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  • Exact and Strongly Exact Filters
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-07-25
    M. A. Moshier, A. Pultr, A. L. Suarez

    A meet in a frame is exact if it join-distributes with every element, it is strongly exact if it is preserved by every frame homomorphism. Hence, finite meets are (strongly) exact which leads to the concept of an exact resp. strongly exact filter, a filter closed under exact resp. strongly exact meets. It is known that the exact filters constitute a frame \({\mathrm{Filt}}_{{\textsf {E}}}(L)\) somewhat

    更新日期:2020-07-25
  • A Categorical Construction for the Computational Definition of Vector Spaces
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-06-23
    Alejandro Díaz-Caro, Octavio Malherbe

    Lambda-\({\mathcal {S}}\) is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi. One is to forbid duplication of variables, while the other is to consider all lambda-terms as algebraic linear functions. The type system of Lambda-\({\mathcal {S}}\) has a constructor S such that a type A is considered as the base of a vector space while S(A) is

    更新日期:2020-06-24
  • The DG-Category of Secondary Cohomology Operations
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-06-23
    Hans-Joachim Baues, Martin Frankland

    We study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up to suitably coherent correction tracks, then it is weakly equivalent to a 1-truncated DG-category. This generalizes work of the first author on the strictification

    更新日期:2020-06-24
  • A Categorical Duality for Semilattices and Lattices
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-06-11
    Sergio A. Celani, Luciano J. González

    The main aim of this article is to develop a categorical duality between the category of semilattices with homomorphisms and a category of certain topological spaces with certain morphisms. The principal tool to achieve this goal is the notion of irreducible filter. Then, we apply this dual equivalence to obtain a topological duality for the category of bounded lattices and lattice homomorphism. We

    更新日期:2020-06-11
  • The Order-Sobrification Monad
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-06-10
    Xiaodong Jia

    We investigate the so-called order-sobrification monad proposed by Ho et al. (Log Methods Comput Sci 14:1–19, 2018) for solving the Ho–Zhao problem, and show that this monad is commutative. We also show that the Eilenberg–Moore algebras of the order-sobrification monad over dcpo’s are precisely the strongly complete dcpo’s and the algebra homomorphisms are those Scott-continuous functions preserving

    更新日期:2020-06-10
  • Hausdorff Coalgebras
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-04-30
    Dirk Hofmann, Pedro Nora

    As composites of constant, finite (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of \(\textsf {Set}\)-functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial functors to the context of quantale-enriched categories. To assume the role of the powerset functor we

    更新日期:2020-04-30
  • Exact Filters and Joins of Closed Sublocales
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-03-07
    R. N. Ball, M. A. Moshier, A. Pultr

    We prove, for a general frame, that the sublocales that can be represented as joins of closed ones are, somewhat surprisingly, in a natural one-to-one correspondence with the filters closed under exact meets, and explain some subfit facts from this perspective. Furthermore we discuss the filters associated in a similar vein with the fitted sublocales.

    更新日期:2020-03-07
  • A Topological Groupoid Representing the Topos of Presheaves on a Monoid
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-03-06
    Jens Hemelaer

    Butz and Moerdijk famously showed that every (Grothendieck) topos with enough points is equivalent to the category of sheaves on some topological groupoid. We give an alternative, more algebraic construction in the special case of a topos of presheaves on an arbitrary monoid. If the monoid is embeddable in a group, the resulting topological groupoid is the action groupoid for a discrete group acting

    更新日期:2020-03-06
  • Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-03-06
    Emily Riehl, Dominic Verity

    In this paper we prove that various quasi-categories whose objects are \(\infty \)-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which we characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories

    更新日期:2020-03-06
  • Universal Central Extensions of Internal Crossed Modules via the Non-abelian Tensor Product
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-03-02
    Davide di Micco, Tim Van der Linden

    In the context of internal crossed modules over a fixed base object in a given semi-abelian category, we use the non-abelian tensor product in order to prove that an object is perfect (in an appropriate sense) if and only if it admits a universal central extension. This extends results of Brown and Loday (Topology 26(3):311–335, 1987, in the case of groups) and Edalatzadeh (Appl Categ Struct 27(2):111–123

    更新日期:2020-03-02
  • Crossed Modules of Monoids II: Relative Crossed Modules
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-02-27
    Gabriella Böhm

    This is the second part of a series of three strongly related papers in which three equivalent structures are studied: Internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans. Crossed modules of monoids relative to this class of spans. Simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of

    更新日期:2020-02-27
  • Word operads and admissible orderings
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-01-18
    Vladimir Dotsenko

    We use Giraudo’s construction of combinatorial operads from monoids to offer a conceptual explanation of the origins of Hoffbeck’s path sequences of shuffle trees, and use it to define new monomial orders of shuffle trees. One such order is utilised to exhibit a quadratic Gröbner basis of the Poisson operad.

    更新日期:2020-01-18
  • Compactly Generated Spaces and Quasi-spaces in Topology
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-01-18
    Willian Ribeiro

    The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category \(\textsf {Top}\) of topological spaces and continuous functions, to study compactly generated spaces and quasi-spaces in this setting. Moreover, for a class \(\mathcal {C}\) of objects we generalize the notion of \(\mathcal {C}\)-generated spaces, from which we derive

    更新日期:2020-01-18
  • Abelian Categories Arising from Cluster Tilting Subcategories
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2020-01-03
    Yu Liu, Panyue Zhou

    For a triangulated category \({\mathcal {T}}\), if \({\mathcal {C}}\) is a cluster-tilting subcategory of \({\mathcal {T}}\), then the factor category \({\mathcal {T}}{/}{\mathcal {C}}\) is an abelian category. Under certain conditions, the converse also holds. This is a very important result of cluster-tilting theory, due to Koenig–Zhu and Beligiannis. Now let \({\mathcal {B}}\) be a suitable extriangulated

    更新日期:2020-01-03
  • The Gray Monoidal Product of Double Categories
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-12-18
    Gabriella Böhm

    The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category \({\mathbb {A}}\), the corresponding internal hom functor sends a double category \({\mathbb {B}}\) to the double category whose 0-cells are the double functors \({\mathbb {A}} \rightarrow {\mathbb {B}}\), whose horizontal and vertical 1-cells are the horizontal and

    更新日期:2019-12-18
  • A Combinatorial-Topological Shape Category for Polygraphs
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-11-30
    Amar Hadzihasanovic

    We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common classes of higher-categorical cell shapes, including globes, cubes, oriented simplices, and a large sub-class of opetopes, and are closed under lax Gray products

    更新日期:2019-11-30
  • Intrinsic Schreier Split Extensions
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-11-27
    Andrea Montoli, Diana Rodelo, Tim Van der Linden

    In the context of regular unital categories we introduce an intrinsic version of the notion of a Schreier split epimorphism, originally considered for monoids. We show that such split epimorphisms satisfy the same homological properties as Schreier split epimorphisms of monoids do. This gives rise to new examples of \({\mathcal {S}}\)-protomodular categories, and allows us to better understand the

    更新日期:2019-11-27
  • Functors and Morphisms Determined by Subcategories
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-10-30
    Shijie Zhu

    We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a \({{\,\mathrm{Hom}\,}}\)-finite hereditary abelian category with enough projectives, we prove that the Auslander–Reiten–Smalø–Ringel formula of the minimal right determiner still holds. As an application, we give a formula of minimal right determiners in the category of finitely presented representations

    更新日期:2019-10-30
  • Pro-compactly Finite MV-Algebras
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-10-17
    Maurice Kianpi, Jean B. Nganou

    We introduce compactly finite MV-algebras and continuous MV-algebras. We also investigate pro-compactly finite MV-algebras, which are the MV-algebras that are inverse limits of systems of compactly finite MV-algebras. We obtain that continuous MV-algebras as well as pro-compactly finite MV-algebras coincide with compact Hausdorff MV-algebras. In addition, further categorical properties of compact Hausdorff

    更新日期:2019-10-17
  • The Vietoris Monad and Weak Distributive Laws
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-10-16
    Richard Garner

    The Vietoris monad on the category of compact Hausdorff spaces is a topological analogue of the power-set monad on the category of sets. Exploiting Manes’ characterisation of the compact Hausdorff spaces as algebras for the ultrafilter monad on sets, we give precise form to the above analogy by exhibiting the Vietoris monad as induced by a weak distributive law, in the sense of Böhm, of the power-set

    更新日期:2019-10-16
  • A Unified Classification Theorem for Mal’tsev-Like Categories
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-10-15
    Nelson Martins-Ferreira

    In this paper we give unified characterizations of categories defined by variations of the Mal’tsev property.

    更新日期:2019-10-15
  • The Universal Property of Infinite Direct Sums in $$\hbox {C}^*$$C∗ -Categories and $$\hbox {W}^*$$W∗ -Categories
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-10-08
    Tobias Fritz, Bas Westerbaan

    When formulating universal properties for objects in a dagger category, one usually expects a universal property to characterize the universal object up to unique unitary isomorphism. We observe that this is automatically the case in the important special case of \(\hbox {C}^*\)-categories, provided that one uses enrichment in Banach spaces. We then formulate such a universal property for infinite

    更新日期:2019-10-08
  • On Integral Structure Types
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-09-12
    James Fullwood

    We introduce integral structure types as a categorical analogue of virtual combinatorial species. Integral structure types then categorify power series with possibly negative coefficients in the same way that combinatorial species categorify power series with non-negative rational coefficients. The notion of an operator on combinatorial species naturally extends to integral structure types, and in

    更新日期:2019-09-12
  • Algebraic Theories and Commutativity in a Sheaf Topos
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-09-09
    Boaz Haberman

    For any site of definition \(\mathcal {C}\) of a Grothendieck topos \(\mathcal {E}\), we define a notion of a \(\mathcal {C}\)-ary Lawvere theory \(\tau : \mathscr {C} \rightarrow \mathscr {T}\) whose category of models is a stack over \(\mathcal {E}\). Our definitions coincide with Lawvere’s finitary theories when \(\mathcal {C}=\aleph _0\) and \(\mathcal {E} = {{\,\mathrm{\mathbf {Set}}\,}}\). We

    更新日期:2019-09-09
  • Further Results on the Structure of (Co)Ends in Finite Tensor Categories
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-09-06
    Kenichi Shimizu

    Let \({\mathcal {C}}\) be a finite tensor category, and let \({\mathcal {M}}\) be an exact left \({\mathcal {C}}\)-module category. The action of \({\mathcal {C}}\) on \({\mathcal {M}}\) induces a functor \(\rho : {\mathcal {C}} \rightarrow \mathrm {Rex}({\mathcal {M}})\), where \(\mathrm {Rex}({\mathcal {M}})\) is the category of k-linear right exact endofunctors on \({\mathcal {M}}\). Our key observation

    更新日期:2019-09-06
  • Extensions of Filtered Ogus Structures
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-08-28
    Bruno Chiarellotto, Nicola Mazzari

    We compute the Ext group of the (filtered) Ogus category over a number field K. In particular we prove that the filtered Ogus realisation of mixed motives is not fully faithful.

    更新日期:2019-08-28
  • Local Presentability of Certain Comma Categories
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-08-10
    Andrew Polonsky, Patricia Johann

    It follows from standard results that if \(\mathcal {A}\) and \(\mathcal {C}\) are locally \(\lambda \)-presentable categories and \(F : \mathcal {A}\rightarrow \mathcal {C}\) is a \(\lambda \)-accessible functor, then the comma category \(\mathsf {Id}_\mathcal {C}{\downarrow }{}F\) is locally \(\lambda \)-presentable. We show that, under the same hypotheses, \(F{\downarrow }{}\mathsf {Id}_\mathcal

    更新日期:2019-08-10
  • Differential Categories Revisited
    Appl. Categor. Struct. (IF 0.552) Pub Date : 2019-07-04
    R. F. Blute, J. R. B. Cockett, J.-S. P. Lemay, R. A. G. Seely

    Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were introduced. The basic approach used a deriving transformation, while a more refined approach, in the presence of a bialgebra modality, used a codereliction. The latter

    更新日期:2019-07-04
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