Recent zbMATH articles in MSC 18 https://zbmath.org/atom/cc/18 2021-11-25T18:46:10.358925Z Werkzeug The nominal/FM Yoneda lemma https://zbmath.org/1472.03065 2021-11-25T18:46:10.358925Z "Crole, R. L." https://zbmath.org/authors/?q=ai:crole.roy-l Summary: This paper explores versions of the Yoneda Lemma in settings founded upon FM sets. In particular, we explore the lemma for three base categories: the category of nominal sets and equivariant functions; the category of nominal sets and all finitely supported functions, introduced in this paper; and the category of FM sets and finitely supported functions. We make this exploration in ordinary, enriched and internal settings. We also show that the finite support of Yoneda natural transformations is a \textit{theorem for free}. Cometic functors and representing order-preserving maps by principal lattice congruences https://zbmath.org/1472.06005 2021-11-25T18:46:10.358925Z "Czédli, Gábor" https://zbmath.org/authors/?q=ai:czedli.gabor Summary: Let $$\mathbf {Lat}^{\text{sd}}_{5}$$ and $$\mathbf {Pos}_{01}^{+}$$ denote the category of selfdual bounded lattices of length 5 with $$\{0,1\}$$-preserving lattice homomorphisms and that of bounded ordered sets with $$\{0,1\}$$-preserving isotone maps, respectively. For an object $$L$$ in $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$, the ordered set of principal congruences of the lattice $$L$$ is denoted by $$\mathrm{Princ}(L)$$. By means of congruence generation, $$\mathrm{Princ}:\mathbf {Lat}^{\mathrm{sd}}_{5}\rightarrow \mathbf {Pos}_{01}^{+}$$ is a functor. We prove that if $$\mathbf {A}$$ is a small subcategory of $$\mathbf {Pos}_{01}^{+}$$ such that every morphism of $$\mathbf {A}$$ is a monomorphism, understood in $$\mathbf {A}$$, then $$\mathbf {A}$$ is the $$\mathrm{Princ}$$-image of an appropriate subcategory of $$\mathbf {Lat}^{\mathrm{sd}}_{5}$$. This result extends G. Grätzer's earlier theorems where $$\mathbf {A}$$ consisted of one or two objects and at most one non-identity morphism, and the author's earlier result where all morphisms of $$\mathbf {A}$$ were 0-separating and no hom-set had more the two morphisms. Furthermore, as an auxiliary tool, we derive some families of maps, also known as functions, from injective maps and surjective maps; this can be useful in various fields of mathematics, not only in lattice theory. Namely, for every small concrete category $$\mathbf {A}$$, we define a functor $$F_{\mathrm{com}}$$, called \textit{cometic functor}, from $$\mathbf {A}$$ to the category $$\mathbf{Set}$$ of sets and a natural transformation $$\boldsymbol\pi^{\mathrm{com}}$$, called \textit{cometic projection}, from $$F_{\mathrm{com}}$$ to the forgetful functor of $$\mathbf {A}$$ into $$\mathbf{Set}$$ such that the $$F_{\mathrm{com}}$$-image of every monomorphism of $$\mathbf {A}$$ is an injective map and the components of $$\boldsymbol{\pi }^{\mathrm{com}}$$ are surjective maps. The category of topological De Morgan molecular lattices https://zbmath.org/1472.06013 2021-11-25T18:46:10.358925Z "Mirhosseinkhani, Ghasem" https://zbmath.org/authors/?q=ai:mirhosseinkhani.ghasem "Nazari, Narges" https://zbmath.org/authors/?q=ai:nazari.narges There exist numerous approaches to point-free topology. For example, [\textit{G. Wang}, Fuzzy Sets Syst. 47, No. 3, 351--376 (1992; Zbl 0783.54032)] considered the theory of topological molecular lattices, where a molecular lattice stands for a complete completely distributive lattice. A point-free analogue of a topological space is then a molecular lattice $$L$$, equipped with a subset $$\tau\subseteq L$$ (called \textit{co-topology} on $$L$$), which is closed under finite joins and arbitrary meets. As an analogue of a continuous map, one considers \textit{continuous generalized order homomorphisms} $$f:(L_1,\tau_1)\rightarrow(L_2,\tau_2)$$, namely, join-preserving maps $$f:L_1\rightarrow L_2$$, which possess a join-preserving right adjoint map $$\hat{f}:L_2\rightarrow L_1$$ such that $$\hat{f}(a)\in\tau_1$$ for every $$a\in\tau_2$$ (\textit{continuity}). The present authors continue the study of \textit{N. Nazari} and \textit{G. Mirhosseinkhani} [Sahand Commun. Math. Anal. 10, No. 1, 1--15 (2018; Zbl 1413.06009)], where they introduced generalized topological molecular lattices, considering subsets $$\tau$$ of molecular lattices closed under finite meets and arbitrary joins (namely, dualizing the setting of Wang [loc. cit.]), which were called \textit{topologies} on $$L$$. Moreover, in the paper under review, they take molecular lattices $$L$$ together with a \textit{pseudocomplementation operation} $$(-)^{*}$$ defined by $$a^{*}=\bigvee\{b\in L\mid a\wedge b=\bot_L\}$$~[\textit{T. S. Blyth}, Lattices and ordered algebraic structures. London: Springer (2005; Zbl 1073.06001)], and, additionally, suppose that such molecular lattices $$L$$ satisfy the first De Morgan law, i.e., $$(\bigvee S)^{*}=\bigwedge_{s\in S}s^{*}$$ for every subset $$S\subseteq L$$. The authors define a particular category of topological De Morgan molecular lattices (\textbf{TDML}), and show that the category of topological spaces and continuous maps is isomorphic to both a reflective and a coreflective subcategory of \textbf{TDML}. In the second part of the paper, the authors provide an explicit description of (co)equalizers and (co)products in the category \textbf{TDML}. The paper is well written (but somewhat technical), provides the most essential parts of its required preliminaries, and will be of interest to all those researchers who study categorical approaches to point-free topology. Variations of the shifting lemma and Goursat categories https://zbmath.org/1472.08009 2021-11-25T18:46:10.358925Z "Gran, Marino" https://zbmath.org/authors/?q=ai:gran.marino "Rodelo, Diana" https://zbmath.org/authors/?q=ai:rodelo.diana "Nguefeu, Idriss Tchoffo" https://zbmath.org/authors/?q=ai:nguefeu.idriss-tchoffo Summary: We prove that Mal'tsev and Goursat categories may be characterized through variations of the Shifting Lemma, that is classically expressed in terms of three congruences $$R$$, $$S$$ and $$T$$, and characterizes congruence modular varieties. We first show that a regular category $$\mathbb{C}$$ is a Mal'tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in $$\mathbb{C}$$. Moreover, we prove that a regular category $$\mathbb{C}$$ is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation $$S$$ and reflexive and positive relations $$R$$ and $$T$$ in $$\mathbb{C}$$. In particular this provides a new characterization of 2-permutable and 3-permutable varieties and quasi-varieties of universal algebras. Formal moduli problems and formal derived stacks https://zbmath.org/1472.14004 2021-11-25T18:46:10.358925Z "Calaque, Damien" https://zbmath.org/authors/?q=ai:calaque.damien "Grivaux, Julien" https://zbmath.org/authors/?q=ai:grivaux.julien Summary: This paper presents a survey on formal moduli problems. It starts with an introduction to pointed formal moduli problems and a sketch of proof of a Theorem (independently proven by \textit{J. P. Pridham} [Adv. Math. 224, No. 3, 772--826 (2010; Zbl 1195.14012)] and \textit{J. Lurie} [Derived algebraic geometry X: formal moduli problems'', (2011)]) which gives a precise mathematical formulation for Drinfeld's derived deformation theory philosophy. This theorem provides a correspondence between formal moduli problems and differential graded Lie algebras. The second part deals with Lurie's general theory of deformation contexts, which we present in a slightly different way than the original paper, emphasizing the (more symmetric) notion of Koszul duality contexts and morphisms thereof. In the third part, we explain how to apply this machinery to the case of non-split formal moduli problems under a given derived affine scheme; this situation has been dealt with recently by \textit{J. Nuiten} [Adv. Math. 354, Article ID 106750, 63 p. (2019; Zbl 1433.14007)], and requires to replace differential graded Lie algebras with differential graded Lie algebroids. In the last part, we globalize this to the more general setting of formal thickenings of derived stacks, and suggest an alternative approach to results of \textit{D. Gaitsgory} and \textit{N. Rozenblyum} [A study in derived algebraic geometry. Volume I: Correspondences and duality. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1408.14001)]. For the entire collection see [Zbl 1471.14005]. Birational geometry of moduli spaces of stable objects on Enriques surfaces https://zbmath.org/1472.14014 2021-11-25T18:46:10.358925Z "Beckmann, Thorsten" https://zbmath.org/authors/?q=ai:beckmann.thorsten The moduli space of stable sheaves on a smooth projective surface is an interesting geometric object and has been studied for a long time. By using the notion of Bridgeland stability condition on triangulate category and its wall-crossing behaviour [\textit{T. Bridgeland}, Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)], there are lots of progress in this direction, including the birational geometry of the moduli space when the surface is $$K3$$ [\textit{A. Bayer} and \textit{E. Macrì}, Invent. Math. 198, No. 3, 505--590 (2014; Zbl 1308.14011)] or $$\mathbb{P}^2$$ [\textit{C. Li, X. Zhao}, Geom. Topol. 23, No. 1, 347--426 (2019; Zbl 1456.14016)]. The paper under review continues the idea for the \textit{generic} Enriques surface. Let $$Y$$ be an Enriques surface and $$\pi: \widetilde{Y} \rightarrow Y$$ be the universal covering map by the $$K3$$ surface $$\widetilde{Y}$$. Assume that $$Y$$ is generic, that is $$\mathrm{Pic}(\widetilde{Y})=\pi^*\mathrm{Pic}(Y)$$. Let $$v$$ be a Mukai vector so that its pullback $$\pi^*(v)$$ is primitive. The main Theorem, Theorem 4.5, established the birational equivalence of two moduli spaces $$M^Y_\sigma(v)$$ and $$M^Y_\tau(v)$$ for two generic stability conditions $$\sigma, \tau \in \mathrm{Stab}^{\dag}(Y)$$ with respect to the Mukai vector $$v$$. To prove Theorem 4.5, the author uses two main ideas. The first idea is to use the notion of constant cycle subvariety. A subvariety is called a constant cycle if all its points become rationally equivalent in the ambient variety. By using the result of [\textit{A. Marian, X. Zhao}, Épijournal de Géom. Algébr., EPIGA Journal Profile 4, Article No. 3, 5 p. (2020; Zbl 1442.14035)], the author shows that the image of the morphism $$\pi^*: M^Y_\sigma(v) \rightarrow M^{\widetilde{Y}}_{\widetilde{\sigma}}(\pi^*(v))$$ is a constant cycle Lagrangian. Here $$\widetilde{\sigma}$$ is the induced Bridgeland stability on $$\widetilde{Y}$$ [\textit{E. Macrì} et al., J. Algebr. Geom. 18, No. 4, 605--649 (2009; Zbl 1175.14010)]. The second idea is to show the corresponding birational morphism $$f: M^{\widetilde{Y}}_{\widetilde{\sigma}_+}(\pi^*(v)) \dashrightarrow M^{\widetilde{Y}}_{\widetilde{\sigma}_-}(\pi^*(v))$$ on the $$K3$$ surface $$\widetilde{Y}$$ is $$i^*$$-equivariant. Here $$i^*\in \mathrm{Aut}(\mathrm{D}^{\mathrm{b}}(\widetilde{Y}))$$ is the induced involution by the map $$\pi$$. The assumption that $$Y$$ is generic is used in the proof of $$i^*$$-equivariance. The moduli spaces $$M^Y_{\sigma_\pm}(v)$$ can be identified as fixed set of the involution $$i^*$$. Since $$f$$ is $$i^*$$-equivariant, then its restriction to the constant cycle Lagrangian $$f|_{M^Y_{\sigma_+}(v)}: M^Y_{\sigma_+}(v) \dashrightarrow M^Y_{\sigma_-}(v)$$ gives the birational morphism. As an application of the main Theorem, the author shows (in Theorem 4.7) that for an arbitrary Enriques surface $$Y$$ and a primitive Mukai vector $$v$$ of odd rank, then $$M^Y_\sigma(v)$$ is birational to some Hilbert scheme of points $$\mathrm{Hilb}^n(Y)$$. As another application, the author shows (in Lemma 4.11) that the existence of global Bayer-Macrì map $$\ell: \mathrm{Stab^{\dag}(Y)} \rightarrow \mathrm{NS}(M_\sigma(v))$$ [\textit{A. Bayer} and \textit{E. Macrì}, Invent. Math. 198, No. 3, 505--590 (2014; Zbl 1308.14011); \textit{W. Liu}, Kyoto J. Math. 58, No. 3, 595--621 (2018; Zbl 1412.14009)]. Moreover, the author shows the nef and semiample divisors $$\ell_{\sigma_0,\pm}\in \mathrm{NS}(M_{\sigma_{\pm}}(v)$$ are big. \textit{H. Nuer} and \textit{K. Yoshioka} [Adv. Math. 372, Article ID 107283, 118 p. (2020; Zbl 1454.14041)] obtained more general results of birational equivalence of $$M^Y_\sigma(v)$$ and $$M^Y_\tau(v)$$ by a different method without assumptions that $$Y$$ is generic and $$v$$ is primitive. An example of birationally inequivalent projective symplectic varieties which are D-equivalent and L-equivalent https://zbmath.org/1472.14040 2021-11-25T18:46:10.358925Z "Okawa, Shinnosuke" https://zbmath.org/authors/?q=ai:okawa.shinnosuke There are many different ways to try to classify algebraic varieties. Among those are \begin{itemize} \item birational equivalence: whether a birational map $$X \dashrightarrow Y$$ exists; \item D-equivalence: whether there is an equivalence $$\mathcal D^b(\mathrm{coh}(X)) \cong \mathcal D^b(\mathrm{coh}(Y))$$; \item L-equivalence: whether $$X$$ and $$Y$$ have the same class in $$K_0(\mathcal{V}ar)[\mathbb L^{-1}]$$, where $$\mathbb L = [\mathbb A_1]$$. \end{itemize} These three attempts are related, but in a mysterious way. The main result of this article is the following. Given a pair $$X,Y$$ of two K3 surfaces of Picard number 1 and degree 2d, which are D- and L-equivalent. Then the Hilbert schemes of points $$X^{[n]}$$ and $$Y^{[n]}$$ are D- and L-equivalent. If $$n>2$$ and if there are integer solutions to the equation $(n-1)x^2 - dy^2 = 1,$ then $$X$$ and $$Y$$ are birationally \emph{in}equivalent. For the proof, the author uses that a birational morphism $$\phi \colon X^{[n]} \dashrightarrow Y^{[n]}$$ induces an isometry of Picard groups, hence preserves the movable cone. The equation of the main result comes now from the description of the movable cone, as obtained in [\textit{A. Bayer} and \textit{E. Macrì}, Invent. Math. 198, No. 3, 505--590 (2014; Zbl 1308.14011)]. To get examples, one can consider a very general K3 surface $$X$$ of degree 12 and its Fourier-Mukai partner $$Y$$, see [\textit{B. Hassett} and \textit{K.-W. Lai}, Compos. Math. 154, No. 7, 1508--1533 (2018; Zbl 1407.14010)] and [\textit{A. Ito} et al., Sel. Math., New Ser. 26, No. 3, Paper No. 38, 27 p. (2020; Zbl 1467.14051)]. If one chooses $$n=6y^2+2$$ for an integer $$y$$, then $$(1,y)$$ is an integer solution of the equation above. Hence, the corresponding hyperkähler varieties $$X^{[n]}$$ and $$Y^{[n]}$$ give examples of varieties, which are D-equivalent (already known by [\textit{D. Ploog}, Adv. Math. 216, No. 1, 62--74 (2007; Zbl 1167.14031)]) and L-equivalent, but not birationally equivalent. More results about birational (in)equivalence of such hyperkähler varieties are obtained in [\textit{C. Meachan} et al., Math. Z. 294, No. 3--4, 871--880 (2020; Zbl 1469.14011)] and [\textit{K. Yoshioka}, Math. Ann. 321, No. 4, 817--884 (2001; Zbl 1066.14013)]. Gromov-Witten theory with derived algebraic geometry https://zbmath.org/1472.14061 2021-11-25T18:46:10.358925Z "Mann, Etienne" https://zbmath.org/authors/?q=ai:mann.etienne "Robalo, Marco" https://zbmath.org/authors/?q=ai:robalo.marco Summary: In this survey we add two new results that are not in our paper [Geom. Topol. 22, No. 3, 1759--1836 (2018; Zbl 1423.14320)]. Using the idea of brane actions discovered by Toën, we construct a lax associative action of the operad of stable curves of genus zero on a smooth variety $$X$$ seen as an object in correspondences in derived stacks. This action encodes the Gromov-Witten theory of $$X$$ in purely geometrical terms. For the entire collection see [Zbl 1471.14005]. Hopf algebras and tensor categories. International workshop, Nanjing University, Nanjing, China, September 9--13, 2019 https://zbmath.org/1472.16001 2021-11-25T18:46:10.358925Z "Andruskiewitsch, Nicolás" https://zbmath.org/authors/?q=ai:andruskiewitsch.nicolas "Liu, Gongxiang" https://zbmath.org/authors/?q=ai:liu.gongxiang "Montgomery, Susan" https://zbmath.org/authors/?q=ai:montgomery.susan "Zhang, Yinhuo" https://zbmath.org/authors/?q=ai:zhang.yinhuo Publisher's description: Articles in this volume are based on talks given at the International Workshop on Hopf Algebras and Tensor Categories, held from September 9--13, 2019, at Nanjing University, Nanjing, China. The articles highlight the latest advances and further research directions in a variety of subjects related to tensor categories and Hopf algebras. Primary topics discussed in the text include the classification of Hopf algebras, structures and actions of Hopf algebras, algebraic supergroups, representations of quantum groups, quasi-quantum groups, algebras in tensor categories, and the construction method of fusion categories. The articles of this volume will be reviewed individually. Envelopes, covers and semidualizing modules https://zbmath.org/1472.16003 2021-11-25T18:46:10.358925Z "Mao, Lixin" https://zbmath.org/authors/?q=ai:mao.lixin Support varieties for finite tensor categories: complexity, realization, and connectedness https://zbmath.org/1472.16010 2021-11-25T18:46:10.358925Z "Bergh, Petter Andreas" https://zbmath.org/authors/?q=ai:bergh.petter-andreas "Plavnik, Julia Yael" https://zbmath.org/authors/?q=ai:plavnik.julia-yael "Witherspoon, Sarah" https://zbmath.org/authors/?q=ai:witherspoon.sarah-j The main motivation of this paper is to obtain advances in support variety theory for finite tensor categories. In the third section of the paper under review, the authors define support varieties for objects of a finite tensor category $${\mathcal C}$$, and conclude some standard properties. They also state the finite generation condition (assumed in most of the results of this paper) on the cohomology of $${\mathcal C}$$. In Section 4, we find the definition of complexity of an object (the rate of growth of a minimal projective resolution as measured by the Frobenius-Perron dimension) and the proof that guarantees that it agrees with the dimension of the support variety. As a consequence, the authors show that an object is projective if and only if its support variety is zero-dimensional. In Section 5, for each homogeneous positive degree element $$\zeta$$ of the cohomology ring of the finite tensor category $${\mathcal C}$$, the authors define an object $$L\zeta$$ whose variety is the zero set of the ideal generated by $$\zeta$$. Then, as a standard consequence of the definition of $$L\zeta$$, they obtain both a tensor product property and a realization result: any conical subvariety of the support variety of the unit object $$1$$ can be realized as the support variety of some object. Finally, in the last section of the paper, the authors show that the variety of an indecomposable object is connected. Mutations and pointing for Brauer tree algebras https://zbmath.org/1472.16012 2021-11-25T18:46:10.358925Z "Schaps, Mary" https://zbmath.org/authors/?q=ai:schaps.mary-e "Zvi, Zehavit" https://zbmath.org/authors/?q=ai:zvi.zehavit Summary: Brauer tree algebras are important and fundamental blocks in the representation theory of finite dimensional algebras. In this research, we present a combination of two main approaches to the tilting theory of Brauer tree algebras. The first approach is the theory initiated by \textit{J. Rickard} [J. Lond. Math. Soc., II. Ser. 39, No. 3, 436--456 (1989; Zbl 0642.16034)], providing a direct link between an ordinary Brauer tree algebra and the Brauer star algebra. This approach was continued by Schaps-Zakay with their theory of pointing the tree. The second approach is the theory developed by \textit{T. Aihara} [Math. J. Okayama Univ. 56, 1--16 (2014; Zbl 1327.16004)], relating to the sequence of mutations from the ordinary Brauer tree algebra to the Brauer star algebra. Our main purpose in this research is to combine these two approaches. We first find an algorithm based on centers which are all terminal edges, for which we are able to obtain a tilting complex constructed from irreducible complexes of length two [the first author and \textit{E. Zakay-Illouz}, Lect. Notes Pure Appl. Math. 224, 187--207 (2002; Zbl 0994.16012)], which is obtained from a sequence of mutations. In [loc. cit.], Aihara gave an algorithm for reducing from tree to star by mutations and showed that it gave a two-term tree-to-star complex. We prove that Aihara's complex is obtained from the corresponding completely folded Rickard tree-to-star complex by a permutation of projectives. Embedding of the derived Brauer group into the secondary $$K$$-theory ring https://zbmath.org/1472.16020 2021-11-25T18:46:10.358925Z "Tabuada, Gonçalo" https://zbmath.org/authors/?q=ai:tabuada.goncalo Summary: In this note, making use of the recent theory of noncommutative motives, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injectivity properties: in the case of a regular integral quasi-compact quasi-separated scheme, it is injective; in the case of an integral normal Noetherian scheme with a single isolated singularity, it distinguishes any two derived Brauer classes whose difference is of infinite order. As an application, we show that the aforementioned canonical map is injective in the case of affine cones over smooth projective plane curves of degree $$\geq 4$$ as well as in the case of Mumford's (famous) singular surface. Frobenius and separable functors for the category of entwined modules over cowreaths. II: applications https://zbmath.org/1472.16028 2021-11-25T18:46:10.358925Z "Bulacu, D." https://zbmath.org/authors/?q=ai:bulacu.daniel "Caenepeel, S." https://zbmath.org/authors/?q=ai:caenepeel.stefaan "Torrecillas, B." https://zbmath.org/authors/?q=ai:torrecillas.blas Throughout the review, $$k$$ denotes a field, $$H$$ denotes a quasi-Hopf $$k$$-algebra, $$A$$ denotes a right $$H$$-comodule $$k$$-algebra with $$\rho\colon A\rightarrow A\otimes H$$, $$\rho (a)=\sum a_{\langle 0\rangle}\otimes a_{\langle 1\rangle}$$, $$C$$ denotes a right $$H$$-module $$k$$-coalgebra, $$\mathcal{M}_k$$ denotes the category of $$k$$-vector spaces, $$\mathcal{M}_A$$ denotes the category of right $$A$$-modules, $$\mathcal{M}(H)^C_A$$ denotes the category of $$(H,A,C)$$-Doi-Hopf modules. The paper under review is intended as the study of the Frobenius and separable properties for the forgetful functor $$F\colon\mathcal{M}(H)^C_A\rightarrow\mathcal{M}_A$$. For $$\psi\colon C\otimes A\rightarrow A\otimes C$$, $$\psi (c\otimes a)=\sum a_{\langle 0\rangle}\otimes c\cdot a_{\langle 1\rangle}$$, $$(C,\psi)$$ is a coalgebra in $$\mathcal{T}(\mathcal{M}_k)^\#_A$$, $$(A,C,\psi)$$ is a cowreath in $$\mathcal{M}_k$$, $$\mathcal{M}(H)^C_A$$ is just the category $$\mathcal{M}_k(\psi)^C_A$$ of right entwined modules over $$(A,C,\psi)$$ in $$\mathcal{M}_k$$. This allowed the authors to apply the results from their previous paper [Part I, Algebr. Represent. Theory 23, No. 3, 1119--1157 (2020; Zbl 1447.16031)]. In the present paper, the authors prove that: (1) If $$C$$ is a Frobenius coalgebra in $$\mathcal{M}_H$$ then $$F\colon\mathcal{M}(H)^C_A\rightarrow\mathcal{M}_A$$ is a Frobenius functor (Proposition 2.3); (2) If $$A=H$$ then $$F\colon\mathcal{M}(H)^C_H\rightarrow\mathcal{M}_H$$ is a Frobenius functor if and only if $$C$$ is a Frobenius coalgebra in $$\mathcal{M}_H$$ (Theorem 2.4); (3) If $$C$$ is a coseparable coalgebra in $$\mathcal{M}_H$$ then $$F\colon\mathcal{M}(H)^C_A\rightarrow\mathcal{M}_A$$ is a separable functor (Proposition 2.6); (4) If $$H$$ has a bijective antipode then there is a bijective correspondence between the set of normalized Casimir morphisms for $$C$$ in $$\mathcal{M}_H$$ and the set of normalized Casimir morphisms for $$(C,\psi)$$ in $$\mathcal{T}(\mathcal{M}_k)^\#_H$$ satisfying some additional condition (Theorem 2.7). Consequently, if $$F\colon\mathcal{M}(H)^C_H\rightarrow\mathcal{M}_H$$ is a separable functor then this does not imply that $$C$$ is a coseparable coalgebra in $$\mathcal{M}_H$$. For a quasi-Hopf $$k$$-algebra $$H$$ with bijective antipode, a right $$H$$-comodule $$k$$-algebra $$A$$, an $$H$$-bimodule $$k$$-coalgebra $$C$$, let $$_H\mathcal{M}_A$$ denote the category of $$(H,A)$$-bimodules, let $$_H\mathcal{M}^C_A$$ denote the category of two-sided $$(H,A)$$-bimodules over $$C$$. In this case, the authors prove that: (1) $$A\otimes H^\mathrm{op}$$ is a right $$H\otimes H^\mathrm{op}$$-comodule $$k$$-algebra, $$C$$ is a a right $$H\otimes H^\mathrm{op}$$-module $$k$$-coalgebra, $$_H\mathcal{M}^C_A$$ is isomorphic to the category $$\mathcal{M}(H\otimes H^\mathrm{op})^C_{A\otimes H^\mathrm{op}}$$ of $$(H\otimes H^\mathrm{op},A\otimes H^\mathrm{op},C)$$-Doi-Hopf modules (Proposition 3.2); (2) If $$C$$ is a Frobenius coalgebra in $$_H\mathcal{M}_H$$ then $$F\colon {_H\mathcal{M}^C_A}\rightarrow {_H\mathcal{M}_A}$$ is a Frobenius functor; (3) If $$A=H$$ then $$F\colon {_H\mathcal{M}^C_H}\rightarrow {_H\mathcal{M}_H}$$ is a Frobenius functor if and only if $$C$$ is a Frobenius coalgebra in $$_H\mathcal{M}_H$$; (4) If $$A=C=H$$ then $$F\colon {_H\mathcal{M}^H_H}\rightarrow {_H\mathcal{M}_H}$$ is a Frobenius functor if and only if $$H$$ is finite dimensional and unimodular (Theorem 3.4). In the case when $$H$$ is finite dimensional, the authors prove that: (1) $$H$$ is coseparable as a coalgebra in $$_H\mathcal{M}_H$$ if and only if $$H$$ is unimodular and cosemisimple (Proposition 3.6); (2) $$F\colon {_H\mathcal{M}^H_H}\rightarrow {_H\mathcal{M}_H}$$ is a separable functor if and only if $$H$$ is unimodular (Theorem 3.7). Consequently, in the case when $$H$$ is finite dimensional: (1) The forgetful functor $$F\colon {_H\mathcal{M}^H_H}\rightarrow {_H\mathcal{M}_H}$$ is a separable if and only if it is Frobenius; (2) If $$F\colon {_H\mathcal{M}^H_H}\rightarrow {_H\mathcal{M}_H}$$ is a separable functor then this does not imply that $$H$$ is coseparable as a coalgebra in $$_H\mathcal{M}_H$$. For a quasi-Hopf $$k$$-algebra $$H$$, an $$H$$-bicomodule $$k$$-algebra $$A$$, an $$H$$-bimodule $$k$$-coalgebra $$C$$, let $$\mathcal{YD}(H)^C_A$$ denote the category of right $$(H,A,C)$$-Yetter-Drinfeld modules. In this case, the authors prove that: (1) If $$C$$ is a Frobenius coalgebra in $$_H\mathcal{M}_H$$ then $$F\colon\mathcal{YD}(H)^C_A\rightarrow\mathcal{M}_A$$ is a Frobenius functor (Proposition 4.2); (2) If $$A=C=H$$ then $$F\colon\mathcal{YD}(H)^H_H\rightarrow\mathcal{M}_H$$ is a Frobenius functor if and only if $$H$$ is finite dimensional and Frobenius as a coalgebra in $$_H\mathcal{M}_H$$ if and only if $$H$$ is finite dimensional and unimodular (Theorem 4.5); (3) If $$A=C=H$$ is finite dimensional then $$F\colon\mathcal{YD}(H)^H_H\rightarrow\mathcal{M}_H$$ is a separable functor if and only if $$H$$ is coseparable as a coalgebra in $$_H\mathcal{M}_H$$ if and only if $$H$$ is unimodular and cosemisimple (Theorem 4.9). Extending structures and classifying complements for left-symmetric algebras https://zbmath.org/1472.17006 2021-11-25T18:46:10.358925Z "Hong, Yanyong" https://zbmath.org/authors/?q=ai:hong.yanyong Summary: Let $$A$$ be a left-symmetric (resp. Novikov) algebra, $$E$$ be a vector space containing $$A$$ as a subspace and $$V$$ be a complement of $$A$$ in $$E$$. The extending structures problem which asks for the classification of all left-symmetric (resp. Novikov) algebra structures on $$E$$ up to an isomorphism which stabilizes $$A$$ such that $$A$$ is a subalgebra of $$E$$ is studied. In this paper, the definition of the unified product for left-symmetric (resp. Novikov) algebras is introduced. It is shown that there exists a left-symmetric (resp. Novikov) algebra structure on $$E$$ such that $$A$$ is a subalgebra of $$E$$ if and only if $$E$$ is isomorphic to a unified product of $$A$$ and $$V$$. A cohomological type object $$\mathcal{H}_A^2(V,A)$$ is constructed to give a theoretical answer to the extending structures problem. Furthermore, given an extension $$A\subset E$$ of left-symmetric (resp. Novikov) algebras, another cohomological type object is constructed to classify all complements of $$A$$ in $$E$$. Several examples are provided in detail. \textsf{Lie}-isoclinism of pairs of Leibniz algebras https://zbmath.org/1472.17011 2021-11-25T18:46:10.358925Z "Riyahi, Zahra" https://zbmath.org/authors/?q=ai:riyahi.zahra "Casas Mirás, José Manuel" https://zbmath.org/authors/?q=ai:casas-miras.jose-manuel Summary: The aim of this paper is to consider the relation between \textsf{Lie}-isoclinism and isomorphism of two pairs of Leibniz algebras. We show that, unlike the absolute case for finite dimensional \textsf{Lie} algebras, these concepts are not identical, even if the pairs of Leibniz algebras are \textsf{Lie}-stem. Moreover, throughout the paper, we provide some conditions under which \textsf{Lie}-isoclinism and isomorphism of \textsf{Lie}-stem Leibniz algebras are equal. In order to get this equality, the concept of factor set is studied as well. Ribbon braided module categories, quantum symmetric pairs and Knizhnik-Zamolodchikov equations https://zbmath.org/1472.17049 2021-11-25T18:46:10.358925Z "De Commer, Kenny" https://zbmath.org/authors/?q=ai:de-commer.kenny "Neshveyev, Sergey" https://zbmath.org/authors/?q=ai:neshveyev.sergey-v "Tuset, Lars" https://zbmath.org/authors/?q=ai:tuset.lars "Yamashita, Makoto" https://zbmath.org/authors/?q=ai:yamashita.makoto According to the Tannaka-Krein principle for quantum groups there is a duality between Hopf algebras and tensor categories with duals. This has been constructed by Drinfeld and verified for type $$A$$ quantum groups. The current paper formulates a framework for type $$B$$ quantum groups in terms of quantum symmetric pairs, the Knizhnik-Zamolodchikov equation and the Drinfeld twisting. Explicitly they construct three tensor categories for modules of quantum groups in type $$B$$ using the aforementioned structures, and show their equivalence for the rank one case. The key is to understand the twist in all three constructions and the authors conjecture that the equivalence holds in general. Fusion categories for affine vertex algebras at admissible levels https://zbmath.org/1472.17089 2021-11-25T18:46:10.358925Z "Creutzig, Thomas" https://zbmath.org/authors/?q=ai:creutzig.thomas Summary: The main result is that the category of ordinary modules of an affine vertex operator algebra of a simply laced Lie algebra at admissible level is rigid and thus a braided fusion category. If the level satisfies a certain coprime property then it is even a modular tensor category. In all cases open Hopf links coincide with the corresponding normalized S-matrix entries of torus one-point functions. This is interpreted as a Verlinde formula beyond rational vertex operator algebras. A preparatory Theorem is a convenient formula for the fusion rules of rational principal W-algebras of any type. Simple current extensions beyond semi-simplicity https://zbmath.org/1472.17090 2021-11-25T18:46:10.358925Z "Creutzig, Thomas" https://zbmath.org/authors/?q=ai:creutzig.thomas "Kanade, Shashank" https://zbmath.org/authors/?q=ai:kanade.shashank "Linshaw, Andrew R." https://zbmath.org/authors/?q=ai:linshaw.andrew-r How strict is strictification? https://zbmath.org/1472.18001 2021-11-25T18:46:10.358925Z "Campbell, Alexander" https://zbmath.org/authors/?q=ai:campbell.alexander-p The study of this relation is motivated by the fact that the bases of $\mathbf{Gray}$-category theory and tricategory theory are distinct. They are respectively the $\mathbf{Gray}$-category of 2-categories and tricategory of bicategories. The relation between underlying categories is given by strictification adjunction: $\mathbf{2-Cat}\to^{\bot}\mathbf{Bicat}_{st}\tag{1.1}$ [\textit{R. Gordon} et al., Coherence for tricategories. Providence, RI: American Mathematical Society (AMS) (1995; Zbl 0836.18001)]. Here \textbf{2-Cat} is the category of 2-categories and 2-functors, and \textbf{Bicat} is the category of bicategories and pseudofunctors, $$\bot$$ is the inclusion of \textbf{2-Cat} into \textbf{Bicat} and \textbf{st} sends a bicategory to its strictification. The higher structure of this adjunction is studied in this paper by A. Campbell. Central role is played by the three-dimensional universal property of strictification (Corollary 3.6). The proof of this property contains, however, several subtle points that have to be mentioned. This universal property states that for every bicategory $$A$$ and 2-category $$B$$, the hom-set bijection \textbf{2-Cat}(\textbf{st}A,B)$$\cong$$\textbf{Hom}(A,B) of the strictification adjunction underlies an isomorphism of 2-categories $\textbf{Ps}(\textbf{st}A,B)\cong\textbf{Hom}(A,B)\,\,(1.2)$ where $$\textbf{Ps}(-,-)$$ and $$\textbf{Hom}(-,-)$$ denote the homs of the textbf{Gray}-category of 2-categories and the tricategory of bicategories respectively (whose morphisms are 2-functors and pseudofunctors respectively, and whose 2-cells and 3-cells are in both cases pseudonatural transformations and modifications)''. One is tempted to look for some kind of a three-dimensional adjunction'' underlied by the strictification adjunction but the weakness'' provided by the definitions of tricategory theory is still required in order to realise the higher structure of the strictification adjunction in this setting''. Alternative approach developed by author permits to realise the same three-dimensional higher structure of the strictification adjunction ... by a {\em strictly} bicategory-enriched adjunction''. (In this approach bicategory-enriched categories are the categories whose hom-objects are bicategories.) In the author's approach a framework for bicategory-enriched categories based on the {\em symmetric monoidal closed multicategory} \textsf{Bicat} of bicategories introduced by Verity''. Main advantage is that the category of bicategories becomes {\em strictly} bicategory-enriched category. From the author's summary: Using standard arguments of enriched category theory generalised in \S2 to the context of enrichment of and over symmetric multicategories, in \S3 we prove our main theorem (Theorem 3.8) as a formal consequence of the three-dimensional universal property of strictification (1.2): Main Theorem. The strictification adjunction (1.1) underlies an adjunction of \textsf{Bicat}-enriched categories and, moreover, an adjunction of \textsf{Bicat}-{\em enriched symmetric multicategories}.'' As an application of the strictness of strictification revealed by this main theorem, we obtain (Proposition 3.11) a hitherto undiscovered \textbf{Gray}-{\em category of bicategories}, whose underlying category is the category \textbf{Bicat} of bicategories and pseudofunctors, and whose hom 2-categories $$\textbf{st\,Hom}(A, B)$$ are the strictifications of the hom bicategories $$\textbf{Hom}(A,B)$$. This \textbf{Gray}-category is triequivalent (via a bijective-on-objects, bijective-on-morphisms trihomomorphism) to the tricategory of bicategories''. Paper includes an appendix: The multicategory of pseudo double categories''. On linear exactness properties https://zbmath.org/1472.18002 2021-11-25T18:46:10.358925Z "Jacqmin, Pierre-Alain" https://zbmath.org/authors/?q=ai:jacqmin.pierre-alain "Janelidze, Zurab" https://zbmath.org/authors/?q=ai:janelidze.zurab The authors elaborate on the conceptual framework they introduced in [\textit{P.-A. Jacqmin} and \textit{Z. Janelidze}, Adv. Math. 377, Article ID 107484, 56 p. (2021; Zbl 1452.18005)]. There, an abstract formalism was developed, based on the notion of sketch, in order to give an abstract account of exactness properties on a small finitely complete category $$\mathbb{C}$$ that are preserved under pro-completion (completion under co-filtered limits, given by the embedding $$\mathbb{C} \hookrightarrow \mathrm{Lex}(\mathbb{C},\mathrm{Set})^{op}).$$ Here the same formalism is used in order to obtain exactness properties in regular categories such that, when the category is algebraic, turn out to be equivalent to the existence of certain Mal'tsev terms in the corresponding algebraic theory. The main characterization theorem of the article (Theorem 3.3) asserts the equivalence of suitable exactness conditions and the existence of Mal'tsev terms and equations in the context of essentially algebraic (hence locally finitely presentable) regular categories. The theorem exploits a reformulation of those exactness conditions on a finitely cocomplete regular category in terms of a certain morphism, in the image of a left Kan extension with values in the finitely cocomplete category, being a regular epimorphism (Theorem 2.1). Purity of the Batanin monad. II https://zbmath.org/1472.18003 2021-11-25T18:46:10.358925Z "Penon, Jacques" https://zbmath.org/authors/?q=ai:penon.jacques This is the sequel to [\textit{J. Penon}, Cah. Topol. Géom. Différ. Catég. 61, No. 1, 57--110 (2020; Zbl 1456.18005)], where the notion of purity of a monad, and syntactic monads, were introduced. The aim of this second part is to utilise the machinery of the first part toward the study of Batanin's monad $$\mathbb B$$. In the first part, the monads were on the category $$\mathbf {Set}$$, but for the purposes of the second part it is necessary to pass to the category $$\mathbf {Glob}$$ of globular sets. The first steps are taken immediately as this part starts, obviously relying on the language of part I. The language of syntactic monads from part I is new and unfamiliar. The author uses Section 2 to present the familiar monad $$\omega$$ of strict $$\infty$$-categories, in terms of the language of part I, in part as a way to exemplify the machinery, as well as towards the weakened version of this monad. The free strict $$\infty$$-category is described in quite some detail. Section 3 introduces the monad $$\mathbb P$$. This is a variant of the author's monad presented in [\textit{J. Penon}, Cah. Topologie Géom. Différ. Catégoriques 40, No. 1, 31--80 (1999; Zbl 0918.18006)], following a technique of the author's of weakening algebraic structures (commented on by \textit{M. A. Batanin} [J. Pure Appl. Algebra 172, No. 1, 1--23 (2002; Zbl 1003.18010)]). The monad $$\mathbb P$$ described in this section is shown to be syntactic and cartesian, but not pure. In a sense, the failure of purity necessitates the passage, carried out in Section 4, to Batanin's monad $$\mathbb B$$. This monad is described in this section in terms of weak trees and using the formalism of the first part of the work. The description as given does not make it clear at all that this is indeed Batanin's monad. That is the aim of Section 5. The presentation given makes it quite straightforward that $$\mathbb B$$ is pure. The rest of the section, and of the work, is making the relation between this definition of $$\mathbb B$$ and the monad arising from free strict $$\omega$$-categories, to arrive at Theorem 5.46, at the identification of $$\mathbb B$$ as Batanin's monad. It is thus pure. Joyal's cylinder conjecture https://zbmath.org/1472.18004 2021-11-25T18:46:10.358925Z "Campbell, Alexander" https://zbmath.org/authors/?q=ai:campbell.alexander-p For each pair of simplicial sets $$A$$ and $$B$$, the category $$\boldsymbol{Cyl} (A,B)$$ of \textit{cylinders from $$A$$ to $$B$$} or \textit{$$(A,B)$$-cylinders} is defined to be the fiber of the functor $(\partial_{0},\partial_{1}):\boldsymbol{sSet}/\Delta\left[ 1\right] \rightarrow\boldsymbol{sSet}\times\boldsymbol{sSet}.$ \textit{A. Joyal} [Notes on quasi-categories'', \url{https://www.math.uchicago.edu/\~may/IMA/Joyal.pdf}, \S 14.6] described what is called the Joyal model structure on $$\boldsymbol{Cyl}(A,B)$$, giving the following conjecture. Conjecture. A cylinder $$X\in\boldsymbol{Cyl}(A,B)$$ is fibrant iff the canonical morphism $$X\rightarrow A\ast B$$ is an inner fibration, and a morphism between fibrant cylinders in $$\boldsymbol{Cyl}(A,B)$$ is a fibration iff it is an inner fibration. The principal objective in this paper is to settle the above conjecture affirmatively (Theorem 5.5). A synopsis of the paper consisting of six sections together with an appendix goes as follows. The proof of Joyal's conjecture is addressed in \S \S 2--5. \begin{itemize} \item \S 2 constructs, for each pair of simplicial sets $$A$$ and $$B$$, both the \textit{Joyal model structure} (Theorem 2.10) and the \textit{ambivaliant model structure} (Theorem 2.11). \item \S 3 makes Joyal's observation that the category $$\boldsymbol{Cyl} (A,B)$$ of $$(A,B)$$-cylinders is equivalent to the category of presheaves over $$\Delta/A\times\Delta/B$$ (Proposition 3.16), which is the basis for a deeper analysis of the ambivariant model structure on $$\boldsymbol{Cyl}(A,B)$$ (\S 3.1). A \textit{Reedy model structure} on $$\boldsymbol{Cyl}(A,B)$$ is induced from the \textit{covariant model structure} on $$\boldsymbol{sSet}/B$$ by making use of the related equivalence of categories $\boldsymbol{Cyl}(A,B)\simeq\left[ (\Delta/A)^{\mathrm{op}},\boldsymbol{sSet}/B\right]$ It is established (Theorem 3.20) that the ambivariant model structure on $$\boldsymbol{Cyl}(A,B)$$ is a \textit{Bousfield localization} of this Reedy model structure, for which the \textit{local objects} are those $$(A,B)$$-cylinders whose corresponding functors $(\Delta/A)^{\mathrm{op}}\rightarrow\boldsymbol{sSet}/B$ send the \textit{final vertex inclusions} in $$\Delta/A$$ to covariant equivalences in $$\boldsymbol{sSet}/B$$. \item \S 4 aims to establish that, for each pair of weak categorical equivalences $$u:A\rightarrow A^{\prime}$$ and $$v:B\rightarrow B^{\prime}$$ in $$\boldsymbol{sSet}$$, the pushforward functor $(u,v)_{!}:\boldsymbol{Cyl}(A,B)\rightarrow \boldsymbol{Cyl}(A^{\prime},B^{\prime})$ \begin{itemize} \item[(a)] preserves weak categorical equivalences (Proposition 4.2), and \item[(b)] reflects ambivariant equivalences (Theorem 4.7). \end{itemize} \item \S 5 aims to establish (Theorem 5.4) that, for any morphism $$f$$ in $$\boldsymbol{Cyl}(A,B)$$ and for each pair of weak categorical equivalences $$u:A\rightarrow A^{\prime}$$ and $$v:B\rightarrow B^{\prime}$$ in $$\boldsymbol{sSet}$$ such that $$A^{\prime}$$ and $$B^{\prime}$$ are quasi-categories, we have \begin{align*} f\text{ is a weak categorical equivalence} & \Rightarrow(u,v)_{!}(f)\text{ is weak categorical equivalence} \Rightarrow\\ (u,v)_{!}(f)\text{ is ambivariant equivalence} & \Rightarrow f\text{ is a ambivariant equivalence} \end{align*} completing the proof of Joyal's conjecture. \item \S 6 gives a new direct proof (Theorem 6.5) of a characterization of covariant equivalences of \textit{J. Lurie} [Higher topos theory. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001), Chapter 2] avoiding the use of the straightening theorem [loc. cit., Theorem 2.2.1.2]. \item Appendix introduces and studies a new family of model structures on the slice category $\boldsymbol{sSet}/B$, which is called the \textit{parametrized Joyal model structures}, whose fibrant objects are the inner fibrations with codomain $B$\ (Theorem A.7). Using this family of model structures, the author defines a new class of morphisms of simplicial sets, called the \textit{absolute weak categorical equivalences}, using which the author establishes some new results concerning inner anodyne extensions and inner fibrations on the basis of [\textit{A. Campbell}, Proc. Am. Math. Soc. 148, No. 1, 37--40 (2020; Zbl 1444.18025)] while a new proof of a theorem [\textit{D. Stevenson}, Theory Appl. Categ. 33, 523--536 (2018; Zbl 1398.55013), Theorem 1.5] is given (Theorem A.27). \end{itemize} Compactness in abelian categories https://zbmath.org/1472.18005 2021-11-25T18:46:10.358925Z "Kálnai, Peter" https://zbmath.org/authors/?q=ai:kalnai.peter "Žemlička, Jan" https://zbmath.org/authors/?q=ai:zemlicka.jan Authors' abstract: We relativize the notion of a compact object in an abelian category with respect to a fixed subclass of objects. We show that the standard closure properties persist to hold in this case. Furthermore, we describe categorical and set-theoretical conditions under which all products of compact objects remain compact. On some cohomological invariants for large families of infinite groups https://zbmath.org/1472.18006 2021-11-25T18:46:10.358925Z "Biswas, Rudradip" https://zbmath.org/authors/?q=ai:biswas.rudradip In this article, the author investigates some cohomological invariants of the class of infinite groups denoted $$LH\mathcal F_{\phi,A}$$. This class is defined using the groups of type $$\Phi$$, introduced by Talelli over $$\mathbb Z$$ in [\textit{O. Talelli}, Arch. Math. 89, No. 1, 24--32 (2007; Zbl 1158.20026)], and groups derived from a hierarchy of groups introduced by \textit{P. H. Kropholler} [J. Pure Appl. Algebra 90, No. 1, 55--67 (1993; Zbl 0816.20042)]. In the article under review, the author extends some results that were known over the ring of integers to arbitrary commutative rings of finite global dimension. In particular, one of the main result states that if a group $$\Gamma$$ is in $$LH\mathcal F_{\phi,A}$$ with $$A$$ of finite global dimension, say $$t$$, then the projective dimension of the module of bounded functions $$B(\Gamma,A)$$, of $$\Gamma$$ on $$A$$ over $$A\Gamma$$, is equal to the Gorenstein cohomological dimension of $$\Gamma$$ over $$A$$. Moreover, this quantity, say $$\theta$$, is bounded above by the finistic dimension of $$A\Gamma$$, which is itself bounded above by $$\theta+t$$. The Gorenstein transpose of comodules https://zbmath.org/1472.18007 2021-11-25T18:46:10.358925Z "Li, Yexuan" https://zbmath.org/authors/?q=ai:li.yexuan "Yao, Hailou" https://zbmath.org/authors/?q=ai:yao.hailou The classical notion of transpose ($$\mathrm{Tr}$$) of a right module was dualized for right comodule over a coalgebra $$\Gamma$$ by \textit{W. Chin} et al. [J. Algebra 249, No. 1, 1--19 (2002; Zbl 1005.16034)] in order to study the existence of almost split sequences in the category of comodules. The authors define the notion of Gorenstein transpose ($$\mathrm{Tr}_G$$) of comodules over a Gorenstein coalgebra, using Gorenstein injective copresentation of comodules instead of injective copresentation. It is shown that for any right $$\Gamma$$-comodule $$M$$, there exist a short exact sequence of left $$\Gamma$$-comodules $$0\to K\to \mathrm{Tr}M \to \mathrm{Tr}_GM\to 0$$, where $$K$$ is Gorenstein injective. Moreover, if $$M$$ has finite injective dimension, then $$\mathrm{Tr}_GM= \mathrm{Tr}M\oplus I$$ with $$I$$ Gorenstein injective. On derived equivalences and homological dimensions https://zbmath.org/1472.18008 2021-11-25T18:46:10.358925Z "Fang, Ming" https://zbmath.org/authors/?q=ai:fang.ming "Hu, Wei" https://zbmath.org/authors/?q=ai:hu.wei "Koenig, Steffen" https://zbmath.org/authors/?q=ai:konig.steffen In representation theory, the global dimension of an algebra is considered to be a fundamental invariant, used to define and to characterise classes of algebras. Dominant dimension, is used to describe, and to measure the quality of, double centraliser properties as in the Morita-Tachikawa correspondence and in Schur-Weyl duality, and to formulate Nakayama's conjecture. The interplay of global and dominant dimensions is crucial in Auslander's correspondence, Auslander's definition of representation dimension and Iyama's higher Auslander correspondence. Derived equivalences are used to connect different areas of mathematics and to compare different situations. In representation theory, derived equivalences from one algebra to another induced by tilting complexes share some homological or $$K$$-theoretic invariance. As is known, Hochschild (co)homology and algebraic $$K$$-theory are derived invariant. But homological dimensions appear to be rarely invariant under derived equivalences. For instance, the difference of global dimensions and finitistic dimensions is bounded above by the length of the tilting complex inducing the derived equivalence. For dominant dimensions, derived equivalent algebras may be have different dominant dimensions, and it seems to be unknown whether finiteness of dominant dimension is preserved. In the paper under review, the authors discuss global dimensions and dominant dimensions are preserved by derived equivalences for a class of algebras with anti-automorphisms preserving simples. Cellular algebras and many finite dimensional algebras in algebraic Lie theory have anti-automorphisms. They also consider derived equivalences between algebras with positive $$\nu$$-dominant dimension always restrict to derived equivalences between their associated self-injective algebras, consequently, two self-injective algebras of finite representation type are seen to be derived equivalent if and only if their Auslander algebras are derived equivalent. Quotients of triangulated categories and equivalences of Buchweitz, Orlov, and Amiot-Guo-Keller https://zbmath.org/1472.18009 2021-11-25T18:46:10.358925Z "Iyama, Osamu" https://zbmath.org/authors/?q=ai:iyama.osamu "Yang, Dong" https://zbmath.org/authors/?q=ai:yang.dong.1 Summary: We give a simple sufficient condition for a Verdier quotient $$\mathcal{T}/\mathcal{S}$$ of a triangulated category $$\mathcal{T}$$ by a thick subcategory $$\mathcal{S}$$ to be realized inside of $$\mathcal{T}$$ as an ideal quotient. As applications, we deduce three significant results by \textit{R.-O. Buchweitz} [Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings'', Preprint, \url{https://hdl.handle.net/1807/16682}], \textit{D. Orlov} [Prog. Math. 270, 503--531 (2009; Zbl 1200.18007)] and \textit{C. Amiot} [Ann. Inst. Fourier 59, No. 6, 2525--2590 (2009; Zbl 1239.16011)], \textit{L. Guo} [J. Pure Appl. Algebra 215, No. 9, 2055--2071 (2011; Zbl 1239.16012)], \textit{B. Keller} [Doc. Math. 10, 551--581 (2005; Zbl 1086.18006); with \textit{M. Van den Bergh}, J. Reine Angew. Math. 654, 125--180 (2011; Zbl 1220.18012)]. On the tensor product of well generated dg categories https://zbmath.org/1472.18010 2021-11-25T18:46:10.358925Z "Lowen, Wendy" https://zbmath.org/authors/?q=ai:lowen.wendy "Ramos González, Julia" https://zbmath.org/authors/?q=ai:ramos-gonzalez.julia Summary: We endow the homotopy category of well generated (pretriangulated) dg categories with a tensor product satisfying a universal property. The resulting monoidal structure is symmetric and closed with respect to the cocontinuous RHom of dg categories (in the sense of \textit{B. Toën} [Invent. Math. 167, No. 3, 615--667 (2007; Zbl 1118.18010)]). We give a construction of the tensor product in terms of localisations of dg derived categories, making use of the enhanced derived Gabriel-Popescu theorem [\textit{M. Porta}, Adv. Math. 225, No. 3, 1669--1715 (2010; Zbl 1227.18011)]. Given a regular cardinal $$\alpha$$, we define and construct a tensor product of homotopically $$\alpha$$-cocomplete dg categories and prove that the well generated tensor product of $$\alpha$$-continuous derived dg categories (in the sense of Porta [loc. cit.]) is the $$\alpha$$-continuous dg derived category of the homotopically $$\alpha$$-cocomplete tensor product. In particular, this shows that the tensor product of well generated dg categories preserves $$\alpha$$-compactness. Relative $$n$$-rigid objects in $$(n+2)$$-angulated categories https://zbmath.org/1472.18011 2021-11-25T18:46:10.358925Z "Xie, Zongyang" https://zbmath.org/authors/?q=ai:xie.zongyang "Liu, Zhongkui" https://zbmath.org/authors/?q=ai:liu.zhong-kui "Di, Zhenxing" https://zbmath.org/authors/?q=ai:di.zhenxing Dendriform algebras relative to a semigroup https://zbmath.org/1472.18012 2021-11-25T18:46:10.358925Z "Aguiar, Marcelo" https://zbmath.org/authors/?q=ai:aguiar.marcelo Recently, numerous generalizations of Rota-Baxter algebras and dendriform algebras appear, where one or two parameters belonging to a semigroup are used: these are family algebras. This paper first shows that similar objects exist for associative, associative commutative, Lie, and many other types of algebras. It then gives a categorical interpretation of these objects: it is shown that they are in fact classical objects in a monoidal category of graded objects over the chosen monoid, satisfying a uniformity condition. On realizing modular data https://zbmath.org/1472.18013 2021-11-25T18:46:10.358925Z "Bonderson, Parsa" https://zbmath.org/authors/?q=ai:bonderson.parsa "Rowell, Eric C." https://zbmath.org/authors/?q=ai:rowell.eric-c "Wang, Zhenghan" https://zbmath.org/authors/?q=ai:wang.zhenghan Summary: We use zesting and symmetry gauging of modular tensor categories to analyze some previously unrealized modular data obtained by Grossman and Izumi. In one case we find all realizations and in the other we determine the form of possible realizations; in both cases all realizations can be obtained from quantum groups at roots of unity. Deligne categories and the periplectic Lie superalgebra https://zbmath.org/1472.18014 2021-11-25T18:46:10.358925Z "Entova-Aizenbud, Inna" https://zbmath.org/authors/?q=ai:aizenbud.inna-entova "Serganova, Vera" https://zbmath.org/authors/?q=ai:serganova.vera-v The authors give a construction of the tensor category $$\text{Rep}(\underline{P})$$, possessing nice universal properties among tensor categories over the category $$\textsf{sVect}$$ of finite-dimensional complex vector superspaces with parity-preserving maps. One way that the category $$\textsf{sVect}$$ is constructed is through an explicit limit of the tensor categories $$\text{Rep}(\mathfrak{p}(n))$$ under Duflo-Serganova functors $$DS_x:\text{Rep}(\mathfrak{p}(n))\rightarrow \text{Rep}(\mathfrak{p}(n-2))$$, where $$n\geq 1$$. That is, let $$x$$ be an odd element in a Lie superalgebra $$\mathfrak{g}$$ satisfying $$[x,x]=0$$. Let $$\mathfrak{g}_x =\mathsf{ker} \:\text{ad}_x/\mathsf{im}\: \text{ad}_x$$, a Lie superalgebra. Define the functor $$DS_x:\text{Rep}(\mathfrak{g})\rightarrow \text{Rep}(\mathfrak{g}_x)$$ to be $$M\mapsto M_x =\mathsf{ker}\: x/\mathsf{im} \: x$$, which is symmetric monoidal. Take $$\mathfrak{g}=\mathfrak{p}(n)$$ and $$x$$ of rank $$2$$ to obtain $$\mathfrak{g}_x\cong \mathfrak{p}(n-2)$$. The second way that one constructs $$\textsf{sVect}$$, inspired by P. Etingof, is by describing $$\text{Rep}(\underline{P})$$ as the category of representations of a periplectic Lie supergroup in the Deligne category $$\textsf{sVect}\boxtimes \text{Rep}(\underline{\text{GL}}_t)$$, where $$\text{Rep}(\underline{\text{GL}}_t)$$ is an example of a non-Tannakian tensor category for $$t\not\in\mathbb{Z}$$; it is not equivalent to the category of representations of any affine algebraic group, nor supergroup. The category $$\text{Rep}(\underline{P})$$ is a lower highest weight category, i.e., there is a filtration $$\text{Rep}(\underline{P})=\bigcup_{k\geq 0}\text{Rep}^k(\underline{P})$$ by full highest weight subcategories, whose standard, costandard and tilting objects play the same role in each $$\text{Rep}^{k'}(\underline{P})$$, where $$k'\geq k$$. The full subcategory of tilting objects in $$\text{Rep}(\underline{P})$$ is precisely a Karoubian additive Deligne category $$\mathfrak{P}$$ in the periplectic case, which is a module category over $$\mathsf{sVect}$$. One nice property of the limit category'' $$\text{Rep}(\underline{P})$$ includes that it is the abelian envelope of the Deligne category corresponding to the periplectic Lie superalgebra. Another nice property is that given a tensor category $$\mathcal{C}$$ over $$\textsf{sVect}$$, exact tensor functors $$\text{Rep}(\underline{P})\rightarrow \mathcal{C}$$ classify pairs $$(X,\omega)$$ in $$\mathcal{C}$$, where $$\omega:X\otimes X\rightarrow \prod \mathbf{1}$$ is a nondegenerate odd symmetric form and $$X$$ cannot be annihilated by any Schur functor (Theorems 1 and 2, page 510). The authors study stabilization properties of finite-dimensional integrable representations of the periplectic Lie superalgebras $$\mathfrak{p}(n)$$ as $$n\rightarrow \infty$$ (Section 5, page 529). Integrals along bimonoid homomorphisms https://zbmath.org/1472.18015 2021-11-25T18:46:10.358925Z "Kim, Minkyu" https://zbmath.org/authors/?q=ai:kim.minkyu The notion of an integral of a bialgebra was introduced in [\textit{R. G. Larson} and \textit{M. E. Sweedler}, Am. J. Math. 91, 75--94 (1969; Zbl 0179.05803)] as a generalization of the Haar measure of a group, which has been used in the study of bialgebras or Hopf algebras [\textit{M. E. Sweedler}, Ann. Math. (2) 89, 323--335 (1969; Zbl 0174.06903); \textit{D. E. Radford}, Bull. Am. Math. Soc. 81, 1103--1105 (1975; Zbl 0326.16008); Am. J. Math. 98, 333--355 (1976; Zbl 0332.16007)]. The notion of a bialgebra was generalized to bimonoids in a symmetric monoidal category $$\mathcal{C}$$ [\textit{M. Aguiar} and \textit{S. Mahajan}, Monoidal functors, species and Hopf algebras. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1209.18002); \textit{S. Mac Lane}, Categories for the working mathematician. New York, NY: Springer (1998; Zbl 0906.18001)], while the integral theory was generalized to the categorical setting to study bimonoids or Hopf monoids [\textit{Y. Bespalov} et al., J. Pure Appl. Algebra 148, No. 2, 113--164 (2000; Zbl 0961.16023)]. This paper introduces a notion of an integral along a bimonoid homomorphism in a symmetric monoidal category $$\mathcal{C}$$, generalizing the notions of integral and cointegral of a bimonoid. The principal objective in this paper is to establish the following theorem. Theorem. Let $$A$$ and $$B$$ be bicommutative Hopf algebras in $$\mathcal{C}$$ with $$\xi:A\rightarrow B$$ a Hopf homomorphism. Then there exists a normalized generator integral $$\mu_{\xi}$$ along $$\xi$$ iff the following conditions are satisfied: \begin{enumerate} \item The kernel Hopf monoid $$Ker(\xi)$$ has a normalized integral. \item The cokernel Hopf monoid $$\mathrm{Cok}(\xi)$$ has a normalized cointegral. \end{enumerate} Moreover, if a normalized integral exists, then it is unique. Some applications are as follows. \begin{itemize} \item The author investigates the category $$\mathrm{Hopf}^{\mathrm{bc},\ast}(\mathcal{C})$$ of bicommutative Hopf monoids with a normalized integral and cointegral, establishing that the category $$\mathrm{Hopf} ^{\mathrm{bc},\ast}(\mathcal{C})$$ is an abelian subcategory of $$\mathrm{Hopf}^{\mathrm{bc}}(\mathcal{C})$$ and closed under short exact sequents. \item The author introduces the notion of volume on an abelian category as a generalization of the dimension of vector spaces and the order of abelian groups, studying basic notions related with it. \item The author constructs an $$\mathrm{End}_{\mathcal{C}}(\boldsymbol{1} )$$-valued volume $$\mathrm{vol}^{-1}$$ on the abelian category $$\mathrm{Hopf} ^{\mathrm{bc},\ast}(\mathcal{C})$$, where $$\boldsymbol{1}$$ is the unit object of $$\mathcal{C}$$ and the endomorphism set $$\mathrm{End}_{\mathcal{C} }(\boldsymbol{1})$$ is an abelian monoid induced by the symmetric monoidal structure of $$\mathcal{C}$$. \item By using the volume $$\mathrm{vol}^{-1}$$, the author introduces a notion of Fredholm homomorphisms between bicommutative Hopf monoids as an analogue of Fredholm operators [\textit{T. Kato}, Perturbation theory for linear operators. Berlin: Springer-Verlag (1995; Zbl 0836.47009)], investigating its index which is robust to some perturbations. A functorial assignment of integrals to Fredholm homomorphisms is constructed. \item The author expects that the result in this paper could be applied to topology. There is a topological invariant of $$3$$-manifolds induced by a finite-dimensional Hopf algebra, called the Kuperberg invariant [\textit{G. Kuperberg}, Int. J. Math. 2, No. 1, 41--66 (1991; Zbl 0726.57016); Non-involutory Hopf algebras and 3-manifold invariants'', Preprint, \url{arXiv:q-alg/9712047}]. \end{itemize} This paper gives a technical preliminary to the author's subsequent paper [\textit{M. Kim}, A pair of homotopy-theoretic version of TQFT's induced by a Brown functor'', Preprint, \url{arXiv:2006.10438}], which uses the results in this paper to give a generalization of the untwisted abelian Dijkgraaf-Witten theory [\textit{R. Dijkgraaf} and \textit{E. Witten}, Commun. Math. Phys. 129, No. 2, 393--429 (1990; Zbl 0703.58011); \textit{M. Wakui}, Osaka J. Math. 29, No. 4, 675--696 (1992; Zbl 0786.57008); \textit{D. S. Freed} and \textit{F. Quinn}, Commun. Math. Phys. 156, No. 3, 435--472 (1993; Zbl 0788.58013)] and the bicommutative Turaev-Viro TQFT [\textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, No. 4, 865--902 (1992; Zbl 0779.57009); \textit{J. W. Barrett} and \textit{B. W. Westbury}, Trans. Am. Math. Soc. 348, No. 10, 3997--4022 (1996; Zbl 0865.57013)], giving a systematic way to construct a sequence of TQFT's from (co)homology theory. The TQFT's are constructed by using path-integral which is formulated by some integral along bimonoid homomorphisms. Monoidal structures on the categories of quadratic data https://zbmath.org/1472.18016 2021-11-25T18:46:10.358925Z "Manin, Yuri Ivanovich" https://zbmath.org/authors/?q=ai:manin.yuri-ivanovich "Vallette, Bruno" https://zbmath.org/authors/?q=ai:vallette.bruno This paper gives several (co)lax 2-monoidal structures on categories of quadratic data, and several constructions of (co)operads are detailed, recovering important examples from the literature. In Section 2, a definition is given for quadratic data'', which is essentially the presentation data of an associative algebra. There are also symmetric and skew-symmetric quadratic data, which present commutative and Lie algebras respectively. Several monoidal structures are defined on each of these categories, as well as several monoidal functors relating them and the categories of algebras they present, including the universal enveloping algebra functor, and Koszul duality functors. In Section 3, they give the required interchange law maps to assemble some of these monoidal structures into lax 2-monoidal structures. A lax 2-monoidal structure on a category is a generalization of duoidal categories which allows the interchange law $$\phi_{A,A',B,B'} \colon (A \otimes A') \boxtimes (B \otimes B') \to (A \boxtimes B) \otimes (A' \boxtimes B')$$ to not necessarily be an isomorphism. In Section 4, they introduce the category of binary operadic quadratic data as well as multiple (co)lax 2-monoidal structures on it. Section 5 discusses how to obtain quadratic data from topological operads. This is done by applying the Magnus construction to the fundamental group. In this way, they are able to construct many operads of interest, such as the genus zero quantum cohomology operad and the operad classiyfing Gerstenhaber algebras. Coquasi-bialgebras with preantipode and rigid monoidal categories https://zbmath.org/1472.18017 2021-11-25T18:46:10.358925Z "Saracco, Paolo" https://zbmath.org/authors/?q=ai:saracco.paolo If $${\mathcal C}$$ is a $$k$$-linear, abelian category over a field $$k$$ and $$\omega:{\mathcal C}\rightarrow kVect_{f}$$ is a $$k$$-linear exact and faithful functor into the category of finite-dimensional vector spaces, \textit{N. Saavedra Rivano} proved in [Categories tannakiennes. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0241.14008)] that there exists a $$k$$-coalgebra $$A$$ such that the category $${\mathcal C}$$ is $$k$$-linearly equivalent to the category of finite-dimensional right A-comodules. If we assume that is symmetric monoidal and rigid, it is possible to prove that A is a commutative Hopf algebra. Moreover, \textit{K. H. Ulbrich} proved in [Isr. J. Math. 72, No. 1--2, 252--256 (1990; Zbl 0727.16029)] that there is still a Hopf algebra structure on $$A$$ if we remove the symmetry on the monoidal structure. \textit{S. Majid} [Contemp. Math. 134, 219--232 (1992; Zbl 0788.17012)] extended the result cited in the previous paragraph to coquasi-bialgebras in the following way: If $${\mathcal C}$$ is an essentially small monoidal category endowed with a functor $$\omega:{\mathcal C} \rightarrow kVect_{f}$$ that respects the tensor product in a suitable way, there is a coquasi-bialgebra $$H$$ such that $$\omega$$ factorizes through a monoidal functor $$\omega^{H}$$ between the category $${\mathcal C}$$ and the category of finite comodules over $$H$$. Finally, all these results can be extended to categories of modules over a commutative ring $${\mathbb K}$$. In the paper under review, the author proves that if the conditions of the result proved by Majid hold and the category $${\mathcal C}$$ is rigid, then the associated coquasi-bialgebra admits a preantipode. If the ring is a field, this allows to characterize coquasi-Hopf algebras in terms of rigidity of finite-dimensional corepresentations. As a consequence, the authord find how to endow the finite dual coalgebra of a quasi-bialgebra with preantipode with a structure of coquasi-bialgebra with preantipode. Braided zesting and its applications https://zbmath.org/1472.18018 2021-11-25T18:46:10.358925Z "Delaney, Colleen" https://zbmath.org/authors/?q=ai:delaney.colleen "Galindo, César" https://zbmath.org/authors/?q=ai:galindo.cesar "Plavnik, Julia" https://zbmath.org/authors/?q=ai:plavnik.julia-yael "Rowell, Eric C." https://zbmath.org/authors/?q=ai:rowell.eric-c "Zhang, Qing" https://zbmath.org/authors/?q=ai:zhang.qing.2|zhang.qing.4|zhang.qing.3|zhang.qing|zhang.qing.1 There are several known ways of constructing new (braided) fusion categories out of old ones. This paper is concerned with a recently developed such construction, called \textit{zesting}. As a construction technique it appeared in [\textit{P. Bruillard} et al., Can. Math. Bull. 57, No. 4, 721--734 (2014; Zbl 1342.18013)] with the purpose to categorify certain fusion rule algebra. Then expected modular structure of the zested fusion category obtained in this way is proved in the present paper. More importantly, the general theory of associative zesting for fusion categories (and further for braided, twist and ribbon zestings for categories possessing these additional structures) is developed. Starting with a $$G$$-graded fusion category $$\mathcal{C}=\sum_{g\in G}\mathcal{C}_g$$ for a group $$G$$, the idea of zesting is to define a new tensor product $X_g\stackrel{\lambda}{\otimes} Y_h:=(X_g\otimes Y_h)\otimes\lambda(g,h),\tag{1}$ where $$X_g\in\mathcal{C}_g, Y_h\in\mathcal{C}_h$$ are simple objects in their corresponding graded components, and $$\lambda(g,h)\in\mathcal{C}_e\cap\mathcal{C}_{pt}$$, that is, it is an invertible object in the trivial component. The zesting of $$\mathcal{C}$$ is thus in particular a $$G$$-graded extension of $$\mathcal{C}_e$$. For a given choice of $$\lambda:G\times G\to\mathcal{C}_e\cap\mathcal{C}_{pt}$$, there are obstructions to $$(\mathcal{C}, \stackrel{\lambda}{\otimes}, 1)$$ admitting a monoidal category structure. That is, when these obstructions vanish, there are choices of associativity constraints. Fixing one such associative zesting, one can moreover study the existence of braiding (leading to \textit{braided zesting}), which leads to new obstructions and choices. Moreover, for a fixed braided zesting of a ribbon fusion category one may search for a ribbon twist structure leading to \textit{ribbon zesting}. Before describing this process in more details, we observe that in the literature two related constructions can be found. The first one is gauging. If $$\mathcal{C}$$ is modular and $$\mathcal{C}_{pt}\cap\mathcal{C}_e\cong\text{Rep}(G)$$ is Tannakian with $$G$$ abelian, one may set $$\mathcal{B}=(\mathcal{C}_e)_G$$ the $$G$$-de-equivariantization of the trivial component, then any modular zesting $$\tilde{\mathcal{C}}$$ will be a $$G$$-gauging of $$\mathcal{B}$$. In the other related construction, if $$\mathcal{C}_{pt}\cap\mathcal{C}_e\cong\text{Rep}(G)$$ as above, one may construct new categories as tensor products over $$G$$ by condensing the diagonal algebra in $$\mathcal{C}\boxtimes\text{Rep}(D^\omega G)$$. In comparision with the latter two constructions, zesting has several features that these do not, like having explicit fusion rules, producing new categories not necessarily modular or braided, zesting only depends on essentially cohomological choices, and if the resulting category is modular there are explicit formulas for the modular data. We now describe zesting more specifically, without entering into full details. Being more specific, but not entering into full details, Given a faithfully $$G$$-graded fusion category, an \textit{associative $$G$$-zesting} consists of a map $$\lambda: G\times G\to \mathcal{C}_e\cap\mathcal{C}_{pt}$$, and for all $$g,h,k\in G$$ an isomorphism $$\lambda_{g,h,k}$$ satisfying certain equation, called \textit{associative zesting constraint}, which is satisfied for every four elements $$g,h,k,l\in G$$, so that both $$\lambda$$'s satisfy normalization conditions. In short, $$\lambda: G\times G\to \mathcal{C}_e\cap\mathcal{C}_{pt}$$ and $$\lambda_{g,h,k}$$ are specific normalized 2- and 3-cocycle, respectively. In Proposition 3.4 it is proved that under these conditions (1) equips $$\mathcal{C}^\lambda=(\mathcal{C}, \stackrel{\lambda}{\otimes}, 1)$$ with a new faithfully $$G$$-graded fusion category structure. In particular, for modular $$\mathbb Z/N$$-graded category $$SU(N)_k$$, taking the trivial 2-cocycle $$\lambda$$, the associative zesting constraint means that $$\lambda_{g,h,k}$$ is a normalized 3-cocycle on $$\mathbb Z/N$$, implying that such associative zestings of $$SU(N)_k$$ recover those obtained in [\textit{D. Kazhdan} and \textit{H. Wenzl}, Reconstructing monoidal categories'', Adv. Soviet Math. 16, 111--136 (1993)] by twisting the associativity morphisms. Dual objects for $$\stackrel{\lambda}{\otimes}$$ are defined in Section 3.3 leading to rigid associative zestings. For a braided faithfully $$G$$-graded fusion category $$\mathcal{B}$$, where $$G$$ is an abelian group, given an associative zesting $$\lambda$$ a \textit{braided zesting} is defined by an isomorphism $$t_{g,h}$$ associated to $$\lambda$$ for each $$g,h\in G$$ and a function $$j: G\to\text{Aut}_\otimes(\text{Id}_{\mathcal{B}})$$ to the abelian group of tensor natural isomorphisms of the identity functor, so that $$(\lambda, j,t)$$ satisfy braided zesting conditions (BZ1) and (BZ2) and normalization conditions. Under these conditions, in Proposition 4.4 a braiding for the fusion category $$\mathcal{B}^\lambda$$ is given. For a premodular tensor category $$\mathcal{B}$$ with a braided zesting $$(\mathcal{B}^\lambda, t)$$ denotes the corresponding braided fusion category. If one takes trivial 2- and 3-cocycles $$\lambda$$, then the braided zestings are simply modifying the braiding by a bicharacter, which is a well-known process. Similarity and equivalence of braided zestings is defined, so that in Proposition 4.7 three claims are proved of the following unifying form. The set of (equivalence classes of) all braided $$G$$-zestings of the form, or similar to, $$(\lambda, j,t)$$ is the torsor over certain groups. First and second partial obstructions to the existence of a function $$j: G\to\text{Aut}_\otimes(\text{Id}_{\mathcal{B}})$$ satisfying (BZ1) are described in Section 4.2. These obstructions automatically vanish when $$G$$ is the universal grading (the notion introduced in [\textit{S. Gelaki} and \textit{D. Nikshych}, Adv. Math. 217, No. 3, 1053--1071 (2008; Zbl 1168.18004)]). For a fixed associative zesting $$\lambda$$ such that the first and second partial obstructions vanish, in order to solve the equation (BZ1) certain 2-cocycle $$O_1(\lambda):G\times G\to k^\times$$ is defined and it is shown in Lemma 4.12 that (BZ1) can be solved if and only if the cohomology class of $$O_1(\lambda)$$ vanishes. Assuming that the latter cohomology class vanishes, certain 2-cocycle $$O_2(\lambda)$$ is defined. In Theorem 4.15 it is proved that given an associative zesting $$\lambda$$, there is to it associated braided zesting if and only if the cohomology class of $$O_1(\lambda)$$ vanishes and $$O_2(\lambda)$$ has a specific computed form. Starting from a finite abelian group $$G$$, a faithfully $$G$$-graded braided tensor category $$\mathcal{B}$$ with twist $$\theta$$, a braided zesting $$(\lambda, j,t)$$ and a function $$f:G\to l^\times$$, in Proposition 5.1 a twist $$\theta^f$$ for $$(\mathcal{B}^\lambda, t)$$ is characterized, as well as when it is a ribbon twist if so is $$\theta$$ for $$\mathcal{B}$$. A quadruple $$(\lambda, j, t, f )$$ where $$f$$ satisfies these two characterizations is called a \textit{ribbon zesting}, and $$(\mathcal{B}^\lambda, t, f)$$ denotes the twist (ribbon) zesting. It is not known if twists or ribbons for braided zesting always exist, though in Remark 5.3 an easy to check requirement for their existence is provided. For a ribbon zesting $$(\lambda, j, t, f )$$ in Proposition 5.5 a quantum trace of an endomorphism in $$(\mathcal{B}^\lambda, t, f)$$ is given, while in Theorem 5.7 the $$T$$- and $$S$$-matrix from the modular data of $$(\mathcal{B}^\lambda, t, f)$$ are computed. In Section 5.2 it is shown that Müger center is not zesting invariant and in Proposition 5.12 sufficient conditions in order for the Müger center of a braided fusion category $$\mathcal{B}$$ and of $$(\mathcal{B}^\lambda, t)$$ to coincide are given. In Section 5.3 the associated braid group representations of a braided fusion category are given, and in Theorem 5.15 the actions of the braid group representations are computed. Section 6 deals with applications to the author's main case of interest, which is for a modular category $$\mathcal{B}$$ and $$G=U(\mathcal{B})$$ the universal grading. In this case $$\mathcal{B}_e\cap\mathcal{B}_{pt}$$ is a symmetric pointed fusion category, that is, it is of the form $$Vec_S$$ with $$S$$ an abelian group and the braiding is given by the twist or minus twist. Given a $$G$$-graded braided fusion category $$\mathcal{B}$$ such that $$Vec_S\subset(\mathcal{B}_e)_{pt}$$, in Proposition 6.1 the obstruction to the existence of an associative zesting is computed, so that it vanishes if the order of $$S$$ is odd. In Section 6.2.2 for a braided pointed fusion category $$\mathcal{C}(C,\Theta)$$ where $$\text{Inv}(\mathcal{B})=C$$ (the group of isomorphism classes of invertible objects in $$\mathcal{B}$$ under the tensor product) is a cyclic group $$C$$ and $$\Theta$$ is a ribbon structure, it is characterized in terms of $$\Theta$$ when $$\mathcal{C}(C,\Theta)$$ is modular, symmetric, Tannakian, and super-Tanankian. In the symmetric case the ribbon is a character and the braiding is the twist or minus twist. Proposition 6.3 gives a parametrization of the equivalence classes of associative zestings of a braided fusion category $$\mathcal{B}$$ under certain conditions involving a cyclic group $$C$$. Furthermore, Proposition 6.4 characterizes when a pair $$(\lambda_a,\lambda_b)$$ of associative zestings constructed in the previous Proposition admits a braided zesting with $$j=id$$. In this case $$N$$ different braided zestings are given explicitly, where $$N$$ is the order of $$C$$. Being $$(\lambda_a,\lambda_b, j=id, t_s)$$ a braided zesting constructed in Proposition 6.4, for $$s=1,\dots,N$$, and if $$\mathcal{B}$$ has a ribbon twist obeying a certain condition, in Proposition 6.5 ribbon zesting and modular data $$T$$ and $$S$$ are constructed. In Section 6.3 it is explained how modular categories $$SU(N)_k$$, which are obtained from quantum groups $$U_Q\mathfrak{sl}_N$$ for $$Q=e^{\pi i/(N+k)}$$, fit the setting of Section 6.2.2 with $$C=\mathbb Z/N$$, then the results of Section 6.1 are applied to $$SU(N)_k$$. Concretely, the category $$SU(N)_k$$ has a (maximally) pointed subcategory $$\mathcal{P}(N, k)$$ with fusion rules like $$\mathbb Z/N$$, and $$SU(N)_k$$ is (universally) $$\mathbb Z/N$$-graded. The subcategory $$\mathcal{P}(N, k)$$ can be identified with the pointed ribbon fusion category $$\mathcal{C}(\mathbb Z/N,\eta)$$, where $$\eta$$ is a quadratic form given by the twists, which are computed using standard techniques. It is argued that $$\mathcal{P}(N, k)$$ is modular if and only if $$(k,N)=1$$. On the other hand, if $$k=\alpha N$$, the following is found. If $$N$$ is odd, the categories $$\mathcal{P}(N, \alpha N)$$ are Tannakian, but if $$N$$ is even, $$\mathcal{P}(N, \alpha N)$$ is Tannakian if and only if $$\alpha$$ is even, and super-Tannakian otherwise. Moreover, if $$N$$ is odd, there are at most $$N^2$$ distinct ribbon twist braided zestings, all of which are modular, while if $$N$$ is even and $$\alpha$$ odd there are at most $$4N^2$$ ribbon braided zestings, each of which is modular. It is pointed out that there can be equivalences among braided zestings, which is illustrated by some examples. For $$SU(3)_3$$ it is argued that from at most 9 modular categories obtained from $$\mathbb Z/3$$-zesting of $$SU(3)_3$$ there are only 3 inequivalent sets of modular data, and presumably only 3 inequivalent modular categories (this is not immediate as modular data is not a complete invariant). Zesting of $$SU(3)_3$$ is compared to gauging construction, and consistently 3 gaugings are obtained. For $$SU(4)_4$$ eight associative zestings are found, each of them admitting 4 braided zestings, and each of these a unique ribbon structure which is also modular. Thus there are at most 32 distinct modular categories obtained as $$\mathbb Z/4$$-zestings of $$SU(4)_4$$. Also $$\mathbb Z/2$$-zestings of $$SU(4)_4$$ are considered, in which case for the whole of two associative zestings it is proved that no braided zesting exists. Finally for $$SU(4)_2$$ also 8 associative zestings are found, there are 16 braided zestings and 32 ribbon zestings. In Section 7 the authors explain the difference between their braided zestings and those studied in [\textit{A. Davydov} and \textit{D. Nikshych}, Braided Picard groups and graded extensions of braided tensor categories'', Preprint, \url{arXiv:2006.08022}]. In the latter reference some particular braided zestings are interpreted as deformations of a braided monoidal 2-functor and its obstruction as an element in an Eilenberg-MacLane cohomology group. This is the obstruction for a symmetric 2-cocycle to admit a braided zesting, while the cohomological obstructions $$O_1(\lambda)$$ and $$O_2(\lambda)$$ in Theorem 4.15 in the present article are the obstructions that a \textit{fixed associative zesting admits a braided zesting}. Thus, to compute the EM-obstruction one needs first to describe the associative zestings and then check that braided zesting obstructions (BZ1) and (BZ2) vanish. The article finishes with prospects for future applications: a) apply associative zesting to fusion categories that do not admit a braiding and braided zesting with respect to non-universal grading groups and non-cyclic grading groups, b) it is worth investigating if zesting has a meaningful physical interpretation, as it is known that symmetry gauging corresponds to phase transitions of topological phases of matter. Simple transitive 2-representations via (co)algebra 1-morphisms https://zbmath.org/1472.18019 2021-11-25T18:46:10.358925Z "Mackaay, Marco" https://zbmath.org/authors/?q=ai:mackaay.marco "Mazorchuk, Volodymyr" https://zbmath.org/authors/?q=ai:mazorchuk.volodymyr "Miemietz, Vanessa" https://zbmath.org/authors/?q=ai:miemietz.vanessa "Tubbenhauer, Daniel" https://zbmath.org/authors/?q=ai:tubbenhauer.daniel Summary: For any fiat $$2$$-category $$\mathcal{C}$$, we show how its simple transitive $$2$$-representations can be constructed using co-algebra $$1$$-morphisms in the injective abelianization of $$\mathcal{C}$$. Dually, we show that these can also be constructed using algebra $$1$$-morphisms in the projective abelianization of $$\mathcal{C}$$. We also extend Morita-Takeuchi theory to our setup and work out several examples, including that of Soergel bimodules for dihedral groups, explicitly. Homology of categories via polygraphic resolutions https://zbmath.org/1472.18020 2021-11-25T18:46:10.358925Z "Guetta, Léonard" https://zbmath.org/authors/?q=ai:guetta.leonard \textit{R. Street} [J. Pure Appl. Algebra 49, 283--335 (1987; Zbl 0661.18005)] defined a nerve functor $N_{\omega}:\omega\boldsymbol{Cat}\rightarrow\widehat{\Delta}$ from the category of strict $$\omega$$-categories (called simply $$\omega$$\textit{-categories}) to the category of simplicial sets, which can be used to transfer the homotopy theory of simplicial sets to $$\omega$$-categories [\textit{D. Ara} and \textit{G. Maltsiniotis}, Adv. Math. 259, 557--654 (2014; Zbl 1308.18004); Adv. Math. 328, 446--500 (2018; Zbl 1390.18011); High. Struct. 4, No. 1, 284--388 (2020; Zbl 07173321); \textit{A. Gagna}, Adv. Math. 331, 542--564 (2018; Zbl 1395.18008); J. Lond. Math. Soc., II. Ser. 100, No. 2, 470--497 (2019; Zbl 1430.18021); \textit{P. Ara} et al., Banach J. Math. Anal. 14, No. 4, 1692--1710 (2020; Zbl 1454.46052); \textit{D. Ara} and \textit{G. Maltsiniotis}, Mém. Soc. Math. Fr., Nouv. Sér. 165, 1--203 (2020; Zbl 07362646)]. In particular, we have the following definition. Definition. Let $$C$$ be an $$\omega$$-category and $$k\in\mathbb{N}$$. The $$k$$-th homology group $$H_{k}(C)$$ of $$C$$ is the $$k$$-th homology group of its nerve $$N_{\omega}(C)$$. On the other hand, \textit{F. Métayer} [Theory Appl. Categ. 11, 148--184 (2003; Zbl 1020.18001)] deined \textit{polygraphic homology groups}, observing that \begin{itemize} \item[(a)] Every $$\omega$$-category admits a polygraphic resolution that is an arrow $u:P\rightarrow C$ of $$\omega\boldsymbol{Cat}$$, such that $$P$$ is a free $$\omega$$-category and $$u$$ abides by some properties of formal similarities with trivial fibrations of topological spaces. \item[(b)] Every free $$\omega$$-category $$P$$ is to be linearized to a chain complex $$\lambda(P)$$. \item[(c)] Given two polygraphic resolutions $$P\rightarrow C$$ and $$P^{\prime}\rightarrow C$$ of the same free $$\omega$$-category, the homology groups of the chain complexes $$\lambda(P)$$ and $$\lambda(P^{\prime})$$ coincide. \end{itemize} Definition. Let $$C$$ be an $$\omega$$-category and $$k\in\mathbb{N}$$. The $$k$$-th polygraphic homology group $$H_{k}^{\mathrm{pol}}(C)$$ of $$C$$ is the $$k$$-th homology group of $$\lambda(P)$$ for any polygraphic resolution $$P\rightarrow C$$. The principal objective in this paper is to establish the following theorem. Theorem. Let $$C$$ be an $$\omega$$-category. For every $$k\in\mathbb{N}$$, we have $H_{k}(C)\simeq H_{k}^{\mathrm{pol}}(C).$ The restriction of the above theorem to the case of monoids is precisely Corollary 3 of [\textit{Y. Lafont} and \textit{F. Métayer}, J. Pure Appl. Algebra 213, No. 6, 947--968 (2009; Zbl 1169.18002), \S 3.4], but the author's novelty lies in his more conceptual proof than theirs. Besides, the actural result in this paper (Theorem 8.3) is more precise than the above theorem in that \begin{itemize} \item[(a)] The homology of an $$\omega$$-category, whether polygraphic or of the nerve, is considered as a chain complex up to quasi-isomorphism, but not only a sequence of abelian groups. \item[(b)] It is established that the polygraphic homology and homology of the nerve of a small category are naturally isomorphic with the natural isomorphism explicitly constructed. \end{itemize} Gray tensor products and lax functors of $$(\infty, 2)$$-categories https://zbmath.org/1472.18021 2021-11-25T18:46:10.358925Z "Gagna, Andrea" https://zbmath.org/authors/?q=ai:gagna.andrea "Harpaz, Yonatan" https://zbmath.org/authors/?q=ai:harpaz.yonatan "Lanari, Edoardo" https://zbmath.org/authors/?q=ai:lanari.edoardo The category of $$2$$-categories carries a monoidal structure, given by the \textit{Gray tensor product} [\textit{J. W. Gray}, Formal category theory: Adjointness for 2-categories. Berlin-Heidelberg-New York: Springer-Verlag (1974; Zbl 0285.18006)]. As is often the case, the monoidal structure comes in a few guises, Gray's original version being a pseudo version while there being a lax and an oplax versions. While the cartesian product is badly-behaved with respect to the folk model category on $$2$$-categories, it was shown by \textit{S. Lack} [$$K$$-Theory 26, No. 2, 171--205 (2002; Zbl 1017.18005)] that the pseudo Gray tensor product is compatible with the folk model category, being also the case for the lax and oplax versions of the Gray tensor product, as was recently demonstrated by \textit{D. Ara} and \textit{M. Lucas} [Theory Appl. Categ. 35, 745--808 (2020; Zbl 1443.18008)]. This paper introduces and studies a particularly well-behaved model of the (oplax) Gray tensor product for $$\left( \infty,2\right)$$-categories, working in the category of scaled simplicial sets equipped with the bicategorical model structure [\textit{J. Lurie}, $$(\infty,2)$$-categories and the Goodwillie calculus. I'', Preprint, \url{arXiv:0905.0462}; the first author et al., On the equivalence of all models for $$(\infty,2)$$-categories'', Preprint, \url{arXiv:1911.01905}] and providing a Gray tensor product which is associative on-the-nose left Quillen bifunctor. The main motivation behind the authors' interest in the Gray tensor product comes from the recent book [A study in derived algebraic geometry. Volume I: Correspondences and duality. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1408.14001); A study in derived algebraic geometry. Volume II: Deformations, Lie theory and formal geometry. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1409.14003)], where \textit{D. Gaitsgory} and \textit{N. Rozenblyum} make use of the Gray tensor product in order to develop a formalism of categories of correspondences. A synopsis of the paper consisting of three sections goes as follows. \begin{itemize} \item \S 1 fixes the notation, recalling the categories of marked and scaled simplicial sets with their respective marked categorical and bicategorical model structures. \item \S 2 introduces the Gray tensor product and the closely related notion of (op)lax transformations, studying some of its basic properties. \S 2.2 compares the definition of the Gray tensor product in this paper with that of \textit{D. R. B. Verity} [Adv. Math. 219, No. 4, 1081--1149 (2008; Zbl 1158.18007); Contemp. Math. 431, 441--467 (2007; Zbl 1137.18005)] within the setting of complicial sets under the Quillen equivalence between the two models established in [\textit{A. Gagna} et al., On the equivalence of all models for $$(\infty,2)$$-categories'', Preprint, \url{arXiv:1911.01905}]. \S 2.3 establishes that the Gray tensor product in this paper is left Quillen bifunctor, which is the main result of the paper. \item \S 3 is bestowed on establishing a universal mapping property for the Gray product in terms of a suitable notion of oplax functors, which is generalized from classical $$2$$-category theory to the setting of $$\infty$$-bicategories. \end{itemize} $$\mathrm{SO}(N)_2$$ braid group representations are Gaussian https://zbmath.org/1472.20075 2021-11-25T18:46:10.358925Z "Rowell, Eric C." https://zbmath.org/authors/?q=ai:rowell.eric-c "Wenzl, Hans" https://zbmath.org/authors/?q=ai:wenzl.hans Summary: We give a description of the centralizer algebras for tensor powers of spin objects in the pre-modular categories $$\mathrm{SO}(N)_2$$ (for $$N$$ odd) and $$\mathrm{O}(N)_2$$ (for $$N$$ even) in terms of quantum $$(n-1)$$-tori, via non-standard deformations of $$\mathrm{U}\mathfrak{so}_N$$. As a consequence we show that the corresponding braid group representations are Gaussian representations, the images of which are finite groups. This verifies special cases of a conjecture that braid group representations coming from weakly integral braided fusion categories have finite image. Categories of two-colored pair partitions. Part II: Categories indexed by semigroups https://zbmath.org/1472.46072 2021-11-25T18:46:10.358925Z "Mang, Alexander" https://zbmath.org/authors/?q=ai:mang.alexander "Weber, Moritz" https://zbmath.org/authors/?q=ai:weber.moritz Summary: Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to half-liberated easy'' interpolations between the unitary group and Wang's quantum counterpart. We complete the classification of all such categories demonstrating that the subcategories of a certain natural halfway point are equivalent to additive subsemigroups of the natural numbers; the categories above this halfway point have been classified in a preceding article. We achieve this using combinatorial means exclusively. Our work reveals that the half-liberation procedure is quite different from what was previously known from the orthogonal case. For Part I; see [\textit{A.~Mang} and \textit{M.~Weber}, Ramanujan J. 53, No.~1, 181--208 (2020; Zbl 07343721)]. Distinguishing open symplectic mapping tori via their wrapped Fukaya categories https://zbmath.org/1472.53095 2021-11-25T18:46:10.358925Z "Kartal, Yusuf Barış" https://zbmath.org/authors/?q=ai:kartal.yusuf-baris Summary: We present partial results towards a classification of symplectic mapping tori using dynamical properties of wrapped Fukaya categories. More precisely, we construct a symplectic manifold $$T_\phi$$ associated to a Weinstein domain $$M\!$$, and an exact, compactly supported symplectomorphism $$\phi$$. The symplectic manifold $$T_\phi$$ is another Weinstein domain and its contact boundary is independent of $$\phi$$. We distinguish $$T_\phi$$ from $$T_{1_M}$$, under certain assumptions (Theorem 1.1). As an application, we obtain pairs of diffeomorphic Weinstein domains with the same contact boundary and whose symplectic cohomology groups are the same, as vector spaces, but that are different as Liouville domains. To our knowledge, this is the first example of such pairs that can be distinguished by their wrapped Fukaya category. Previously, we have suggested a categorical model $$M_\phi$$ for the wrapped Fukaya category $$\mathcal{W}(T_\phi)$$, and we have distinguished $$M_\phi$$ from the mapping torus category of the identity. We prove $$\mathcal{W}(T_\phi)$$ and $$M_\phi$$ are derived equivalent (Theorem 1.9); hence, deducing the promised Theorem 1.1. Theorem 1.9 is of independent interest as it preludes an algebraic description of wrapped Fukaya categories of locally trivial symplectic fibrations as twisted tensor products. The localic compact interval is an Escardó-Simpson interval object https://zbmath.org/1472.54015 2021-11-25T18:46:10.358925Z "Vickers, Steven" https://zbmath.org/authors/?q=ai:vickers.steven Summary: The locale corresponding to the real interval $$[ - 1, 1]$$ is an interval object, in the sense of Escardó and Simpson, in the category of locales. The map $$c : 2^\omega \to [-1, 1]$$, mapping a stream $$s$$ of signs $$\pm 1$$ to $$\sum_{i = 1}^\infty s_i 2^{- i}$$, is a proper localic surjection; it is also expressed as a coequalizer. The proofs are valid in any elementary topos with natural numbers object. Ephemeral persistence modules and distance comparison https://zbmath.org/1472.55003 2021-11-25T18:46:10.358925Z "Berkouk, Nicolas" https://zbmath.org/authors/?q=ai:berkouk.nicolas "Petit, François" https://zbmath.org/authors/?q=ai:petit.francois The paper studies multiparameter persistence modules by a sheaf-theoretic approach. Extracting appropriate features from multiparameter persistence modules is difficult and many approaches have been proposed, for example, using polynomial rings and quiver representations. Another approach using sheaf theory is proposed by [\textit{J. Curry}, Sheaves, cosheaves and applications. (Ph.D. thesis) University of Pennsylvania (2014), \url{arXiv:1303.3255}] and [\textit{M. Kashiwara} and \textit{P. Schapira}, J. Appl. Comput. Topol. 2, No. 1-2, 83--113 (2018; Zbl 1423.55013)]. The current paper reveals more about the significance of sheaf theory, in particular, $$\gamma$$-sheaves, for studying persistent homology. One of the aims of the paper is to compare the two approaches of Curry and Kashiwara-Schapira, where the former uses Alexandrov topology and the latter uses $$\gamma$$-topology, where $$\gamma$$ is a closed convex proper cone in a finite-dimensional real vector space $$\mathbb{V}$$. The authors consider a morphism of sites between Alexandrov and $$\gamma$$-topology spaces $$\beta \colon \mathbb{V}_\mathfrak{a} \to \mathbb{V}_\gamma$$, which induces a functor between the categories of sheaves of $$\mathbf{k}$$-vector spaces $$\beta_* \colon \mathrm{Mod}(\mathbf{k}_{\mathbb{V}_\mathfrak{a}}) \to \mathrm{Mod}(\mathbf{k}_{\mathbb{V}_\gamma})$$. The category of ephemeral modules $$\mathrm{Eph}(\mathbf{k}_{\mathbb{V}_\mathfrak{a}})$$ is now defined as the kernel of the functor. In this way, the notion of ephemeral modules can be defined in arbitrary dimension and coincides with the definition of [\textit{F. Chazal} et al., The structure and stability of persistence modules. Cham: Springer (2016; Zbl 1362.55002)] in the one-dimensional case. An equivalence between $$\mathrm{Mod}(\mathbf{k}_{\mathbb{V}_\mathfrak{a}})/\mathrm{Eph}(\mathbf{k}_{\mathbb{V}_\mathfrak{a}})$$ and $$\mathrm{Mod}(\mathbf{k}_{\mathbb{V}_\gamma})$$ is also obtained, which shows the equivalence between the observable category and $$\gamma$$-sheaves. The authors extend the results to the derived setting. The latter half of the paper studies the distances on the two categories. For a fixed vector $$v$$, both categories admit interleaving distances to the $$v$$-direction. The authors show an isometry theorem on these metrics. Finally, the paper investigates the relation between the convolution distance defined by Kashiwara-Schapira and the interleaving distance on the category of sheaves on $$\mathbb{V}_\gamma$$. The convolution distance depends on the norm on $$\mathbb{V}$$, whereas the interleaving distance depends on $$v$$. The authors introduce a preferred norm and prove an isometry theorem on these metrics under a mild assumption. Persistent local systems https://zbmath.org/1472.55004 2021-11-25T18:46:10.358925Z "MacPherson, Robert" https://zbmath.org/authors/?q=ai:macpherson.robert-d "Patel, Amit" https://zbmath.org/authors/?q=ai:patel.amit-j|patel.amit-m|patel.amit-d|patel.amit-k This paper presents a generalization of persistent homology to situations where the rank function takes values in a manifold. For this purpose a persistent local system is constructed which depends on information of a sheaf and a cosheaf combined in a bisheaf. A new tool, the isobisheaf stack, provides functorial properties. The authors show the values are stable under small pertubation of maps. In conclusion a few examples are computed. Smoothness filtration of the magnitude complex https://zbmath.org/1472.55006 2021-11-25T18:46:10.358925Z "Gomi, Kiyonori" https://zbmath.org/authors/?q=ai:gomi.kiyonori Let $$(X, d)$$ be a metric space. A sequence of points $$\langle x_0, x_1, \dots, x_n\rangle$$ ($$x_i\in X$$) is said to be a proper $$n$$-chain of length $$\ell$$ if $$x_{i-1}\neq x_i$$ ($$i=1, \dots, n$$) and $$d(x_0, x_1)+d(x_1, x_2)+\dots +d(x_{n-1}, x_n)=\ell$$. Let $$C_n^\ell(X)$$ be the free abelian group generated by proper $$n$$-chains of length $$\ell$$. One can naturally define the boundary map $$\partial : C_n^\ell(X)\longrightarrow C_{n-1}^\ell(X)$$ whose homology group $$H_n^\ell(X):=H_n(C_*^\ell(X))$$ is called the \textit{magnitude homology} of $$(X, d)$$. The notion of magnitude homology was first introduced by \textit{R. Hepworth} and \textit{S. Willerton} [Homology Homotopy Appl. 19, No. 2, 31--60 (2017; Zbl 1377.05088)] for a finite metric space defined by a graph. Later, it was generalized to a metric space (furthermore, enriched category) by \textit{T. Leinster} and \textit{M. Shulman} [Magnitude homology of enriched categories and metric spaces'', Preprint, \url{arXiv:1711.00802}]. The computation of magnitude homology is, in general, difficult. In particular, if there exists a \textit{$$4$$-cut}, that is a chain $$\langle x_0, x_1, x_2, x_3\rangle$$ satisfying $\begin{split} d(x_0, x_3) &< d(x_0, x_1) + d(x_1, x_2) + d(x_2, x_3)\\ &= d(x_0, x_2) + d(x_2, x_3) = d(x_0, x_1) + d(x_1, x_3), \end{split}$ then the computation becomes complicated. Indeed, a previous work by \textit{R. Kaneta} and \textit{M. Yoshinaga} [Bull. Lond. Math. Soc. 53, No. 3, 893--905 (2021; Zbl 1472.55007)] showed that if the metric space $$(X, d)$$ does not have $$4$$-cuts, then the computation of the magnitude homology is reduced to that of the order complexes for posets. The present paper extends and refines the previous works by using spectral sequences. In a proper chain $$\langle x_0, x_1, \dots, x_n\rangle$$, a point $$x_i$$ is said to be a \textit{smooth point} if $$d(x_{i-1}, x_{i})+d(x_{i}, x_{i+1})=d(x_{i-1}, x_{i+1})$$ (otherwise, it is called a \textit{singular point}). Denote the number of smooth points in $$x$$ by $$\sigma(x)$$ and the submodule of $$C_n^\ell(X)$$ generated by proper chains with $$\sigma(x)\leq p$$ by $$F_pC_n^\ell(X)$$. Then, $$F_pC_*^\ell(X)$$ defines a filtration on the magnitude chain complex $$C_*^\ell(X)$$. The associated spectral sequence is the main object of the present paper. The author completely describes at which page the spectral sequence degenerates. Precise results are as follows. \begin{itemize} \item[(a)] The spectral sequence always degenerates at $$E^4$$. \item[(b)] The spectral sequence degenerates at $$E^2$$ if and only if the metric space $$(X, d)$$ does not contain $$4$$-cuts. \item[(c)] The spectral sequence degenerates at $$E^3$$ if and only if there does not exist a chain $$x=\langle x_0, x_1, x_2, x_3, x_4\rangle$$ such that both $$\langle x_0, x_1, x_2, x_3\rangle$$ and $$\langle x_1, x_2, x_3, x_4\rangle$$ are $$4$$-cuts. \end{itemize} The author also applies the spectral sequence to a number of concrete examples. A simplicial approach to stratified homotopy theory https://zbmath.org/1472.55014 2021-11-25T18:46:10.358925Z "Douteau, Sylvain" https://zbmath.org/authors/?q=ai:douteau.sylvain The author constructs a combinatorial model structure of stratified spaces. It uses the adjunction between stratified spaces and simplicial sets filtered over a poset. Applications are given, for instance to computations of filtered homotopy groups. Staggered and affine Kac modules over $$A_1^{(1)}$$ https://zbmath.org/1472.81115 2021-11-25T18:46:10.358925Z "Rasmussen, Jørgen" https://zbmath.org/authors/?q=ai:rasmussen.jorgen-born|rasmussen.jorgen|rasmussen.jorgen-h Summary: This work concerns the representation theory of the affine Lie algebra $$A_1^{(1)}$$ at fractional level and its links to the representation theory of the Virasoro algebra. We introduce affine Kac modules as certain finitely generated submodules of Wakimoto modules. We conjecture the existence of several classes of staggered $$A_1^{(1)}$$-modules and provide evidence in the form of detailed examples. We extend the applicability of the Goddard-Kent-Olive coset construction to include the affine Kac and staggered modules. We introduce an exact functor between the associated category of $$A_1^{(1)}$$-modules and the corresponding category of Virasoro modules. At the level of characters, its action generalises the Mukhi-Panda residue formula. We also obtain explicit expressions for all irreducible $$A_1^{(1)}$$-characters appearing in the decomposition of Verma modules, re-examine the construction of Malikov-Feigin-Fuchs vectors, and extend the Fuchs-Astashkevich theorem from the Virasoro algebra to $$A_1^{(1)}$$. Twisted differential $$K$$-characters and D-branes https://zbmath.org/1472.81192 2021-11-25T18:46:10.358925Z "Ferrari Ruffino, Fabio" https://zbmath.org/authors/?q=ai:ferrari-ruffino.fabio "Rocha Barriga, Juan Carlos" https://zbmath.org/authors/?q=ai:rocha-barriga.juan-carlos Summary: We analyse in detail the language of partially non-abelian Deligne cohomology and of twisted differential $$K$$-theory, in order to describe the geometry of type II superstring backgrounds with D-branes. This description will also provide the opportunity to show some mathematical results of independent interest. In particular, we begin classifying the possible gauge theories on a D-brane or on a stack of D-branes using the intrinsic tool of long exact sequences. Afterwards, we recall how to construct two relevant models of differential twisted $$K$$-theory, paying particular attention to the dependence on the twisting cocycle within its cohomology class. In this way we will be able to define twisted $$K$$-homology and twisted Cheeger-Simons $$K$$-characters in the category of simply-connected manifolds, eliminating any unnatural dependence on the cocycle. The ambiguity left for non simply-connected manifolds will naturally correspond to the ambiguity in the gauge theory, following the previous classification. This picture will allow for a complete characterization of D-brane world-volumes, the Wess-Zumino action and topological D-brane charges within the $$K$$-theoretical framework, that can be compared step by step to the old cohomological classification. This has already been done for backgrounds with vanishing $$B$$-field; here we remove this hypothesis.