Abstract: 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. PubDate: 2020-02-01

Abstract: 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, \(F{\downarrow }{}\mathsf {Id}_\mathcal {C}\) is also locally \(\lambda \)-presentable. PubDate: 2020-02-01

Abstract: 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 light of their ‘negativity’ we define the notion of the commutator of two operators on integral structure types. We then extend integral structure types to the setting of stuff types as introduced by Baez and Dolan, and then conclude by using integral structure types to give a combinatorial description for Chern classes of projective hypersurfaces. PubDate: 2020-02-01

Abstract: In this paper, we introduce a notion of categorified cyclic operad for set-based cyclic operads with symmetries. Our categorification is obtained by relaxing defining axioms of cyclic operads to isomorphisms and by formulating coherence conditions for these isomorphisms. The coherence theorem that we prove has the form “all diagrams of canonical isomorphisms commute”. Our coherence results come in two flavours, corresponding to the “entries-only” and “exchangeable-output” definitions of cyclic operads. Our proof of coherence in the entries-only style is of syntactic nature and relies on the coherence of categorified non-symmetric operads established by Došen and Petrić. We obtain the coherence in the exchangeable-output style by “lifting” the equivalence between entries-only and exchangeable-output cyclic operads, set up by the second author. Finally, we show that a generalization of the structure of profunctors of Bénabou provides an example of categorified cyclic operad, and we exploit the coherence of categorified cyclic operads in proving that the Feynman category for cyclic operads, due to Kaufmann and Ward, admits an odd version. PubDate: 2020-02-01

Abstract: In universal algebra, it is well known that varieties admitting a majority term admit several Mal’tsev-type characterizations. The main aim of this paper is to establish categorical counterparts of some of these characterizations for regular categories. We prove a categorical version of Bergman’s Double-projection Theorem: a regular category is a majority category if and only if every subobject S of a finite product \(A_1 \times A_2 \times \cdots \times A_n\) is uniquely determined by its twofold projections. We also establish a categorical counterpart of the Pairwise Chinese Remainder Theorem for algebras, and characterize regular majority categories by the classical congruence equation \(\alpha \cap (\beta \circ \gamma ) = (\alpha \cap \beta ) \circ (\alpha \cap \gamma )\) due to A.F. Pixley. PubDate: 2020-02-01

Abstract: A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as ‘machines’ with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special cases include continuous and discrete dynamical systems (e.g. Moore machines). Additionally, morphisms between the different types of systems allow their translation in a common framework. A central goal is to understand the systems that result from arbitrary interconnection of component subsystems, possibly of different types, as well as establish conditions that ensure totality and determinism compositionally. The fundamental categorical tools used here include lax monoidal functors, which provide a language of compositionality, as well as sheaf theory, which flexibly captures the crucial notion of time. PubDate: 2020-02-01

Abstract: 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. PubDate: 2020-01-18

Abstract: 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, for instance, a general concept of Alexandroff spaces. Furthermore, as done for \(\textsf {Top}\), we also study, in our level of generality, the relationship between compactly generated spaces and quasi-spaces. PubDate: 2020-01-18

Abstract: 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 category, which is a simultaneous generalization of triangulated categories and exact categories. We introduce the notion of pre-cluster tilting subcategory \({\mathcal {C}}\) of \({\mathcal {B}}\), which is a generalization of cluster tilting subcategory. We show that \({\mathcal {C}}\) is cluster tilting if and only if the factor category \({\mathcal {B}}{/}{\mathcal {C}}\) is abelian. Our result generalizes the related results on a triangulated category and is new for an exact category case. PubDate: 2020-01-03

Abstract: 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 vertical pseudo transformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure. PubDate: 2019-12-18

Abstract: This paper concerns the problem of lifting a KZ doctrine P to the 2-category of pseudo T-algebras for some pseudomonad T. Here we show that this problem is equivalent to giving a pseudo-distributive law (meaning that the lifted pseudomonad is automatically KZ), and that such distributive laws may be simply described algebraically and are essentially unique [as known to be the case in the (co)KZ over KZ setting]. Moreover, we give a simple description of these distributive laws using Bunge and Funk’s notion of admissible morphisms for a KZ doctrine (the principal goal of this paper). We then go on to show that the 2-category of KZ doctrines on a 2-category is biequivalent to a poset. We will also discuss here the problem of lifting a locally fully faithful KZ doctrine, which we noted earlier enjoys most of the axioms of a Yoneda structure, and show that a bijection between oplax and lax structures is exhibited on the lifted “Yoneda structure” similar to Kelly’s doctrinal adjunction. We also briefly discuss how this bijection may be viewed as a coherence result for oplax functors out of the bicategories of spans and polynomials, but leave the details for a future paper. PubDate: 2019-12-01

Abstract: We give conditions on an inclusion \({\mathbb {C}}\hookrightarrow {\mathbb {D}}\) where \({\mathbb {C}}\) is a Mal’tsev (resp. protomodular) subcategory in order to produce on \({\mathbb {D}}\) a partial \(\Sigma \) -Mal’tsev (resp. \(\Sigma \) -protomodular) structure. PubDate: 2019-12-01

Abstract: 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 and joins. We define constructible polygraphs to be presheaves on a category of atoms and inclusions, and extend the monoidal structures. We show that constructible directed complexes are a well-behaved subclass of Steiner’s directed complexes, which we use to define a realisation functor from constructible polygraphs to \(\omega \)-categories. We prove that the realisation of a constructible polygraph is a polygraph in restricted cases, and in all cases conditionally to a conjecture. Finally, we define the geometric realisation of a constructible polygraph, and prove that it is a CW complex with one cell for each of its elements. PubDate: 2019-11-30

Abstract: 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 homological behaviour of monoids from a categorical perspective. PubDate: 2019-11-27

Abstract: 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 of strongly locally finite quivers. PubDate: 2019-10-30

Abstract: 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 MV-algebras such as co-completeness, injective objects, (co)-Hopfian objects are considered and studied. PubDate: 2019-10-17

Abstract: We define and discuss the notions of additivity and idempotency for neighbourhood and interior operators. We then propose an order-theoretic description of the notion of convergence that was introduced by D. Holgate and J. Šlapal with the help of these two properties. This will provide a rather convenient setting in which compactness and completeness can be studied via neighbourhood operators. We prove, among other things, a Frolík-type theorem with the introduction of reflecting morphisms. PubDate: 2019-10-11

Abstract: For a subfit frame L, let \({\mathcal {S}}_{\mathfrak {c}}(L)\) denote the complete Boolean algebra whose elements are the sublocales of L that are joins of closed sublocales. Identifying every element of L with the open sublocale it determines allows us to view L as a subframe of \({\mathcal {S}}_{\mathfrak {c}}(L)\) . With this backdrop, we say L is maximal Lindelöf if it is Lindelöf and whenever \(L\subseteq M\) , for some Lindelöf subframe M of \({\mathcal {S}}_{\mathfrak {c}}(L)\) , then \(L=M\) . Recall that a topological space \((X,\tau )\) is maximal Lindelöf if it is Lindelöf, and there is no strictly finer topology \(\rho \) on X such that \((X,\rho )\) is a Lindelöf space. We show that a space is maximal Lindelöf if and only if the frame of its open subsets is maximal Lindelöf. We then characterize maximal Lindelöf frames internally. Among regular frames, we show that the maximal Lindelöf frames are precisely the Lindelöf ones in which every \(F_\sigma \) -sublocale (meaning a join of countably many closed sublocales) is closed. PubDate: 2019-08-20

Abstract: We consider an alternate form of the equivalence between the category of compact Hausdorff spaces and continuous functions and a category formed from Gleason spaces and certain relations. This equivalence arises from the study of the projective cover of a compact Hausdorff space. This line leads us to consider the category of compact Hausdorff spaces with closed relations, and the corresponding subcategories with continuous and interior relations. Various equivalences of these categories are given extending known equivalences of the category of compact Hausdorff spaces and continuous functions with compact regular frames, de Vries algebras, and also with a category of Gleason spaces that we introduce. Study of categories of compact Hausdorff spaces with relations is of interest as a general setting to consider Gleason spaces, for connections to modal logic, as well as for the intrinsic interest in these categories. PubDate: 2019-08-13

Abstract: This is the first 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 monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this first part the theory of relative pullbacks is worked out leading to the definition of a relative category. PubDate: 2019-08-10