Recent zbMATH articles in MSC 18Dhttps://www.zbmath.org/atom/cc/18D2021-02-12T15:23:00+00:00WerkzeugFiltered cocategories.https://www.zbmath.org/1452.180182021-02-12T15:23:00+00:00"Lyubashenko, Volodymyr"https://www.zbmath.org/authors/?q=ai:lyubashenko.volodymyr-vThis paper is concerned mostly with \(\mathbb{L}\)-filtered \(\mathbb{Z}\)-graded cocategories over a graded commutative complete \(\mathbb{L}\)-filtered ring \(A\), combining the features of [\textit{K. Fukaya}, Adv. Stud. Pure Math. 34, 31--127 (2002; Zbl 1030.53087)]; \textit{K. Fukaya} et al., Lagrangian intersection Floer theory. Anomaly and obstruction. I. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53002); Lagrangian intersection Floer theory. Anomaly and obstruction. II. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53003)] on the one hand and those of [\textit{O. De Deken} and \textit{W. Lowen}, Appl. Categ. Struct. 26, No. 5, 943--996 (2018; Zbl 1409.18019)] on the other.
A synopsis of the paper, consisting of three sections, goes as follows. \S 1 deals with non-filtered graded cocategories, largely recollecting notions and results from [\textit{V. Lyubashenko}, Homology Homotopy Appl. 5, No. 1, 1--48 (2003; Zbl 1026.18003)].
\S 2 is devoted to \(\mathbb{L}\)-filtered \(\mathbb{Z}\)-graded cocategories, especially, to completed conilpotent cocategories, beginning with conditions on a commutative partially ordered monoid \(\mathbb{L}\). \(\mathbb{L}\)-filtered \(\mathbb{Z}\)-graded abelian groups and later (complete) \(\mathbb{L}\)-filtered \(\Lambda\)-modules are investigated, where \(\Lambda\) is a graded commutative complete \(\mathbb{L}\)-filtered ring, for instance, the universal Novikov ring. Completed conilpotent cocategories and their morphisms (cofunctors) are defined in \S 2.11. Cofunctors with values in a completed tensor cocategory are described in Theorem 2.23. Coderivations are investigated in \S 2.30.
Coderivations with values in a completed tensor cocategory are described in Proposition 2.32. The evaluation cofunctor, whose property justifying the name is established in Theorem 2.38, is defined in \S 2.37.
\S 3 applies these results to differential graded completed tensor cocategories, aka filtered \(A_{\infty}\)-categories. It is shown in Proposition 3.2 that the coderivation quiver for two filtered \(A_{\infty}\)-categories is a filtered \(A_{\infty}\)-category itself.
Reviewer: Hirokazu Nishimura (Tsukuba)Quantitative semantics of the lambda calculus. Some generalisations of the relational model.https://www.zbmath.org/1452.030462021-02-12T15:23:00+00:00"Ong, C.-H. Luke"https://www.zbmath.org/authors/?q=ai:ong.c-h-luke|ong.chih-hao-lukeCocycle deformations and Galois objects of semisimple Hopf algebras of dimension 16.https://www.zbmath.org/1452.160322021-02-12T15:23:00+00:00"Xiong, Rongchuan"https://www.zbmath.org/authors/?q=ai:xiong.rongchuan"Yu, Zhiqiang"https://www.zbmath.org/authors/?q=ai:yu.zhiqiangGalois objects are related to fiber functors, cocycle deformation, and Drinfeld twists.
\textit{V. Ostrik} [Transform. Groups 8, No. 2, 177--206 (2003; Zbl 1044.18004); Int. Math. Res. Not. 2003, No. 27, 1507--1520 (2003; Zbl 1044.18005)] and \textit{S. Natale} [SIGMA, Symmetry Integrability Geom. Methods Appl. 13, Paper 042, 9 p. (2017; Zbl 1437.18011)] classified the fiber functors over the group-theoretical fusion categories in terms of subgroups and cohomology data. One of the main ingredient of these classifications is to find groups and 3-cocycles with certain conditions. In Section 3.1, summarized in Lemma 3.1, the authors, on a case-by-case basis, find them for semisimple Hopf algebras of dimension 16, which was classified by \textit{Y. Kashina} [J. Algebra 232, No. 2, 617--663 (2000; Zbl 0969.16014)] (she also describes the Grothendieck rings of these algebras).
The other ingredient is deciding whether or not the class of a certain 2-cocycle, introduced by Natale [loc. cit.], is symmetric. With these, the authors find the number of Galois objects (Theorem 3.18) of each of these algebras.
To describe the cocycle deformations, the authors use higher Frobenius-Schur indicators (Lemma 4.1) and Grothendieck rings (Theorem 4.2). Finally, they determine whether one of these algebras is a Drinfeld twist of some group algebras (Theorem 4.4).
Reviewer: Luz Adriana Mejia (Barranquilla)Some classes of abstract simplicial complexes motivated by module theory.https://www.zbmath.org/1452.130272021-02-12T15:23:00+00:00"Chiaselotti, Giampiero"https://www.zbmath.org/authors/?q=ai:chiaselotti.giampiero"Infusino, F."https://www.zbmath.org/authors/?q=ai:infusino.federico-gAuthors' abstract: In this paper we analyze some classes of abstract simplicial complexes relying on algebraic models arising from module theory. To this regard, we consider a leftmodule on a unitary ring and find models of abstract complexes and related set operators having specific regularity properties, which are strictly interrelated to the algebraic properties of both the module and the ring. Next, taking inspiration from the aforementioned models, we carry out our analysis from modules to arbitrary sets. In such a more general perspective, we start with an abstract simplicial complex and an associated set operator. Endowing such a set operator with the corresponding properties obtained in our module instances, we investigate in detail and prove several properties of three subclasses of abstract complexes. More specifically, we provide uniformity conditions in relation to the cardinality of the maximal members of such classes. By means of the notion of OSS-bijection, we prove a correspondence theorem between a subclass of closure operators and one of the aforementioned families of abstract complexes, which is similar to the classic correspondence theorem between closure operators and Moore systems. Next, we show an extension property of a binary relation induced by set systems when they belong to one of the above families. Finally, we provide a representation result in terms of pairings between sets for one of the three classes of abstract simplicial complexes studied in this work.
Reviewer: Amir Mafi (Sanandaj and Tehran)Cartesian closedness of a category of non-frame valued complete fuzzy orders.https://www.zbmath.org/1452.180062021-02-12T15:23:00+00:00"Liu, Min"https://www.zbmath.org/authors/?q=ai:liu.minSummary: Let \(H = \{0, \frac{ 1}{ 2}, 1 \}\) with the natural order and \(p \operatorname{\&} q = \max \{p + q - 1, 0 \}\) for all \(p, q \in H\). We know that the category of liminf complete \(H\)-ordered sets is Cartesian closed. In this paper, it is proved that the category of conically cocomplete \(H\)-ordered sets with liminf continuous functions as morphisms is Cartesian closed. More importantly, a counterexample is given, which shows that the function spaces consisting of liminf continuous functions of complete \(H\)-ordered sets need not be complete. Thus, the category of complete \(H\)-ordered sets with liminf continuous functions as morphisms is not Cartesian closed.Two complete axiomatisations of pure-state qubit quantum computing.https://www.zbmath.org/1452.810792021-02-12T15:23:00+00:00"Hadzihasanovic, Amar"https://www.zbmath.org/authors/?q=ai:hadzihasanovic.amar"Ng, Kang Feng"https://www.zbmath.org/authors/?q=ai:ng.kang-feng"Wang, Quanlong"https://www.zbmath.org/authors/?q=ai:wang.quanlong