove that the fundamental group of Open image in new window coincides with the maximum group image of T. We explain the connection between E-unitary inverse semigroups and locally decidable toposes, characterize E-unitary inverse semigroups in terms of a kind of geometric morphism called a spread, characterize F-inverse semigroups, and interpret McAlister’s “P-theorem” in terms of the universal covering.

Notes on Pointfree Disconnectivity with a Ring-theoretic Slant

Abstract: We give characterizations of extremally disconnected frames, basically disconnected frames and F-frames L in terms of ring-theoretic properties of the ring \(\mathcal{R}L\) of continuous real-valued functions on L. Emanating from these are new (and purely ring-theoretic) proofs that a frame is extremally disconnected, basically disconnected or an F-frame iff the same holds for its Cech-Stone compactification.

Crossed Products in Weak Contexts

Abstract: We define the general notion of crossed products in a weak context, which generalizes the ones defined by Blattner, Cohen and Montgomery, Doi and Takeuchi in the context of Hopf algebras and the one given by Brzezinski. Also, the crossed products obtained by the authors, for weak Hopf algebras living in a symmetric monoidal category and weak C-cleft extensions associated to weak entwined structures, are particular instances of this theory.

Note on the Construction of Free Monoids

Abstract: We construct free monoids in a monoidal category Open image in new window with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains. In particular this will be the case if tensoring preserves sifted colimits.

Sheaves as Modules

Abstract: We revisit sheaves on locales by placing them in the context of the theory of quantale modules. The local homeomorphisms p:X→B are identified with the Hilbert B-modules that are equipped with a natural notion of basis. The homomorphisms of these modules are necessarily adjointable, and the resulting self-dual category yields a description of the equivalence between local homeomorphisms and sheaves whereby morphisms of sheaves arise as the “operator adjoints” of the inverse images of the maps of local homeomorphisms.

Constructing New Braided T-Categories over Weak Hopf Algebras

Abstract: Let Aut weak Hopf (H) denote the set of all automorphisms of a weak Hopf algebra H with bijective antipode in the sense of Bohm et al. (J Algebra 221:385–438, 1999) and let G be a certain crossed product group Aut weak Hopf (H)×Aut weak Hopf (H). The main purpose of this paper is to provide further examples of braided T-categories in the sense of Turaev (1994, 2008). For this, we first introduce a class of new categories $ _{H}{\mathcal {WYD}}^{H}(\alpha, \beta)$ of weak (α, β)-Yetter-Drinfeld modules with α, β ∈ Aut weak Hopf (H) and we show that the category ${\mathcal WYD}(H) =\{{}_{H}\mathcal {WYD}^{H}(\alpha, \beta)\}_{(\alpha , \beta )\in G}$ becomes a braided T-category over G, generalizing the main constructions by Panaite and Staic (Isr J Math 158:349–365, 2007). Finally, when H is finite-dimensional we construct a quasitriangular weak T-coalgebra WD(H) = {WD(H)(α, β)}(α, β) ∈ G in the sense of Van Daele and Wang (Comm Algebra, 2008) over a family of weak smash product algebras $\{\overline{H^{*cop}\# H_{(\alpha,\beta)}}\}_{(\alpha , \beta)\in G}$ , and we obtain that ${\mathcal {WYD}}(H)$ is isomorphic to the representation category of the quasitriangular weak T-coalgebra WD(H).

Graded and Koszul Categories

Abstract: Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there is always a naturally associated Koszul theory. To obtain this, the notions of Koszul algebras, linear modules and Koszul duality are extended to additive (graded) categories over a field. The main focus of this paper is to provide these generalizations and the necessary preliminaries.

Essentiality and Injectivity

Abstract: Essentiality is an important notion closely related to injectivity. Depending on a class M of morphisms of a category A, three different types of essentiality are considered in literature. Each has its own benefits in regards with the behaviour of M-injectivity. In this paper we intend to study these different notions of essentiality and to investigate their relations to injectivity and among themselves. We will see, among other things, that although these essential extensions are not necessarily equivalent, they behave almost equivalently with regard to injectivity.

A Representation Theorem for Geometric Morphisms

Abstract: It it shown that geometric morphisms between elementary toposes can be represented as certain adjunctions between the corresponding categories of locales These adjunctions are characterized by (i) they preserve the order enrichment and the Sierpinski locale, and (ii) they satisfy Frobenius reciprocity

Homotopy Gerstenhaber structures and vertex algebras

Abstract: We provide a simple construction of a G ∞ -algebra structure on an important class of vertex algebras V, which lifts the Gerstenhaber algebra structure on BRST cohomology of V introduced by Lian and Zuckerman. We outline two applications to algebraic topology: the construction of a sheaf of G ∞ algebras on a Calabi–Yau manifold M, extending the operations of multiplication and bracket of functions and vector fields on M, and of a Lie ∞ structure related to the bracket of Courant (Trans Amer Math Soc 319:631–661, 1990).

Bisets as Categories and Tensor Product of Induced Bimodules

Abstract: Bisets can be considered as categories. This note uses this point of view to give a simple proof of a Mackey-like formula expressing the tensor product of two induced bimodules.

TTF Triples in Functor Categories

Abstract: We characterize the hereditary torsion pairs of finite type in the functor category of a ring R that are associated to tilting torsion pairs in the category of R-modules. Moreover, we determine a condition under which they give rise to TTF triples.

Exact Sequences and Closed Model Categories

Abstract: For every closed model category with zero object, Quillen gave the construction of Eckman-Hilton and Puppe sequences. In this paper, we remove the hypothesis of the existence of zero object and construct (using the category over the initial object or the category under the final object) these sequences for unpointed model categories. We illustrate the power of this result in abstract homotopy theory given some interesting applications to group cohomology and exterior homotopy groups.

Sequentially Dense Essential Monomorphisms of Acts Over Semigroups

Abstract: The class Md of sequentially dense monomorphisms were first defined and studied by Giuli, Ebrahimi, and Mahmoudi for projection algebras (acts over the monoid (N ∞ , min), of interest to computer scientists, as studied by Herrlich, Ehrig, and some others) and generalized to acts over arbitrary semigroups. Md- injectivity has been shown by some of the above authors to be also useful in the study of ordinary injectivity of acts. Essentiality is an important notion closely related to injectivity. In this paper, we study essentiality with respect to sequentially dense monomorphisms of acts. We will show that the three different definitions of essentiality usually used in literature with respect to a subclass of monomorphisms are equivalent for the class of sequentially dense monomorphisms and, among other things, give some criterion to characterize and describe them explicitly. Also, we show the existence and the explicit description of a maximal such essential extension for any given act.

Automorphisms and Homotopies of Groupoids and Crossed Modules

Abstract: This paper is concerned with the algebraic structure of groupoids and crossed modules of groupoids. We describe the group structure of the automorphism group of a finite connected groupoid C as a quotient of a semidirect product. We pay particular attention to the conjugation automorphisms of C, and use these to define a new notion of groupoid action. We then show that the automorphism group of a crossed module of groupoids \(\mathcal{C}\), in the case when the range groupoid is connected and the source group totally disconnected, may be determined from that of the crossed module of groups \(\mathcal{C}_u\) formed by restricting to a single object u. Finally, we show that the group of homotopies of \(\mathcal{C}\) may be determined once the group of regular derivations of \(\mathcal{C}_u\) is known.

The (Tetra) Category of Pseudocategories in an Additive 2-category with Kernels

Abstract: We describe the (tetra) category of pseudo-categories, pseudo-functors, natural transformations, pseudo-natural transformations, and modifications, as introduced in Martins-Ferreira (JHRS 1:47–78, 2006), internal to an additive 2-category with kernels, as formalized in Martins-Ferreira (Fields Inst Commun 43:387–410, 2004). In the context of a 2-Ab-category, we introduce the notion of a pseudo-morphism and prove the equivalence of categories: PsCat(A)~PsMor(A) between pseudo-categories and pseudo-morphisms in an additive 2-category, A, with kernels– extending thus the well known equivalence Cat(Ab)~Mor(Ab) between internal categories and morphisms of abelian groups. The leading example of an additive 2-category with kernels is Cat(Ab). In the case A=Cat(Ab) we obtain a description of the (tetra) category of internal pseudo-double categories in Ab, and particularize it to a description of the (tetra) category of internal bicategories in abelian groups. As expected, pseudo-natural transformations coincide with homotopies of 2-chain complexes (as in Bourn, J Pure Appl Algebra 66:229–249, 1990).

Girard couples of quantales

Abstract: We introduce the concept of a Girard couple, which consists of two (not necessarily unital) quantales linked by a strong form of duality. The two basic examples of Girard couples arise in the study of endomorphism quantales and of the spectra of operator algebras. We construct, for an arbitrary sup-lattice S, a Girard quantale whose right-sided part is isomorphic to S.

Well-behaved Epireflections for Kan Extensions

Abstract: Let \(K:\mathbb{B}\rightarrow \mathbb{A}\) be a functor such that the image of the objects in \(\mathbb{B}\) is a cogenerating set of objects for \(\mathbb{A}\). Then, the right Kan extensions adjunction \(\mathbf{Set}^K\dashv Ran_K\) induces necessarily an epireflection with stable units and a monotone-light factorization. This result follows from the one stating that an adjunction produces an epireflection in a canonical way, provided there exists a prefactorization system which factorizes all of its unit morphisms through epimorphisms. The stable units property, for the so obtained epireflections, is thereafter equivalently restated in such a manner it is easily recognizable in the examples. Furthermore, having stable units, there is a strong but quite simple sufficient condition for the existence of an associated monotone-light factorization, which has proved to be fruitful.

Fibrations of Simplicial Sets

Abstract: There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen’s axioms for a homotopy model category. The combinatorics underlying these fibrations is purely finitary and seems interesting both for its own sake and for its interaction with homotopy types. To show that these notions of fibration are indeed distinct, one needs to understand how iterates of Kan’s Ex functor act on graphs and on nerves of small categories.

Quasi-Commutative Algebras

Abstract: We characterise algebras commutative with respect to a Yang-Baxter operator (quasi-commutative algebras) in terms of certain cosimplicial complexes. In some cases this characterisation allows the classification of all possible quasi-commutative structures.

On Proper and Exterior Sequentiality

Abstract: In this article a sequential theory in the category of spaces and proper maps is described and developed. As a natural extension a sequential theory for exterior spaces and maps is obtained.

Läuchli’s Completeness Theorem from a Topos-Theoretic Perspective

Abstract: We prove a variant of Lauchli’s completeness theorem for intuitionistic predicate calculus. The formulation of the result relies on the observation (due to Lawvere) that Lauchli’s theorem is related to the logic of the canonical indexing of the atomic topos of \(\mathbb{Z}{\text{ - sets}}\). We show that the process that transforms Kripke-counter-models into Lauchli-counter-models is (essentially) the inverse image of a geometric morphism. Completeness follows because this geometric morphism is an open surjection.

The General Theory of Diads

Abstract: A diad is a generalisation of a monad and a comonad. The idea is that we ignore the unit or counit, and consider only the natural transformations between T and T2. It turns out that almost all the constructions that we form for a monad or comonad can also be constructed from a related diad. Diads were introduced in Kenney (Appl. Categ. Structures, 2008), where they give a generalisation of the results that the category of coalgebras for a finite-limit preserving comonad on a topos is another topos, and that the category of algebras for a finite-limit preserving idempotent monad on a topos is another topos. In that paper, we were only interested in a special class of diads called codistributive diads, and we considered only the part of the theory of diads necessary to prove the result about finite-limit preserving diads in topoi. Here, we will study general diads in greater detail. We will develop the general theory with constructions that extend the standard constructions for monads and comonads.

2-Filteredness and The Point of Every Galois Topos

Abstract: A connected locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bi-limits of topoi, we show that every Galois topos has a point.

Constructing Dynamical Twists Over a Non-abelian Base

Abstract: We give examples of dynamical twists in finite-dimensional Hopf algebras over an arbitrary Hopf subalgebra. The construction is based on the categorical approach of dynamical twists introduced by Donin and Mudrov (Commun Math Phys 254:719–760, 2005).