Abstract: In this paper, we introduce the notions of stable future, past and total component systems on a directed space with no loops. Then, we associate the stable component category to a stable (future, past or total) component system. Stable component categories are enriched in some monoidal category, eg. the homotopy category of spaces, and carry information about the spaces of directed paths between particular points. It is shown that the geometric realizations of finite pre-cubical sets with no loops admit unique minimal stable (future/past/total) component systems. These constructions provide a new family of invariants for directed spaces. PubDate: 2019-06-01

Abstract: We review the theory of Co-Gorenstein algebras, which was introduced in Beligiannis (Commun Algebra 28(10):4547–4596, 2000). We show a connection between Co-Gorenstein algebras and the Nakayama and Generalized Nakayama conjecture. PubDate: 2019-06-01

Abstract: We extend the homotopy theories based on point reduction for finite spaces and simplicial complexes to finite acyclic categories and \(\varDelta \) -complexes, respectively. The functors of classifying spaces and face posets are compatible with these homotopy theories. In contrast with the classical settings of finite spaces and simplicial complexes, the universality of morphisms and simplices plays a central role in this paper. PubDate: 2019-06-01

Abstract: Just as the presheaf category is the free cocompletion of any small category, there is an analogous notion of free cocompletion for any small restriction category. In this paper, we extend the work on restriction presheaves to presheaves over join restriction categories, and show that the join restriction category of join restriction presheaves is equivalent to some partial map category of sheaves. We then use this to show that the Yoneda embedding exhibits the category of join restriction presheaves as the free cocompletion of any small join restriction category. PubDate: 2019-06-01

Abstract: It is shown that every d-frame admits a complete lattice of quotients. Quotienting may be triggered by a binary relation on one of the two constituent frames, or by changes to the consistency or totality structure, but as these are linked by the reasonableness conditions of d-frames, the result in general will be that both frames are factored and both consistency and totality are increased. PubDate: 2019-06-01

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: 2019-05-24

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: 2019-05-17

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: 2019-04-05

Abstract: We study the category of Reedy diagrams in a \(\mathscr {V}\) -model category. Explicitly, we show that if K is a small category, \(\mathscr {V}\) is a closed symmetric monoidal category and \(\mathscr {C}\) is a closed \(\mathscr {V}\) -module, then the diagram category \(\mathscr {V}^K\) is a closed symmetric monoidal category and the diagram category \(\mathscr {C}^K\) is a closed \(\mathscr {V}^K\) -module. We then prove that if further K is a Reedy category, \(\mathscr {V}\) is a monoidal model category and \(\mathscr {C}\) is a \(\mathscr {V}\) -model category, then with the Reedy model category structures, \(\mathscr {V}^K\) is a monoidal model category and \(\mathscr {C}^K\) is a \(\mathscr {V}^K\) -model category provided that either the unit 1 of \(\mathscr {V}\) is cofibrant or \(\mathscr {V}\) is cofibrantly generated. PubDate: 2019-04-04

Abstract: We show that in an ideal-determined unital category the Higgins commutator can be characterized as the largest binary operation C on subobjects (defined on all subobjects of each object) satisfying the following conditions: (a) C is order-preserving; (b) C(H, K) is always less or equal to the meet of normal closures of H and K; (c) \(C(f(H),f(K))=f(C(H,K))\) for every pair of subobjects H and K of an object X, and every morphism f whose domain is X. PubDate: 2019-04-01

Abstract: Given the pair of a dualizing k-variety and its functorially finite subcategory, we show that there exists a recollement consisting of their functor categories of finitely presented objects. We provide several applications for Auslander’s formulas: the first one realizes a module category as a Serre quotient of a suitable functor category. The second one shows a close connection between Auslander–Bridger sequences and recollements. The third one gives a new proof of the higher defect formula which includes the higher Auslander–Reiten duality as a special case. PubDate: 2019-04-01

Abstract: It is proved that the category of extended multisets is dually equivalent to the category of compact Hausdorff MV-algebras with continuous homomorphisms, which is in turn equivalent to the category of complete and completely distributive MV-algebras with homomorphisms that reflect principal maximal ideals. Urysohn–Strauss’s Lemma, Gleason’s Theorem, and projective objects are also investigated for topological MV-algebras. Among other things, it is proved that the only MV-algebras in which Urysohn–Strauss’s Lemma holds are Boolean algebras and that the projective objects in the category of compact Hausdorff MV-algebras are precisely the ones having the 2-element Boolean algebras as factor. PubDate: 2019-04-01

Abstract: Let L be a Lie algebra over a field of arbitrary characteristic. In this paper, we give a necessary and sufficient condition for the existence of universal central extensions in the category of crossed modules of Lie algebras over L. Also, we determine the structure of the universal central extension of a crossed L-module and show that the kernel of this extension is related to the first non-abelian homology of L. PubDate: 2019-04-01

Abstract: Around the year 1988, Joyal and Street established a graphical calculus for monoidal categories, which provides a firm foundation for many explorations of graphical notations in mathematics and physics. For a deeper understanding of their work, we consider a similar graphical calculus for semi-groupal categories. We introduce two frameworks to formalize this graphical calculus, a topological one based on the notion of a processive plane graph and a combinatorial one based on the notion of a planarly ordered processive graph, which serves as a combinatorial counterpart of a deformation class of processive plane graphs. We demonstrate the equivalence of Joyal and Street’s graphical calculus and the theory of upward planar drawings. We introduce the category of semi-tensor schemes, and give a construction of a free monoidal category on a semi-tensor scheme. We deduce the unit convention as a kind of quotient construction, and show an idea to generalize the unit convention. Finally, we clarify the relation of the unit convention and Joyal and Street’s construction of a free monoidal category on a tensor scheme. PubDate: 2019-04-01

Abstract: The paper is devoted to homology groups of cubical sets with coefficients in contravariant systems of abelian groups. The study is based on the proof of the assertion that the homology groups of the category of cubes with coefficients in the diagram of abelian groups are isomorphic to the homology groups of the normalized complex of the cubical abelian group corresponding to this diagram. The main result shows that the homology groups of a cubical set with coefficients in a contravariant system of abelian groups are isomorphic to the values of left derived functors of the colimit functor on this contravariant system. This is used to obtain the isomorphism criterion for homology groups of cubical sets with coefficients in contravariant systems, and also to construct spectral sequences for the covering of a cubical set and for a morphism between cubical sets. The use of the main result leads to an isomorphism between homology groups of small category with coefficients in the diagram of abelian groups and normalized homology groups of its cubical nerve with coefficients in the corresponding contravariant system. PubDate: 2019-04-01

Abstract: We classify certain subcategories in quotients of exact categories. In particular, we classify the triangulated and thick subcategories of an algebraic triangulated category, i.e. the stable category of a Frobenius category. PubDate: 2019-03-11

Abstract: The present note has three aims. First, to complement the theory of cofibrant generation of algebraic weak factorisation systems (awfss) to cover some important examples that are not locally presentable categories. Secondly, to prove that cofibrantly kz-generated awfss (a notion we define) are always lax orthogonal. Thirdly, to show that the two known methods of building lax orthogonal awfss, namely cofibrantly kz-generation and the method of “simple adjunctions”, construct different awfss. We study in some detail the example of cofibrant kz-generation that yields representable multicategories, and a counterexample to cofibrant generation provided by continuous lattices. PubDate: 2019-02-27

Authors:Joost Nuiten Abstract: We construct Quillen equivalent semi-model structures on the categories of dg-Lie algebroids and \(L_\infty \) -algebroids over a commutative dg-algebra in characteristic zero. This allows one to apply the usual methods of homotopical algebra to dg-Lie algebroids: for example, every Lie algebroid can be resolved by dg-Lie algebroids that arise from dg-Lie algebras, i.e. whose anchor map is zero. As an application, we show how Lie algebroid cohomology is represented by an object in the homotopy category of dg-Lie algebroids. PubDate: 2019-02-18 DOI: 10.1007/s10485-019-09563-z

Authors:Ren Wang Abstract: Let \({\mathscr {C}}\) be a finite projective EI category and k be a field. The singularity category of the category algebra \(k{\mathscr {C}}\) is a tensor triangulated category. We compute its spectrum in the sense of Balmer. PubDate: 2019-02-13 DOI: 10.1007/s10485-019-09562-0

Authors:Anthony Bordg Abstract: In this article, the author endows the functor category \([\mathbf {B}(\mathbb {Z}_2),\mathbf {Gpd}]\) with the structure of a type-theoretic fibration category with a univalent universe, using the so-called injective model structure. This gives a new model of Martin-Löf type theory with dependent sums, dependent products, identity types and a univalent universe. This model, together with the model (developed by the author in another work) in the same underlying category and with the same universe, which turns out to be provably not univalent with respect to projective fibrations, provide an example of two Quillen equivalent model categories that host different models of type theory. Thus, we provide a counterexample to the model invariance problem formulated by Michael Shulman. PubDate: 2019-02-01 DOI: 10.1007/s10485-019-09558-w