Recent zbMATH articles in MSC 18https://www.zbmath.org/atom/cc/182021-02-27T13:50:00+00:00WerkzeugCancellative conjugation semigroups and monoids.https://www.zbmath.org/1453.200852021-02-27T13:50:00+00:00"Garrão, A. P."https://www.zbmath.org/authors/?q=ai:garrao.a-paula"Martins-Ferreira, N."https://www.zbmath.org/authors/?q=ai:martins-ferreira.nelson"Raposo, M."https://www.zbmath.org/authors/?q=ai:raposo.maria-cristina|raposo.marta"Sobral, M."https://www.zbmath.org/authors/?q=ai:sobral.manuelaA \textit{conjugation semigroup} \((S, +,\overline{()})\) is defined as a semigroup \((S,+)\) equipped with a unary operation \(\overline{()}\) such that the following identities are satisfied:
\begin{itemize}
\item[1)] \(\overline{x}+x =x+\overline{x}\),
\item[2)] \(x+\overline{y}+y=y+\overline{y}+x\),
\item[3)] \(\overline{x+y}=\overline{y}+\overline{x}\).
\end{itemize}
Let
\(A \xrightarrow{f} B\xleftarrow{g} C\) and \(A\xleftarrow{r} B\xrightarrow{s} C\) together with homomorphisms \(\alpha: A\to D\), \(\beta:B\to D\), \(\gamma: C\to D\) be a diagram such that \(fr=1_B=gs\) and \(\alpha r=\beta=\gamma s\). Then this diagram is called \textit{admissible} if there exists a unique morphism \(\varphi: A\times_B C\to D\) such that \(\varphi e_1=\alpha\) and \(\varphi e_2=\beta\), where \(e_1=\langle 1_A, sf\rangle\) and \(e_2=\langle rg, 1_C\rangle\). A finitely complete category \(\mathcal C\) is called \textit{weakly Mal'tsev} if the morphisms \(e_1\) and \(e_2\) are jointly epimorphic. A split epimorphism \((f,r)\), \(f: A\to B\), \(fr=1_B,\) is called a \textit{Schreier split epimorphism} if there exists a unique set-theoretical map \(q: A\to X\) (\(X\) -- the kernel of \(f\)) such that, for every \(a\in A\), \(a=kq(a)+rf(a)\), where \(k: X\to A\) is the kernel morphism. Admissible diagrams in the category \(\mathcal S\) of cancelative conjugation semigroups as well as Schreier split epimorphisms in the category \(\mathcal M\) of cancellative conjugation monoids are described. The category \(\mathcal S\) turnes out to be weakly Mal'tsev. Conditions are found under which for a given Schreier split epimorphism, a morphism \(X\to B\) induces a reflexive graph, an internal category or an internal groupoid. Two Schreier equivalence relations on an object in \(\mathcal M\) commute if and only if their normalizations commute.
Reviewer: Peeter Normak (Tallinn)Twisting structures and morphisms up to strong homotopy.https://www.zbmath.org/1453.180222021-02-27T13:50:00+00:00"Hess, Kathryn"https://www.zbmath.org/authors/?q=ai:hess.kathryn-p"Parent, Paul-Eugène"https://www.zbmath.org/authors/?q=ai:parent.paul-eugene"Scott, Jonathan"https://www.zbmath.org/authors/?q=ai:scott.jonathan-aSummary: We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the ``strong homotopy'' morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads \(\mathcal{A}^\perp \rightarrow \mathbf{B}\mathcal{A}\), which is exactly the two-sided Koszul resolution of the associative operad \(\mathcal{A}\), also known as the Alexander-Whitney co-ring.Direct limits of adèle rings and their completions.https://www.zbmath.org/1453.111512021-02-27T13:50:00+00:00"Kelly, James P."https://www.zbmath.org/authors/?q=ai:kelly.james-pierre"Samuels, Charles L."https://www.zbmath.org/authors/?q=ai:samuels.charles-lFor a Galois extension \(E/F\), with F a global field, the paper defines a topological ring, denoted by \(\overline{\mathbb{V}}_E\), called the generalized adèle ring of \(E\).
Denote by \(\mathcal{J}_E\) the set \(\{K \subseteq E : K/F\text{ finite Galois}\}\), by \(\mathbb{A}_K\) the adèle ring of \(K\) and by \(\overline{\mathbb{A}}_K\) its completion with respect to some (any) invariant metric on \(\mathbb{A}_K\).
Main theorems are now stated in a short form.
Theorem 1. If \(E/F\) is a Galois extension, then the following hold:
\begin{itemize}
\item[i)] \(\overline{\mathbb{V}}_E\) is a metrizable topological ring which is complete with respect to any invariant metric on \(\overline{\mathbb{V}}_E\).
\item[ii)] If \(\mathbb{V}_E = \bigcup_{K\in\mathcal J_E} \mathbb{V}_K\), then \(\overline{\mathbb{V}}_E\) equals the closure of \(\mathbb{V}_E\) in \(\overline{\mathbb{V}}_E\).
\item[iii)] There exists a topological ring isomorphism \(\phi: \overline{\mathbb{A}}_E \to \overline{\mathbb{V}}_E\) such that \(\phi(\mathbb{A}_E) = \mathbb{V}_E\).
\end{itemize}
Theorem 2.
If \(E/F\) is an infinite Galois extension, then \(\mathbb{A}_E\) has empty interior in \(\overline{\mathbb{A}}_E\).
Reviewer: Stelian Mihalas (Timişoara)Some remarks on connectors and groupoids in Goursat categories.https://www.zbmath.org/1453.180042021-02-27T13:50:00+00:00"Gran, Marino"https://www.zbmath.org/authors/?q=ai:gran.marino"Rodelo, Diana"https://www.zbmath.org/authors/?q=ai:rodelo.diana"Tchoffo Nguefeu, Idriss"https://www.zbmath.org/authors/?q=ai:nguefeu.idriss-tchoffoSummary: We prove that connectors are stable under quotients in any (regular) Goursat category. As a consequence, the category \(\mathsf{Conn}(\mathbf{C})\) of connectors in \(\mathbf{C}\) is a Goursat category whenever \(\mathbf C\) is. This implies that Goursat categories can be characterised in terms of a simple property of internal groupoids.A ghost at \(\omega_1\).https://www.zbmath.org/1453.030622021-02-27T13:50:00+00:00"Levy, Paul Blain"https://www.zbmath.org/authors/?q=ai:levy.paul-blainSummary: In the final chain of the countable powerset functor, we show that the set at index \(\omega_1\), regarded as a transition system, is not strongly extensional because it contains a ``ghost'' element that has no successor even though its component at each successor index is inhabited. The method, adapted from a construction of \textit{M. Forti} and \textit{F. Honsell} [in: Partial differential equations and the calculus of variations. Essays in honor of Ennio De Giorgi. Volume I and II. Boston, MA etc.: Birkhäuser. 473--518 (1989; Zbl 0709.03030)], also gives ghosts at larger ordinals in the final chain of other subfunctors of the powerset functor. This leads to a precise description of which sets in these final chains are strongly extensional.On the tensor structure of modules for compact orbifold vertex operator algebras.https://www.zbmath.org/1453.170172021-02-27T13:50:00+00:00"McRae, Robert"https://www.zbmath.org/authors/?q=ai:mcrae.robertSuppose \(V\) is a vertex operator algebra and \(G\) is a compact Lie group acting faithfully and continuously (and by vertex operator algebra automorphisms) on \(V\); write \(V^G\) for the fixed subalgebra. \textit{C. Dong} et al. provided in [Int. Math. Res. Not. 1996, No. 18, 913--921 (1996; Zbl 0873.17028)] a Schur-Weyl type duality statement: \(V\) is semisimple as a \((V^G \times G)\)-module, and decomposes as \(V = \bigoplus_I V_I \otimes I\), where \(I\) ranges over the irreducible finite-dimensional \(G\)-modules, and \(V_I\) are nonzero distinct irreducible \(V^G\) modules. In particular, this theorem provides an identification of linear semisimple categories between \(\mathrm{Rep}(G)\) and the subcategory \(\mathcal{C}_V \subset \mathrm{Rep}(V^G)\) consisting of direct sums of the \(V_I\)'s.
The main result of the present paper improves this to an identification of braided monoidal categories. (The theorem is proved for arbitrary abelian intertwining algebras, which are a mild generalization of vertex operator algebras.) The only assumption needed is \(V^G\) indeed has a category of modules, containing \(\mathcal{C}_V\), with a braided tensor structure. This assumption is fairly deep in general: except in very special cases (unitary, strongly rational, etc.), the construction of braided tensor category structures on modules for vertex operator algebras requires the sophisticated theory of logarithmic vertex tensor categories developed by \textit{Y.-Z. Huang} and \textit{J. Lepowsky} [J. Phys. A, Math. Theor. 46, No. 49, Article ID 494009, 21 p. (2013; Zbl 1280.81125)], which in turn depends on subtle convergence properties of 4-point functions. Nevertheless, the assumption is known to hold for a vast assortment of examples. The proof uses a nice mixture of vertex-algebraic and tensor-categorical techniques.
Reviewer: Theo Johnson-Freyd (Waterloo)Diagonal \(p\)-permutation functors.https://www.zbmath.org/1453.180052021-02-27T13:50:00+00:00"Bouc, Serge"https://www.zbmath.org/authors/?q=ai:bouc.serge"Yılmaz, Deniz"https://www.zbmath.org/authors/?q=ai:yilmaz.denizSummary: Let \(k\) be an algebraically closed field of positive characteristic \(p\), and let \(\mathbb{F}\) be an algebraically closed field of characteristic 0. We consider the \(\mathbb{F} \)-linear category \(\mathbb{F} p p_k^{\Delta}\) of finite groups, in which the set of morphisms from \(G\) to \(H\) is the \(\mathbb{F} \)-linear extension \(\mathbb{F} T^{\Delta}(H, G)\) of the Grothendieck group \(T^{\Delta}(H, G)\) of \(p\)-permutation \((k H, k G)\)-bimodules with (twisted) diagonal vertices. The \(\mathbb{F} \)-linear functors from \(\mathbb{F} p p_k^{\Delta}\) to \(\mathbb{F} \text{-Mod}\) are called \textit{diagonal p-permutation functors}. They form an abelian category \(\mathcal{F}_{p p_k}^{\Delta} \). We study in particular the functor \(\mathbb{F} T^{\Delta}\) sending a finite group \(G\) to the Grothendieck group \(\mathbb{F} T(G)\) of \(p\)-permutation \textit{kG}-modules, and show that \(\mathbb{F} T^{\Delta}\) is a semisimple object of \(\mathcal{F}_{p p_k}^{\Delta} \), equal to the direct sum of specific simple functors parametrized by isomorphism classes of pairs \((P, s)\) of a finite \(p\)-group \(P\) and a generator \(s\) of a \(p^\prime \)-subgroup acting faithfully on \(P\). This leads to a precise description of the evaluations of these simple functors. In particular, we show that the simple functor indexed by the trivial pair \((1, 1)\) is isomorphic to the functor sending a finite group \(G\) to \(\mathbb{F} K_0(k G)\), where \(K_0(k G)\) is the Grothendieck group of projective \textit{kG}-modules.Homotopical inverse diagrams in categories with attributes.https://www.zbmath.org/1453.180072021-02-27T13:50:00+00:00"Kapulkin, Krzysztof"https://www.zbmath.org/authors/?q=ai:kapulkin.krzysztof"Lumsdaine, Peter LeFanu"https://www.zbmath.org/authors/?q=ai:lumsdaine.peter-lefanuThe present article deals with categories with attributes (CwA). This is one of several categorical formalisms that allow an algebraic treatment of type theories. They constitute a formalism suitable for the treatment of dependent types. CwA's come along with extra structure that is intended to capture the extensions of contexts, so that for a context of variables \(\Gamma\) and a type \(A\) in context \(\Gamma\) one can form the context \(\Gamma . A\) allowing for fresh variables of type \(A\) to occur. In particular a CwA \(\mathcal{C}\) is a category with terminal object (corresponding to the empty context) equipped with a presheaf \(Ty \colon \mathcal{C} ^{op} \to \mathrm{Set}\) (types of context \(\Gamma \in \mathcal{C}\)) and an assignment, for each type \(A\in Ty(\Gamma),\) of an object \(\Gamma . A \in \mathcal{C} \) and a projection map \( \Gamma . A \to \Gamma\) so that, for a change of contexts \(\Gamma \to \Delta ,\) an obvious square becomes a pullback.
The authors examine the behaviour of such structures under the formation of the category of diagrams \(\mathcal{C} ^\mathcal{I}\) for inverse categories
\(\mathcal{I}.\) The latter are categories with the feature that, in the dual category, the relation induced by the existence of a non-identity arrow between two objects, is a well-founded ordering and each object has finitely-many predecessors with respect to it. The category of diagrams is then equipped with a structure of a CwA (this is the content of section 3 of the article). Some natural questions arise in this framework. First, if the CwA \(\mathcal{C}\) carries identity, or \(\Sigma\)- or \(\Pi\)-types, does also \(\mathcal{C} ^\mathcal{I}\) do the same? The authors answer affirmatively this question (the answer is section 4 of the article). Secondly, when \(\mathcal{C}\) and \(\mathcal{I}\) have a homotopical structure (meaning here that they both have a distinguished class \(\mathcal{W}\) of morphisms called equivalences, containing the identities and closed under \(2\) out of \(6\)) do homotopical diagrams (those that take equivalences to equivalences) carry a CwA structure? The authors show that they form a sub-CwA of that of ordinary diagrams (section 5). Moreover it carries identity, \(\Sigma\)- and unit types if \(\mathcal{C}\) does do, while it carries \(\Pi\)-types if \(\mathcal{C}\) does so and all maps in \(\mathcal{I}\) are equivalences (Proposition 5.14). Finally, they give conditions on a functor \(\mathcal{J} \to \mathcal{I}\) so that the induced \(\mathcal{C} ^\mathcal{I} \to \mathcal{C} ^\mathcal{J}\) to be a local fibration or local equivalence in the sense of [\textit{K. Kapulkin} and \textit{P. L. Lumsdaine}, Adv. Math. 337, 1--38 (2018; Zbl 1397.18015)]. In particular they show that for a CwA \(\mathcal{C},\) an ordered homotopical discrete opfibration \(\mathcal{J} \to \mathcal{J}\) induces contravariantly a local fibration between the respective diagram categories when it is moreover injective, while it induces a local equivalence when it is a homotopical equivalence.
Reviewer: Panagis Karazeris (Patras)Galois connections between lattices of preradicals induced by ring epimorphisms.https://www.zbmath.org/1453.160322021-02-27T13:50:00+00:00"Fernández-Alonso, Rogelio"https://www.zbmath.org/authors/?q=ai:fernandez-alonso.rogelio"Magaña, Janeth"https://www.zbmath.org/authors/?q=ai:magana.janethThe authors introduced the concept of a Galois connection between the lattices of predradicals \(R\)-pr and \(S\)-pr of two rings \(R\) and \(S\) induced by any adjoint pair of functors between the categories in an earlier paper [the authors, Appl. Categ. Struct. 24, No. 3, 241--268 (2016; Zbl 1345.16030)]). The paper under review continues this study. For the Galois connection \(\langle \phi , \psi \rangle\) induced by the adjoint pair \(\langle F , G \rangle\) of functors, several results concerning preradicals are found. Let \(R\) and \(S\) be associative rings with identity and let \(\tau\) be any preradical over \(R\) and \(\sigma\) any preradical over \(S\). Then the pretorsion-free class associated with \(\phi (\tau)\) and the pretorsion class associated with \(\psi (\sigma )\) are described explicitly. It is also shown that \(\phi\) preserves idempotency and \(\psi\) preserves radicals.
If \(f:A \rightarrow B\) is a ring homomorphism, then two pairs of adjoint functors \(\langle F,G \rangle\) and \(\langle G,H \rangle\) exist with induced Galois connections \(\langle \phi , \psi \rangle \) and \(\langle \zeta , \xi \rangle\) respectively. If \(\tau\) is a preradical on \(A\) and \(N\) is a \(B\)-submodule, then \(\psi(\tau)(N)\) is shown to be the least \(B\)-submodule containing \(\tau(_{A}N)\) and \(\xi(\tau)(N)\) is shown to be the greatest \(B\)-submodule of \(N\) that is contained in \(\tau(_{A}N)\). This leads to a description of the pretorsion and pretorsion-free classes that correspond to the induced Galois connections.
Whereas the earlier paper of the authors [loc. cit.] focused on the case where \(S\) is a quotient ring, a slightly more general setting is considered here, viz. the case where \(f:A \rightarrow B\) is a ring epimorphism. In this case, it is shown that \(\psi (\sigma)\) is the greatest and \(\zeta(\sigma)\) is the least extension of any preradical \(\sigma\) on \(B\) and it follows that \(\phi\) and \(\xi\) are surjective and \(\psi\) and \(\zeta\) are injective. The class of all extensions of any preradical \(\sigma\) on \(A\) is characterized. If \(\tau\) is an idempotent preradical (resp. radical) on \(B\), then it is proved that the interval \([\zeta(\tau), \psi(\tau)]\) is closed under products (resp. coproducts). Characterizations are also found for the equalities \(\phi = \xi\) and \(\psi = \zeta\) to be true.
Reviewer: Frieda Theron (Potchefstroom)Normality and quotient in crossed modules over groupoids and 2-groupoids.https://www.zbmath.org/1453.180212021-02-27T13:50:00+00:00"Temel, Sedat"https://www.zbmath.org/authors/?q=ai:temel.sedatBy the paper reference [\textit{R. Brown} and \textit{C. B. Spencer}, Nederl. Akad. Wet., Proc., Ser. A 79, 296--302 (1976; Zbl 0333.55011)] it is known that the category of crossed modules and the category of group-groupoids which are also known as 2-groups are equivalent. Using this equivalence of the categories in the paper [\textit{O. Mucuk} et al., Appl. Categ. Struct. 23, No. 3, 415--428 (2015; Zbl 1316.18011)] the concept of normal subgroup-groupoids and of quotient group-groupoids are obtained.
In this paper normal and quotient objects in the category of crossed modules over groupoids and the category of 2-groupoids are compared and the corresponding objects in the category of 2-groupoids are characterized using the well-known categorical equivalence between 2-groupoids and crossed modules over groupoids. Since a 2-groupoid with one object is a group-groupoid (i.e. 2-group), the results of this paper are generalizations of the work for normality and quotient in the category of group-groupoids and the category of crossed modules over groups.
Reviewer: Osman Mucuk (Kayseri)Separated monic representations. II: Frobenius subcategories and RSS equivalences.https://www.zbmath.org/1453.160122021-02-27T13:50:00+00:00"Zhang, Pu"https://www.zbmath.org/authors/?q=ai:zhang.pu"Xiong, Bao-Lin"https://www.zbmath.org/authors/?q=ai:xiong.bao-linSummary: This paper looks for Frobenius subcategories, via the separated monomorphism category \( \operatorname {smon}(Q, I, \mathcal {X})\), and on the other hand, aims to establish an RSS equivalence from \( \operatorname {smon}(Q, I, \mathcal {X})\) to its dual \( \operatorname {sepi}(Q, I, \mathcal {X})\). For a bound quiver \( (Q, I)\) and an algebra \( A\), where \( Q\) is acyclic and \( I\) is generated by monomial relations, let \( \Lambda =A\otimes _k kQ/I\). For any additive subcategory \( \mathcal {X}\) of \( A\)-mod, we introduce \( \operatorname {smon}(Q, I, \mathcal {X})\) combinatorially. It describes Gorenstein-projective \( \Lambda \)-modules as \( \mathcal {GP}(\Lambda ) = \operatorname {smon}(Q, I, \mathcal {GP}(A))\). It admits a homological interpretation and enjoys a reciprocity \( \operatorname {smon}(Q, I, {}^\bot T)= {}^\bot (T\otimes kQ/I)\) for a cotilting \( A\)-module \( T\). As an application, \( \operatorname {smon}(Q, I, \mathcal {X})\) has Auslander-Reiten sequences if \( \mathcal {X}\) is resolving and
contravariantly finite with \( \widehat {\mathcal {X}}=A\)-mod. In particular, \( \operatorname {smon}(Q, I, A)\) has Auslander-Reiten sequences. It also admits a filtration interpretation as \( \operatorname {smon}(Q, I, \mathcal {X})=\operatorname {Fil}(\mathcal {X}\otimes \mathcal P(kQ/I))\), provided that \( \mathcal {X}\) is extension-closed. As an application, \( \operatorname {smon}(Q, I, \mathcal {X})\) is an extension-closed Frobenius subcategory if and only if so is \( \mathcal {X}\). This gives ``new'' Frobenius subcategories of \( \Lambda \)-mod in the sense that they may not be \( \mathcal {GP}(\Lambda )\). Ringel-Schmidmeier-Simson equivalence \( \operatorname {smon}(Q, I, \mathcal {X})\cong \operatorname {sepi}(Q, I, \mathcal {X})\) is introduced and the existence is proved for arbitrary extension-closed subcategories \( \mathcal {X}\). In particular, the Nakayama functor \( \mathcal N_\Lambda \) gives an RSS equivalence \( \operatorname {smon}(Q, I, A)\cong \operatorname {sepi}(Q, I, A)\) if and only if \( A\) is Frobenius. For a chain \( Q\) with arbitrary \( I\), an explicit formula of an RSS equivalence is found for arbitrary additive subcategories \( \mathcal {X}\).Towards a categorical boson-fermion correspondence.https://www.zbmath.org/1453.180152021-02-27T13:50:00+00:00"Tian, Yin"https://www.zbmath.org/authors/?q=ai:tian.yin|tian.yin.1The boson-fermion correspondence establishes an isomorphism between a bosonic Fock space and a fermionic Fock space, a Heisenberg algebra acting on the bosonic Fock space \(V_{B}=\mathbb{Z}[x_{1},x_{2},\dots]\) (a ring of polynomials of infinite variables), while a Clifford algebra acting on the fermionic Fock space \(V_{F}\) (a free abelian group with a basis of semi-infinite monomials). The correspondence also provides maps between the Heisenberg and Clifford algebras via vertex operators [\textit{I. B. Frenkel}, J. Funct. Anal. 44, 259--327 (1981; Zbl 0479.17003)].
The author has already constructed a DG categorification of a Clifford algebra [\textit{Y. Tian}, Int. Math. Res. Not. 2015, No. 21, 10872--10928 (2015; Zbl 1344.18010)]. This paper aims to give an algebraic interpretation of the geometric structure underlying the Clifford categorification. \textit{M. Khovanov} [Fundam. Math. 225, 169--210 (2014; Zbl 1304.18019)] constructed a \(\mathbf{k}\)-linear additive categorification of the Heisenberg algebra, where \(\mathbf{k}\) is a field of characteristic zero. The Heisenberg category acts on the category of \(\bigoplus_{n=0}^{\infty}\mathbf{k}[S(n)]\)-modules, where \(S(n)\) is the \(n\)-th symmetric group. This paper provides a modification of of Khovanov's Heisenberg category, showing that it is the Heisenberg counterpart of the Clifford category [\textit{Y. Tian}, Int. Math. Res. Not. 2015, No. 21, 10872--10928 (2015; Zbl 1344.18010)] under a categorical boson-fermion correspondence.
On the Heisenberg facet, a \(\mathbf{k}\)-algebra \(B\) containing \(\bigoplus_{n=0}^{\infty}\mathbf{k}[S(n)]\) as a subalgebra is constructed with generators of \(B\) not in \(\bigoplus_{n=0}^{\infty}\mathbf{k}[S(n)]\) being closely related to contact structures on \((\mathbb{R}\times[0,1])\times[0,1]\). The homotopy category \(\mathcal{B}=\mathsf{Kom}(B)\) of finite-dimensional projective \(B\)-modules admits a monoidal structure given by the derived tensor product over \(B\), there being two distinguished bimodules \(P\) and \(Q\) which correspond to the induction and restriction functors of \(B\). The Heisenberg category \(\mathcal{DH}\) is defined as a full triangulated monoidal subcategory of the derived category \(D(B^{e})\) which is generated by \(B\), \(P\) and \(Q\).
On the Clifford facet, the construction in the author's previous paper is generalized from \(\mathbb{F}_{2}\) to \(\mathbf{k}\), a DG \(\mathbf{k}\)-algebra \(R=\bigoplus_{k\in\mathbb{Z}}R_{k}\) being defined where all \(R_{k}\)'s are isomorphic to each other. A homotopy category of certain DG \(R\)-modules categorifies the fermionic Fock space, there being a family of distinguished DG \(R\)-bimodules \(T(i)\) for \(i\in\mathbb{Z}\) which correspond to certain contact geometric objects. The Clifford category \(\mathcal{CL}\) is defined as a full triangulated monoidal subcategory of the derived category \(D(R^{e})\) which is generated by \(R\) and \(T(i)\)'s.
The main results of the paper goes as follows.
\begin{itemize}
\item The DG algebra \(R_{0}\) is quasi-isomorphic to its cohomology algebra \(H(R_{0})\) with the trivial differential, which is derived Morita equivalent to a DG algebra \(\widetilde{H}(R_{0})\) with the trivial differential and concentrated in degree zero. It is shown that it is isomorphic to a quiver algebra \(F\), and that the algebras \(B\) and \(F\) are Morita equivalent. Certain categories of \(B\)-modules and \(R_{0}\)-modules are equivalent, which categorifies the isomorphism of the Fock spaces (Theorem 5.1).
\item There are some \(B\)-bimodule homomorphisms and extensions between \(B\), \(P\) and \(Q\) which do not exist in Khovanov's Heisenberg category. These extra morphisms enable one to construct an infinite chain of adjoint pairs in \(\mathcal{DH}\) containing the bimodules \(P\) and \(Q\) (Theorem 3.28).
\item The bimodules \(T(i)\) for \(i\in\mathbb{Z}\) form a chain of adjoint pairs in the Clifford category, their classes \(t_{i}=[T(i)]\) in the Grothendieck group generating a Clifford algebra \(Cl\) with the relation
\[
t_{i}t_{j}+t_{j}t_{i}=\delta_{\left\vert i-j\right\vert,1}1
\]
Using a variation of vertex operator construction, one can express the Heisenberg generators \(p,q\) abiding by
\[
qp-pq=1
\]
in terms of the Clifford generators as
\begin{align*}
g(q)& =\sum_{i\leq0}t_{2i}t_{2i-1}-\sum_{i>0}t_{2i-1}t_{2i}\\
g(p)& =\sum_{i\leq0}t_{2i+1}t_{2i}-\sum_{i>0}t_{2i+1}t_{2i}
\end{align*}
One constructs two objects \(\overline{Q},\overline{P}\) in \(D(R_{0}^{e})\) lifting the expressions \(g(q),g(p)\). The chain
\[
R_{0}\longleftrightarrow H(R_{0})\longleftrightarrow \widetilde{H}(R_{0})\cong F\longleftrightarrow B
\]
induces an equivalence
\[
\mathcal{G}:D(B^{e})\rightarrow D(R_{0}^{e})
\]
of categories. It is shown that \(\mathcal{G}(Q)\) and \(\mathcal{G}(P)\) are isomorphic to \(\overline{Q}\) and \(\overline{P}\), respectively (Theorem 5.3).
\item One can consider two generating series
\begin{align*}
\overline{t}(z)& =\sum_{i\in\mathbb{Z}}t_{2i+1}z^{i}\\
t(z)& =\sum_{i\in\mathbb{Z}}t_{2i}z^{-i}
\end{align*}
associated to \(Cl\). The expresions \(\overline{t}(z)\mid_{z=-1}\) and \(t(z)\mid_{z=-1}\) define two linear operators of the Fock space, which are categorified to certain endofunctors of the Fock space categorification \(\mathcal{B}\) (Theorem 6.5 and 6.9).
\end{itemize}
This review closes with comments on related works.
\begin{itemize}
\item Frenkel, Penkov and Serganova [Zbl 1355.17032] gave a categorification of the boson-fermion correspondece via the representation theory of \(\mathfrak{sl}(\infty)\).
\item Based upon the previous work of \textit{S. Cautis} and \textit{A. Licata} [Duke Math. J. 161, No. 13, 2469--2547 (2012; Zbl 1263.14020); ``Vertex operators and 2-representations of quantum affine algebras'', Preprint, \url{arXiv:1112.6189}], \textit{S. Cautis} and \textit{J. Sussan} [Commun. Math. Phys. 336, No. 2, 649--669 (2015; Zbl 1327.17009)] constructed another categorical version of the correspondence whose Heisenberg facet is Khovanov's categoricfication.
\item The algebra \(B\) has already appeared in work on the stability of representation of symmetric groups, e.g., in [\textit{T. Church} et al., Duke Math. J. 164, No. 9, 1833--1910 (2015; Zbl 1339.55004)].
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)\(E_1\)-degeneration of the irregular Hodge filtration.https://www.zbmath.org/1453.320102021-02-27T13:50:00+00:00"Esnault, Hélène"https://www.zbmath.org/authors/?q=ai:esnault.helene"Sabbah, Claude"https://www.zbmath.org/authors/?q=ai:sabbah.claude"Yu, Jeng-Daw"https://www.zbmath.org/authors/?q=ai:yu.jeng-dawSummary: For a regular function \(f\) on a smooth complex quasi-projective variety, \textit{J.-D. Yu} introduced in [Manuscr. Math. 144, No. 1--2, 99--133 (2014; Zbl 1291.14040)] a filtration (the irregular Hodge filtration) on the de Rham complex with twisted differential \(\mathrm{d}+\mathrm{d}f\), extending a definition of Deligne in the case of curves. In this article, we show the degeneration at \(E_1\) of the spectral sequence attached to the irregular Hodge filtration, by using the method of \textit{C. Sabbah} [Adv. Stud. Pure Math. 59, 289--347 (2010; Zbl 1264.14011)]. We also make explicit the relation with a complex introduced by M. Kontsevich and give details on his proof of the corresponding \(E_1\)-degeneration, by reduction to characteristic \(p\), when the pole divisor of the function is reduced with normal crossings. In Appendix E, M. Saito gives a different proof of the \(E_1\)-degeneration.Recollements associated to cotorsion pairs.https://www.zbmath.org/1453.130492021-02-27T13:50:00+00:00"Chen, Wenjing"https://www.zbmath.org/authors/?q=ai:chen.wenjing"Liu, Zhongkui"https://www.zbmath.org/authors/?q=ai:liu.zhong-kui"Yang, Xiaoyan"https://www.zbmath.org/authors/?q=ai:yang.xiaoyanThe homological projective dual of \(\operatorname{Sym}^2\mathbb{P}(V)\).https://www.zbmath.org/1453.140532021-02-27T13:50:00+00:00"Rennemo, Jørgen Vold"https://www.zbmath.org/authors/?q=ai:rennemo.jorgen-voldSummary: We study the derived category of a complete intersection \(X\) of bilinear divisors in the orbifold \(\operatorname{Sym}^2\mathbb{P}(V)\). Our results are in the spirit of Kuznetsov's theory of homological projective duality, and we describe a homological projective duality relation between \(\operatorname{Sym}^2\mathbb{P}(V)\) and a category of modules over a sheaf of Clifford algebras on \(\mathbb{P}(\operatorname{Sym}^2V^{\vee })\). The proof follows a recently developed strategy combining variation of geometric invariant theory (VGIT) stability and categories of global matrix factorisations. We begin by translating \(D^b(X)\) into a derived category of factorisations on a Landau-Ginzburg (LG) model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of non-birational Calabi-Yau 3-folds have equivalent derived categories.On one class of commutative operads.https://www.zbmath.org/1453.180202021-02-27T13:50:00+00:00"Gaynullina, Alina"https://www.zbmath.org/authors/?q=ai:gaynullina.alina-rThe universal Lie \(\infty\)-algebroid of a singular foliation.https://www.zbmath.org/1453.530332021-02-27T13:50:00+00:00"Laurent-Gengoux, Camille"https://www.zbmath.org/authors/?q=ai:laurent-gengoux.camille"Lavau, Sylvain"https://www.zbmath.org/authors/?q=ai:lavau.sylvain"Strobl, Thomas"https://www.zbmath.org/authors/?q=ai:strobl.thomasThe treatment of singular foliations in [\textit{I. Androulidakis} and \textit{G. Skandalis}, J. Reine Angew. Math. 626, 1--37 (2009; Zbl 1161.53020)] defines them as submodules of the module of vector fields over the ring of smooth functions, which are involutive and locally finitely generated. The principal result of the paper under review is that any projective resolution of a module as such admits a Lie-infinity algebra structure. Also, any two such Lie-infinity algebras are quasi-isomorphic. Assuming that the vector fields are analytic is necessary in this treatment.
The authors use the cohomology associated with this Lie-infinity algebra to study the geometry of a singular foliation. In particular, they give a cohomological obstruction to the existence of a Lie algebroid which defines the given foliation. However, they show that every singular foliation is defined by some Leibniz algebroid. Finally, they realize the universal cover of the holonomy groupoid constructed in [loc. cit.] as the fundamental groupoid associated to the universal Lie-infinity algebroid associated to the given foliation.
Reviewer: Iakovos Androulidakis (Athína)Ring-theoretic blowing down. I.https://www.zbmath.org/1453.140072021-02-27T13:50:00+00:00"Rogalski, Daniel"https://www.zbmath.org/authors/?q=ai:rogalski.daniel"Sierra, Susan J."https://www.zbmath.org/authors/?q=ai:sierra.susan-j"Stafford, J. Toby"https://www.zbmath.org/authors/?q=ai:stafford.j-tobySummary: One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). Earlier work of the authors classified the connected graded noetherian subalgebras of Sklyanin algebras using a noncommutative analogue of blowing up. In order to understand other algebras birational to a Sklyanin algebra, one also needs a notion of blowing down. This is achieved in this paper, where we give a noncommutative analogue of Castelnuovo's classic theorem that \((-1)\)-lines on a smooth surface can be contracted. The resulting noncommutative blown-down algebra has pleasant properties; in particular it is always noetherian and is smooth if the original noncommutative surface is smooth.
In a companion paper we will use this technique to construct explicit birational transformations between various noncommutative surfaces which contain an elliptic curve.Quillen-Segal algebras and stable homotopy theory.https://www.zbmath.org/1453.180232021-02-27T13:50:00+00:00"Bacard, Hugo"https://www.zbmath.org/authors/?q=ai:bacard.hugo-vSummary: Let \(\mathscr{M}\) be a monoidal model category that is also combinatorial. If \(\mathcal{O}\) is a monad, operad, properad, or a PROP; following Segal's ideas, we develop a theory of Quillen-Segal \(\mathcal{O}\)-algebras and show that we have a Quillen equivalence between usual \(\mathcal{O}\)-algebras and Quillen-Segal algebras. We also introduce Quillen-Segal theories, and we use them to get the stable homotopy category by a similar method as Hovey.An algebraic theory of Markov processes.https://www.zbmath.org/1453.080022021-02-27T13:50:00+00:00"Bacci, Giorgio"https://www.zbmath.org/authors/?q=ai:bacci.giorgio"Mardare, Radu"https://www.zbmath.org/authors/?q=ai:mardare.radu"Panangaden, Prakash"https://www.zbmath.org/authors/?q=ai:panangaden.prakash"Plotkin, Gordon"https://www.zbmath.org/authors/?q=ai:plotkin.gordon-dAdjoint functors, preradicals and closure operators in module categories.https://www.zbmath.org/1453.160082021-02-27T13:50:00+00:00"Kashu, Alexei I."https://www.zbmath.org/authors/?q=ai:kashu.aleksei-iSummary: In this article preradicals and closure operators are studied in an adjoint situation, defined by two covariant functors between the module categories \(R\)-Mod and \(S\)-Mod. The mappings which determine the relationship between the classes of preradicals and the classes of closure operators of these categories are investigated. The goal of research is to elucidate the concordance (compatibility) of these mappings. For that some combinations of them, consisting of four mappings, are considered and the commutativity of corresponding diagrams (squares) is studied. The obtained results show the connection between considered mappings in adjoint situation.Finite quasi-quantum groups of diagonal type.https://www.zbmath.org/1453.160362021-02-27T13:50:00+00:00"Huang, Hua-Lin"https://www.zbmath.org/authors/?q=ai:huang.hua-lin"Liu, Gongxiang"https://www.zbmath.org/authors/?q=ai:liu.gongxiang"Yang, Yuping"https://www.zbmath.org/authors/?q=ai:yang.yuping"Ye, Yu"https://www.zbmath.org/authors/?q=ai:ye.yuSummary: In this paper, we give a classification of finite-dimensional radically graded elementary quasi-Hopf algebras of diagonal type, or equivalently, finite-dimensional coradically graded pointed Majid algebras of diagonal type. By a Tannaka-Krein type duality, this determines a big class of pointed finite tensor categories. Some efficient methods of construction are also given.Basic structures on derived critical loci.https://www.zbmath.org/1453.140092021-02-27T13:50:00+00:00"Vezzosi, Gabriele"https://www.zbmath.org/authors/?q=ai:vezzosi.gabrieleSummary: We review the derived algebraic geometry of derived zero loci of sections of vector bundles, with particular emphasis on derived critical loci. In particular we some of the structures carried by derived critical loci: the homotopy Batalin-Vilkovisky structure, the action of the 2-monoid of the self-intersection of the zero section, and the derived symplectic structure of degree \(-1\). We also show how this structure exists, more generally, on derived lagrangian intersections inside a symplectic algebraic manifold.Weighted projective lines and rational surface singularities.https://www.zbmath.org/1453.320332021-02-27T13:50:00+00:00"Iyama, Osamu"https://www.zbmath.org/authors/?q=ai:iyama.osamu"Wemyss, Michael"https://www.zbmath.org/authors/?q=ai:wemyss.michaelSummary: We study rational surface singularities \(R\) with star shaped dual graphs, and under very mild assumptions on the self-intersection numbers we give an explicit description of all their special Cohen-Macaulay modules. We do this by realising \(R\) as a certain \(Z\)-graded Veronese subring \(S^x\) of the homogeneous coordinate ring \(S\) of the Geigle-Lenzing weighted projective line \(X\), and we realise the special CM modules as explicitly described summands of the canonical tilting bundle on \(X\). We then give a second proof that these are special CM modules by comparing \(\mathrm{qgr} S^x\) and \(\mathrm{coh} X\), and we also give a necessary and sufficient combinatorial criterion for these to be equivalent categories. In turn, we show that \(\mathrm{qgr} S^x\) is equivalent to qgr of the reconstruction algebra, and that the degree zero piece of the reconstruction algebra coincides with Ringel's canonical algebra. This implies that the reconstruction algebra contains the canonical algebra, and furthermore its qgr category is derived equivalent to the canonical algebra, thus linking the reconstruction algebra of rational surface singularities to the canonical algebra of representation theory.Fundamentals of soft category theory.https://www.zbmath.org/1453.180032021-02-27T13:50:00+00:00"Sardar, Sujit Kumar"https://www.zbmath.org/authors/?q=ai:sardar.sujit-kumar"Gupta, Sugato"https://www.zbmath.org/authors/?q=ai:gupta.sugato"Davvaz, Bijan"https://www.zbmath.org/authors/?q=ai:davvaz.bijanSummary: The soft category theory offers a way to study soft theories developed so far more generally. The main purpose of this paper is to introduce the basic algebraic operations in soft categories, and for that we introduce some algebraic operations, like intersection and union, in categories. Also, the notion of composition of soft functors is introduced to form category of all soft categories.Totally reflexive modules over rings that are close to Gorenstein.https://www.zbmath.org/1453.130032021-02-27T13:50:00+00:00"Kustin, Andrew R."https://www.zbmath.org/authors/?q=ai:kustin.andrew-r"Vraciu, Adela"https://www.zbmath.org/authors/?q=ai:vraciu.adela-nOver a commutative Noetherian ring, \textit{M. Auslander} and \textit{M. Bridger} [Stable module theory. Providence, RI: American Mathematical Society (AMS) (1969; Zbl 0204.36402)] extended the notion of finitely generated projective modules to totally reflexive modules, i.e. modules occurring as the cokernels of differentials in exact chain complexes of finitely generated free modules whose dual complexes are also exact. There has been a growing interest in identifying the local rings over which every totally reflexive module is projective or equivalently free, i.e. the so-called ``G-regular rings'' coined by Takahashi. Regular local rings and Golod local rings that are not hypersurfaces are among the examples of G-regular rings. However, a singular Gorenstein local ring is never G-regular. In the paper under review, the authors prove that any non-Gorenstein quotient of small colength of a deeply embedded equicharacteristic Artinian Gorenstein local ring is G-regular. Their investigation relies on the fact that an Artinian local ring is G-regular if there exists a projective-test module that happens to be a direct summand of a syzygy of the canonical module of the ring. Overall, this is an interesting and readable paper.
Reviewer: Hossein Faridian (Clemson)Generic base change, Artin's comparison theorem, and the decomposition theorem for complex Artin stacks.https://www.zbmath.org/1453.140052021-02-27T13:50:00+00:00"Sun, Shenghao"https://www.zbmath.org/authors/?q=ai:sun.shenghaoSummary: We prove the generic base change theorem for stacks and give an exposition on the lisse-analytic topos of complex analytic stacks, proving some comparison theorems between various derived categories of complex analytic stacks. This enables us to deduce the decomposition theorem for perverse sheaves on complex Artin stacks with affine stabilizers from the case over finite fields.Some properties of the Schur multiplier and stem covers of Leibniz crossed modules.https://www.zbmath.org/1453.170032021-02-27T13:50:00+00:00"Casas, José Manuel"https://www.zbmath.org/authors/?q=ai:casas-miras.jose-manuel"Ravanbod, Hajar"https://www.zbmath.org/authors/?q=ai:ravanbod.hajarLeibniz crossed modules are usual generalization of Leibniz algebras and also modules and ideals over Leibniz algebras. So it is a natural to search for similar concepts and results of Leibniz algebras in a higher level.
In this paper, the well-known notions related to the Schur multiplier of groups, Lie algebras, Leibniz algebras are studied in the category \textbf{XLb} of Leibniz algebras crossed modules. At first, authors recalled some basic categorical concepts such as the commutator of two ideals, the center and central extensions of Leibniz crossed modules. Moreover, they showed that the category of Leibniz crossed modules has enough projective objects. Hence, the Baer invariant associated with a projective presentation of \((\mathfrak{n},\mathfrak{q},\delta)\) can be defined and is called the Schur multiplier of the Leibniz crossed module \((\mathfrak{n},\mathfrak{q},\delta)\) which is denoted by \(\mathcal{M}(\mathfrak{n},\mathfrak{q},\delta)\).
As an interesting description of the Scur multiplier, authors proved that
\[\mathcal{M}(\mathfrak{n},\mathfrak{q},\delta)\cong \mathrm{Ker}((\mathfrak{q}\curlywedge\mathfrak{n}, \mathfrak{q}\curlywedge\mathfrak{q}, id\curlywedge\delta)\to (\mathfrak{n}, \mathfrak{q}, \delta)), \]
where the symbol \(\curlywedge\) is the non-abelian exterior product of Leibniz algebras. They used this result to construct a Stallings type of six-term exact sequence of Leibniz crossed modules associated with a central extension of Leibniz crossed modules.
Also, the authors study basic properties of stem covers of Leibniz crossed modules. In particular, they prove the existence of stem covers for an arbitrary Leibniz crossed module and determined the common structure of all stem covers of a Leibniz crossed modules whose Schur multiplier is finite dimensional.
Finally, in the last section, they study the connections between the stem covers of a Lie crossed module in the categories of Lie and Leibniz crossed modules.
Reviewer: Behrouz Edalatzadeh (Kermanshah)Lie algebroid cohomology and Lie algebroid extensions.https://www.zbmath.org/1453.320252021-02-27T13:50:00+00:00"Aldrovandi, E."https://www.zbmath.org/authors/?q=ai:aldrovandi.ettore"Bruzzo, U."https://www.zbmath.org/authors/?q=ai:bruzzo.ugo"Rubtsov, V."https://www.zbmath.org/authors/?q=ai:rubtsov.vladimir-nSummary: We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid \(\mathcal{Q}\) over a scheme \(X\), and a coherent sheaf of Lie \(\mathcal{O}_X\)-algebras \(\mathcal{L}\), we determine the obstruction to the existence of extensions \(0 \rightarrow \mathcal{L} \rightarrow \mathcal{E} \rightarrow \mathcal{Q} \rightarrow 0\), and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology.On descent cohomology.https://www.zbmath.org/1453.160352021-02-27T13:50:00+00:00"Mesablishvili, B."https://www.zbmath.org/authors/?q=ai:mesablishvili.b-n|mesablishvili.bachukiSummary: The zeroth and first descent cohomology sets for a (co)monad on arbitrary base category with coefficients in a (co)algebra are introduced and their basic properties are studied. These sets generalize those for a coring with coefficients in a comodule. It is shown that under this generalization, essential properties and relationships are preserved.Castelnuovo-Mumford regularity of representations of certain product categories.https://www.zbmath.org/1453.180102021-02-27T13:50:00+00:00"Gan, Wee Liang"https://www.zbmath.org/authors/?q=ai:gan.wee-liang"Li, Liping"https://www.zbmath.org/authors/?q=ai:li.lipingSummary: We show in this paper that representations of a finite product of categories satisfying certain combinatorial conditions have finite Castelnuovo-Mumford regularity if and only if they are presented in finite degrees, and hence the category consisting of them is abelian. These results apply to examples such as the categories \(\operatorname{FI}^m\) and \(\operatorname{FI}_G^m\).The Grothendieck group of a tower of the Temperley-Lieb algebras.https://www.zbmath.org/1453.160092021-02-27T13:50:00+00:00"Wang, Pei"https://www.zbmath.org/authors/?q=ai:wang.peiGiven a tower of finite-dimensional algebras with Axiom (T), the Grothendieck group of this tower has structure of algebra and coalgebra (Proposition 3.1).
Consider a tower of Temperley-Lieb algebras. The author considers some restrictions of the cell modules, called wall modules, with which he obtains a certain chain of modules, to prove the main theorem; this result establishes the structure constants of the multiplication. As a corollary, the Grothendieck group in this tower is not a bialgebra.
Reviewer: Luz Adriana Mejia (Barranquilla)Exceptional cycles for perfect complexes over gentle algebras.https://www.zbmath.org/1453.180132021-02-27T13:50:00+00:00"Guo, Peng"https://www.zbmath.org/authors/?q=ai:guo.peng"Zhang, Pu"https://www.zbmath.org/authors/?q=ai:zhang.puLet \(\mathcal T\) be a \(k\)-linear triangulated category with Serre functor \(\mathsf S\). Then an exceptional cycle \((E_1,\ldots, E_n)\) with \(n \geq 2\) consists of exceptional objects \(E_i \in \mathcal T\) such that \(E_{i+1} = \mathsf S(E_i)[m_i]\) for some integer \(m_i\) and all \(i \in \{1,\ldots,n\} = \mathbb Z/n\mathbb Z\).
In the case \(n=1\), one says that a spherical objects forms a (degenerate) exceptional cycle.
This notion was introduced by \textit{N. Broomhead} et al. [Math. Z. 285, No. 1--2, 39--89 (2017; Zbl 1409.18012)], but appeared already under the name spherical sequences in the article [``Braid group actions on triangulated categories'', Möbius contest (2007)] by \textit{A. I. Efimov}, see also [``Groups generated by two twists along spherical sequences'', Preprint, \url{arXiv:1901.10904}] by \textit{Y. V. Volkov}.
Such sequences are interesting, because the give rise to a twist functor which turns out to be an autoequivalence of \(\mathcal T\), thus generalising spherical twists.
In the article cited above by Broomhead et al., exceptional cycles were completely classfied in the case of \(\mathcal T = \mathcal D^b(\Lambda(r,n,m)\operatorname{-mod})\), where \(\Lambda(r,n,m)\) is a derived discrete algebra, as studied by \textit{D. Vossieck} [J. Algebra 243, No. 1, 168--176 (2001; Zbl 1038.16010)].
Derived discrete algebras are a subclass of gentle algebras and the article under review gives a classification of exceptional cycles in \(K^b(A\operatorname{-proj})\) where \(A\) is an indecomposable finite-dimensional gentle \(k\)-algebra.
For the main result (Theorem 1.4), the authors have to exclude the case that the underlying graph of \(A\) is of type \(A_3\), but in that case a classification of exceptional cycles is given by the same authors in [Acta Math. Sin., Engl. Ser. 36, No. 3, 207--223 (2020; Zbl 1436.18011)].
The proof of the main result goes roughly along the following lines.
The first observation is that any object in an exceptional cycle has to lie at the mouth.
Then for \(X\) an indecomposable object in a characteristic component \(C\) of \(K^b(A\operatorname{-proj})\) with AG-invariant \((n,m)\) gives rise to the exceptional \(n\)-cycle \((X, \tau X, \ldots, \tau^{n-1} X)\).
Up to shift and cyclic permutation, this is the only such cycle in \(C\).
For \(n \geq 2\), these turn out to be all exceptional \(n\)-cycles.
For \(n = 1\), that is, when we are speaking about spherical objects,
a string complex is spherical, if and only if it is at the mouth of characteristic component with AG-invariant \((1,m)\).
Whereas for a band complex \(E\), the authors show that, if \(E\) is spherical, then it has to be at the mouth of a homogeneous component.
But they give an example that the converse implication is not true in general.
Reviewer: Andreas Hochenegger (Milano)Operadic categories and décalage.https://www.zbmath.org/1453.180192021-02-27T13:50:00+00:00"Garner, Richard"https://www.zbmath.org/authors/?q=ai:garner.richard"Kock, Joachim"https://www.zbmath.org/authors/?q=ai:kock.joachim"Weber, Mark"https://www.zbmath.org/authors/?q=ai:weber.mark\textit{Operads} originated in [\textit{J. M. Boardman} and \textit{R. M. Vogt}, Bull. Am. Math. Soc. 74, 1117--1122 (1968; Zbl 0165.26204)] under the name ``category of operators in standard form'', their modern name and modern development having begun with [\textit{J. P. May}, The geometry of iterated loop spaces. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0244.55009)]. As the use of operads grew out, their re-expression in various more abstract ways emerged [\textit{J. C. Baez} and \textit{J. Dolan}, Adv. Math. 135, No. 2, 145--206 (1998; Zbl 0909.18006); \textit{A. Joyal}, Lect. Notes Math. 1234, 126--159 (1986; Zbl 0612.18002); \textit{G. M. Kelly}, Repr. Theory Appl. Categ. 2005, No. 13, 1--13 (2005; Zbl 1082.18009)], leading to a rich profusion of operad-like structures. Various authors have proposed unifying frameworks to bring order to this proliferation, one such framework being that of \textit{operadic categories} [\textit{M. Batanin} and \textit{M. Markl}, Adv. Math. 285, 1630--1687 (2015; Zbl 1360.18009)], which inspired \textit{S. Lack} [High. Struct. 2, No. 1, 1--29 (2018; Zbl 1410.18012)] to draw a intimate correspondence between operadic categories and the \textit{skew-monoidal categories} of \textit{K. Szlachányi} [Adv. Math. 231, No. 3--4, 1694--1730 (2012; Zbl 1283.18006)]. This paper gives another reconfiguration of the definition of operadic category linking it to the (upper) \textit{décalage} construction.
An \textit{operadic category} consists of the following entities.
\begin{itemize}
\item a small category \(\mathcal{C}\) with a chosen terminal object in each connected component;
\item a cardinality functor \(\left\vert -\right\vert :\mathcal{C}\rightarrow\mathcal{S}\) into the category of finite ordinals and arbitrary mappings;
\item an operation assigning to every \(f:Y\rightarrow X\) in \(\mathcal{C}\) and \(i\in\left\vert X\right\vert \) an abstract fiber \(f^{-1}\left( i\right) \in\mathcal{C}\), functorially in \(Y\).
\end{itemize}
There are two main aspects to the close relationship between operadic categories and décalage. The first connection comes from the fact that the décalage construction on categories underlies a comonad \textsf{D} on \(\mathcal{C}\mathrm{at}\), whose coalgebras may be identified with categories endowed with a choice of terminal object in each connected component. The second connection arises through the functorial assignment of abstract fibers \(f\mapsto f^{-1}\left( i\right) \) in an operadic category with functoriality claiming that, for a fixed \(X\in\) \(\mathcal{C}\) and \(i\in\left\vert X\right\vert \), this assignment is the action on objects of a functor \(\varphi_{X,i}:\mathcal{C}/X\rightarrow\mathcal{C}\), so that the totality of the abstract fibers is to be expressed via a single functor
\[
\varphi:\sum_{\substack{X\in\mathcal{C}\\ i\in\left\vert X\right\vert}}\mathcal{C}/X\rightarrow\mathcal{C}
\]
The domain of this functor is clearly related to the décalage of \(\mathcal{C}\), and the authors explain it in terms of a \textit{modified décalage} construction on categories equipped with a functor to \(\mathcal{S}\) but that the operadic category is unary, where an operadic category is called \textit{unary} if each \(\left\vert X\right\vert \) is a singleton.
A synopsis of the paper consisting of seven sections goes as follows. \S 2 recalls Batanin and Markl's definition of operadic category [\textit{M. Batanin} and \textit{M. Markl}, Adv. Math. 285, 1630--1687 (2015; Zbl 1360.18009)].
\S 3 recalls the décalage construction, establishing the first of the two links with the notion of operadic category. \S 4 establishes the first main theorem characterizing unary operadic categories in terms of décalage.
\S 5 is engaged in describing the modified décalage construction required to capture general operadic categories. \S 6 and \S 7 establish the second and third main theorems giving the characterization of lax-operadic categories and, finally, of operadic categories themselves.
Reviewer: Hirokazu Nishimura (Tsukuba)Homotopy invariant presheaves with framed transfers.https://www.zbmath.org/1453.140662021-02-27T13:50:00+00:00"Garkusha, Grigory"https://www.zbmath.org/authors/?q=ai:garkusha.grigory"Panin, Ivan"https://www.zbmath.org/authors/?q=ai:panin.ivanSummary: The category of framed correspondences \(Fr_{\ast} (k)\), framed presheaves and framed sheaves were invented by \textit{V. Voevodsky} in his unpublished notes [``Notes on framed correspondences'', unpublished, available at \url{www.math.ias.edu/vladimir/files/framed.pdf}]. Based on the notes [loc. cit.] a new approach to the classical Morel-Voevodsky motivic stable homotopy theory was developed in [the authors, J. Am. Math. Soc. 34, No. 1, 261--313 (2021; Zbl 07304882)]. This approach converts the classical motivic stable homotopy theory into an equivalent local theory of framed bispectra. The main result of the paper is the core of the theory of framed bispectra. It states that for any homotopy invariant quasi-stable radditive framed presheaf of Abelian groups \(\mathcal{F} \), the associated Nisnevich sheaf \(\mathcal{F}_{\mathrm{nis}}\) is strictly homotopy invariant and quasi-stable whenever the base field \(k\) is infinite perfect of characteristic different from 2.Standard conjecture \(D\) for matrix factorizations.https://www.zbmath.org/1453.140062021-02-27T13:50:00+00:00"Brown, Michael K."https://www.zbmath.org/authors/?q=ai:brown.michael-kegan"Walker, Mark E."https://www.zbmath.org/authors/?q=ai:walker.mark-eSummary: We prove the non-commutative analogue of Grothendieck's standard conjecture \(D\) for the dg-category of matrix factorizations of an isolated hypersurface singularity in characteristic 0. Along the way, we show the Euler pairing for such dg-categories of matrix factorizations is positive semi-definite.Absolute sets and the decomposition theorem.https://www.zbmath.org/1453.140542021-02-27T13:50:00+00:00"Wang, Botong"https://www.zbmath.org/authors/?q=ai:wang.botong"Budur, Nero"https://www.zbmath.org/authors/?q=ai:budur.neroSummary: We show that any natural (derived) functor on constructible sheaves on smooth complex algebraic varieties can be used to construct a special kind of constructible sets of local systems, called absolute sets. We conjecture that the absolute sets satisfy a `` special varieties package,'' similar to the André-Oort conjecture. The conjecture gives a simple proof of the Decomposition Theorem for all semi-simple perverse sheaves, assuming it for the geometric ones. We prove the conjecture in the rank-one case: the closed absolute sets are finite unions of torsion-translated affine tori. This extends a structure result of the authors for cohomology jump loci to any other natural jump loci. For example, to jump loci of intersection cohomology and Leray filtrations. We also show that the Leray spectral sequence for the open embedding in a good compactification degenerates for all rank one local systems at the usual page, not just for unitary local systems.Ribbon tensorial logic.https://www.zbmath.org/1453.030692021-02-27T13:50:00+00:00"Melliès, Paul-André"https://www.zbmath.org/authors/?q=ai:mellies.paul-andreFormal Abel-Jacobi maps.https://www.zbmath.org/1453.140272021-02-27T13:50:00+00:00"Fiorenza, Domenico"https://www.zbmath.org/authors/?q=ai:fiorenza.domenico"Manetti, Marco"https://www.zbmath.org/authors/?q=ai:manetti.marcoSummary: We realize the infinitesimal Abel-Jacobi map as a morphism of formal deformation theories, realized as a morphism in the homotopy category of differential graded Lie algebras. The whole construction is carried out in a general setting, of which the classical Abel-Jacobi map is a special example.Categorified duality in Boij-Söderberg theory and invariants of free complexes.https://www.zbmath.org/1453.130422021-02-27T13:50:00+00:00"Eisenbud, David"https://www.zbmath.org/authors/?q=ai:eisenbud.david"Erman, Daniel"https://www.zbmath.org/authors/?q=ai:erman.danielSummary: We present a robust categorical foundation for the duality theory introduced by Eisenbud and Schreyer to prove the Boij-Söderberg conjectures describing numerical invariants of syzygies. The new foundation allows us to extend the reach of the theory substantially.
More explicitly, we construct a pairing between derived categories that simultaneously categorifies all the functionals used by Eisenbud and Schreyer. With this new tool, we describe the cone of Betti tables of finite, minimal free complexes having homology modules of specified dimensions over a polynomial ring, and we treat many examples beyond polynomial rings. We also construct an analogue of our pairing between derived categories on a toric variety, yielding toric/multigraded analogues of the Eisenbud-Schreyer functionals.Split epimorphisms as a productive tool in universal algebra.https://www.zbmath.org/1453.080032021-02-27T13:50:00+00:00"Bourn, Dominique"https://www.zbmath.org/authors/?q=ai:bourn.dominiqueSummary: By various examples, we show how, in Universal Algebra, the choice of a class of split epimorphisms \(\Sigma\), defined by specific equations or local operations in their fibres, can be used as a productive and flexible tool determining \(\Sigma\)-partial properties. We focus, here, our attention on \(\Sigma\)-partial congruence modular and \(\Sigma\)-partial congruence distributive formulae.A classification theorem for \(t\)-structures.https://www.zbmath.org/1453.180122021-02-27T13:50:00+00:00"Fiorot, Luisa"https://www.zbmath.org/authors/?q=ai:fiorot.luisa"Mattiello, Francesco"https://www.zbmath.org/authors/?q=ai:mattiello.francesco"Tonolo, Alberto"https://www.zbmath.org/authors/?q=ai:tonolo.albertoSummary: We give a classification theorem for a relevant class of \(t\)-structures in triangulated categories, which includes, in the case of the derived category of a Grothendieck category, a large class of \(t\)-structures whose hearts have at most \(n\) fixed consecutive non-zero cohomologies. Moreover, by this classification theorem, we deduce the construction of the \(t\)-\textit{tree}, a new technique which generalizes the filtration induced by a torsion pair. At last we apply our results in the tilting context generalizing the 1-tilting equivalence proved by \textit{D. Happel} et al. [Tilting in abelian categories and quasitilted algebras. Providence, RI: American Mathematical Society (AMS) (1996; Zbl 0849.16011)]. The last section provides applications to classical \(n\)-tilting objects, examples of \(t\)-trees for modules over a path algebra, and new developments on compatible \(t\)-structures [\textit{B. Keller} and \textit{D. Vossieck}, C. R. Acad. Sci., Paris, Sér. I 307, No. 11, 543--546 (1988; Zbl 0663.18005); \textit{B. Keller}, ``Derived categories and tilting'', in: Handbook of tilting theory. Cambridge: Cambridge University Press. 49--104 (2007; Zbl 1106.16300)].Crossed products of Calabi-Yau algebras by finite groups.https://www.zbmath.org/1453.160272021-02-27T13:50:00+00:00"Le Meur, Patrick"https://www.zbmath.org/authors/?q=ai:le-meur.patrickSummary: Let a finite group \(G\) act on a differential graded algebra \(A\). This article presents necessary conditions and sufficient conditions for the skew group algebra \(A \ast G\) to be Calabi-Yau. In particular, when \(A\) is the Ginzburg dg algebra of a quiver with an invariant potential, then \(A \ast G\) is Calabi-Yau and Morita equivalent to a Ginzburg dg algebra. Some applications of these results are derived to compare the generalised cluster categories of \(A\) and \(A \ast G\) when they are defined and to compare the higher Auslander-Reiten theories of \(A\) and \(A \ast G\) when \(A\) is a finite dimensional algebra.A strictly commutative model for the cochain algebra of a space.https://www.zbmath.org/1453.550112021-02-27T13:50:00+00:00"Richter, Birgit"https://www.zbmath.org/authors/?q=ai:richter.birgit"Sagave, Steffen"https://www.zbmath.org/authors/?q=ai:sagave.steffenThis paper introduces a strictly commutative algebraic model \(A^\mathcal{I}(X)\) for spaces (simplicial sets) over any ring \(k\). This model is shown to be an integral version of the Sullivan algebra \(A_\mathrm{PL}(X)\) of polynomial differential forms [\textit{D. Sullivan}, Publ. Math., Inst. Hautes Étud. Sci. 47, 269--331 (1977; Zbl 0374.57002)]. Results by \textit{M. A. Mandell} [ibid. 103, 213--246 (2006; Zbl 1105.55003)] imply that it determines the homotopy type of nilpotent spaces of finite type when working over the integers.
Let \(\mathcal{I}\) be the category with objects the finite sets \(\mathbf{m}:=\{1,\ldots,m\}\), \(m\ge0\), and the injective maps as morphisms. By convention, \(\mathbf{0} = \emptyset\). The authors consider the category \(\mathrm{Ch}_k^\mathcal{I}\) of \(\mathcal{I}\)-chain complexes, i.e. functors \(\mathcal{I}\to\mathrm{Ch}_k\), with the symmetric monoidal product \(\boxtimes\) induced by concatenation in \(\mathcal{I}\). A commutative \(\mathcal{I}\)-dga is a commutative monoid in this symmetric monoidal category; the resulting category is denoted \(\mathrm{Ch}_k^\mathcal{I}[\mathcal{C}]\).
The authors construct an explicit homotopy colimit functor
\[
(-)_{h\mathcal{I}}:\mathrm{Ch}_k^\mathcal{I}\longrightarrow\mathrm{Ch}_k
\]
and prove:
Theorem 2.13. For every commutative \(\mathcal{I}\)-dga \(E\), the chain complex \(E_{h\mathcal{I}}\) has a natural action of the Barrat-Eccles operad \(\mathcal{E}\).
Thus, homotopy colimits of commutative \(\mathcal{I}\)-dgas are \(E_\infty\)-algebras.
The authors use techniques inspired by the behavior of the Sullivan forms to construct a simplicial commutative \(\mathcal{I}\)-dga \(A^\mathcal{I}_\bullet:\Delta^\mathrm{op}\to\mathrm{Ch}_k^\mathcal{I}[\mathcal{C}]\) which induces the desired model by
\[
A^\mathcal{I}(X):=\mathrm{sSet}(X, A^\mathcal{I}_\bullet).
\]
This defines a functor \(A^\mathcal{I}:\mathrm{sSet}\to\mathrm{Ch}_k^\mathcal{I}[\mathcal{C}]\) called the \textit{commutative \(\mathcal{I}\)-dga of polynomial forms} which has a contravariant right adjoint \(\langle-\rangle_\mathcal{I}\) called the \textit{Sullivan realization}.
By taking the homotopy colimit one obtains an \(E_\infty\)-algebra \(A^\mathcal{I}(X)_{h\mathcal{I}}\) which can be compared to the usual cochains of spaces.
Theorem 1.1. The contravariant functors \(X\mapsto A^\mathcal{I}(X)_{h\mathcal{I}}\) and \(X\mapsto C(X; k)\) from simplicial sets to \(E_\infty\)-algebras are naturally quasi-isomorphic.
If \(k\) is a field of characteristic \(0\), then there is a natural quasi-isomorphism \(A^\mathcal{I}(X)_{h\mathcal{I}}\to A_\mathrm{PL}(X)\) relating the constructions of the paper with the classical polynomial forms of Sullivan.
Writing \(A^\mathcal{I}(X;\mathbb{Z})\) for \(A^\mathcal{I}(X)\) when working over \(k=\mathbb{Z}\), one obtains the following reformulation of a theorem by Mandell [loc. cit.], showing that this model is a complete homotopy invariant for nilpotent spaces of finite type.
Theorem 1.2. Two nilpotent spaces of finite type \(X, Y\) are weakly equivalent if and only if \(A^\mathcal{I}(X;\mathbb{Z})\) and \(A^\mathcal{I}(Y;\mathbb{Z})\) are weakly equivalent in \(\mathrm{Ch}_\mathbb{Z}^\mathcal{I}[\mathcal{C}]\)
This paper provides an important new object in homotopy theory in the strictly commutative model \(A^\mathcal{I}(X)\), which is expected to be a useful replacement of the usual chains \(C(X; k)\) of a space, which are an \(E_\infty\)-algebra. Additionally, the constructions in the paper provide a simple construction for an \(E_\infty\)-model for the cochain algebra of a space \(X\) in the homotopy colimit \(A^\mathcal{I}(X)_{h\mathcal{I}}\).
Reviewer: Daniel Robert-Nicoud (Zürich)Replicators, Manin white product of binary operads and average operators.https://www.zbmath.org/1453.180172021-02-27T13:50:00+00:00"Pei, Jun"https://www.zbmath.org/authors/?q=ai:pei.jun"Bai, Chengming"https://www.zbmath.org/authors/?q=ai:bai.chengming"Guo, Li"https://www.zbmath.org/authors/?q=ai:guo.li"Ni, Xiang"https://www.zbmath.org/authors/?q=ai:ni.xiangStructure theorems for dendriform and tridendriform algebras.https://www.zbmath.org/1453.180182021-02-27T13:50:00+00:00"Burgunder, Emily"https://www.zbmath.org/authors/?q=ai:burgunder.emily"Delcroix-Oger, Bérénice"https://www.zbmath.org/authors/?q=ai:delcroix-oger.bereniceWith the help of a method discovered by the same authors in a preceding paper [``Rigidity theorem, freeness of algebras and applications'', Preprint, \url{arXiv:1701.01323}], several rigidity theorems are proved and then used to show the freeness of several well-known combinatorial Hopf algebras as a dendriform or tridendriform algebras. In all these cases, the considered objects are given a coalgebraic and an algebraic structure on two operads, with a certain compatibility between them; it is proved that, under a connectivity condition, such an object is free and cofree on the corresponding operads.
In the context of planar binary trees, skew duplicial algebras are defined as a sort of deformation of duplicial algebras. The pairs of operads with a rigidity theorem are duplicial-dendriform, dendriform-dendriform, skew duplicial-dendriform, skew duplicial-duplicial and skew duplicial-skew duplicial. This is used to give a new proof that the Hopf algebras of surjections and of parking functions are free as dendriform algebras.
In the context of planar reduced trees, the notion of terplicial algebra is introduced by analogy with skew duplicial algebras. It differs from the notion of triduplicial algebra defined by Novelli and Thibon. The pairs of operads with a rigidity theorem are terplicial-terplicial, terplical-tridendriform and tridendrifom-tridendriform. This is used to show that the Solomon-Tits is a free tridendriform algebra.
A combinatorial description of the products and the coproducts induced by these structures on planar binary trees and on reduced trees is given, using paths and cuts.
For the entire collection see [Zbl 1437.05003].
Reviewer: Loïc Foissy (Calais)A theory of linear typings as flows on 3-valent graphs.https://www.zbmath.org/1453.030122021-02-27T13:50:00+00:00"Zeilberger, Noam"https://www.zbmath.org/authors/?q=ai:zeilberger.noamC-systems defined by universe categories: presheaves.https://www.zbmath.org/1453.030672021-02-27T13:50:00+00:00"Voevodsky, Vladimir"https://www.zbmath.org/authors/?q=ai:voevodsky.vladimir-aleksandrovichSummary: The main result of this paper may be stated as a construction of ``almost representations'' \(\mu_n\) and \(\tilde{\mu}_n\) for the presheaves \(\mathcal Ob_n\) and \(\widetilde{\mathcal{O}b}_n\) on the C-systems defined by locally cartesian closed universe categories with binary product structures and the study of the behavior of these ``almost representations'' with respect to the universe category functors.
In addition, we study a number of constructions on presheaves on C-systems and on universe categories that are used in the proofs of our main results, but are expected to have other applications as well.Facets of congruence distributivity in Goursat categories.https://www.zbmath.org/1453.180062021-02-27T13:50:00+00:00"Gran, Marino"https://www.zbmath.org/authors/?q=ai:gran.marino"Rodelo, Diana"https://www.zbmath.org/authors/?q=ai:rodelo.diana"Tchoffo Nguefeu, Idriss"https://www.zbmath.org/authors/?q=ai:nguefeu.idriss-tchoffoMal'tsev categories extend 2-permutable varieties of universal algebras and include many non-varietal examples. The paper does the same for Trapezoid Lemma and Triangular Lemma studied in universal algebra and introduces equivalence distributive Mal'tsev and Goursat categories.
Reviewer: Jiří Rosický (Brno)GAGA theorems in derived complex geometry.https://www.zbmath.org/1453.140042021-02-27T13:50:00+00:00"Porta, Mauro"https://www.zbmath.org/authors/?q=ai:porta.mauroThe author develops some aspects of derived analytic geometry as introduced by \textit{J. Lurie} in [``Derived Algebraic Geometry V: Structured Spaces'', preprint, \url{arXiv:0905.0459}]. Therefore, a basic knowledge of Lurie's work and the language of \(\infty\)-categories is required for reading the paper.
The paper starts by reviewing the notion of derived analytic space in the sense of Lurie's work. Then, it is shown that there is a functor of points interpretation of the derived spaces obtained on a suitable site of derived Stein spaces. Next, the derived version of the analytication functor is introduced. This is a generalization of the functor studied by Serre extended to the category of derived stacks (more precisely derived Deligne-Mumford stack locally almost of finite presentation) over \(\mathbb{C}\), sending them to derived analytic stacks over \(\mathbb{C}\).
This derived analytification functor is used to prove the main results of the paper. These are a series of GAGA statements that compare algebraic and analytic notions on derived stacks. In particular, it is proved that for a proper morphism of analytic derived Artin stacks \(f: X \to Y\) the push-forward functor induces a functor between the \(\infty\)-categories of coherent sheaves \(f_*: \mathrm{Coh}^-(X) \to \mathrm{Coh}^-(Y)\), that for a morphism of algebraic derived Artin stacks \(f: X \to Y\) the identity \((-)^{\mathrm{an}} \circ f_* = f_* \circ (-)^{\mathrm{an}}\) holds, where \((-)^{\mathrm{an}}\) is the analytification functor, and that for an algebraic proper derived Artin stack \(X\) one has that \(\mathrm{Coh}(X) \cong\mathrm{Coh}(X^{\mathrm{an}})\).
These results are then applied for studying the essential image of the analytification functor and the deformation theory of derived analytic spaces. In particular, it is proved that a proper derived analytic space is in the image of \((-)^{\mathrm{an}}\) if and only if its classical underlying analytic space is, i.e. if and only if its \(0\)-th truncation is the analytification of an algebraic variety over \(\mathbb{C}\). Then, it proved that for any derived analytic space \(X\) and any \(x \in X\) the shifted tangent complex at \(x\), usually denoted \(\mathbb{T}_x X[-1]\), admits a differential graded Lie algebra structure.
Reviewer: Federico Bambozzi (Oxford)The defect recollement, the MacPherson-Vilonen construction, and pp formulas.https://www.zbmath.org/1453.180092021-02-27T13:50:00+00:00"Dean, Samuel"https://www.zbmath.org/authors/?q=ai:dean.samuelSummary: For any abelian category \(\mathcal{A}\), Auslander constructed a localisation \(w : \mathrm{fp}(\mathcal{A}^{\mathrm{op}}, \mathrm{Ab}) \rightarrow \mathcal{A}\) called the defect, which is the left adjoint to the Yoneda embedding \(Y : \mathcal{A} \rightarrow \mathrm{fp}(\mathcal{A}^{\mathrm{op}}, \mathrm{Ab})\). If \(\mathcal{A}\) has enough projectives, then this localisation is part of a recollement called the defect recollement. We show that this recollement is an instance of the MacPherson-Vilonen construction if and only if \(\mathcal{A}\) is hereditary. We also discuss several subcategories of \(\mathrm{fp}(\mathcal{A}^{\mathrm{op}}, \mathrm{Ab})\) which arise as canonical features of the defect recollement, and characterise them by properties of their projective presentations and their orthogonality with other subcategories. We apply some parts of the defect recollement to the model theory of modules. Let \(R\) be a ring and let \(\phi / \psi\) be a pp-pair. When \(R\) is an artin algebra, we show that there is a smallest pp formula \(\rho\) such that \(\psi \leqslant \rho \leqslant \phi\) which agrees with \(\phi\) on injectives, and that there is a largest pp formula \(\mu\) such that \(\psi \leqslant \mu \leqslant \phi\) and \(\psi R = \mu R\). When \(R\) is left coherent, we show that there is a largest pp formula \(\sigma\) such that \(\psi \leqslant \sigma \leqslant \phi\) which agrees with \(\psi\) on injectives, and that the pp-pair \(\psi / \phi\) is isomorphic to a pp formula if and only if \(\psi = \sigma\), and that there is a smallest pp formula \(\nu\) such that \(\psi \leqslant \nu \leqslant \phi\) and \(\phi R = \nu R\). We also show that, for any pp-pair \(\phi / \psi\), \(w(\phi / \psi) \cong(D \psi) R /(D \phi) R\), where \(D\) is the elementary duality of pp formulas. We also give expressions for \(w(\phi / \psi)\) in terms of free realisation of \(\phi\) and \(\psi\).Allegories: decidability and graph homomorphisms.https://www.zbmath.org/1453.180022021-02-27T13:50:00+00:00"Pous, Damien"https://www.zbmath.org/authors/?q=ai:pous.damien"Vignudelli, Valeria"https://www.zbmath.org/authors/?q=ai:vignudelli.valeriaSymmetric monoidal \(g\)-categories and their strictification.https://www.zbmath.org/1453.180142021-02-27T13:50:00+00:00"Guillou, Bertrand J."https://www.zbmath.org/authors/?q=ai:guillou.bertrand-j"May, J. Peter"https://www.zbmath.org/authors/?q=ai:may.j-peter"Merling, Mona"https://www.zbmath.org/authors/?q=ai:merling.mona"Osorno, Angélica M."https://www.zbmath.org/authors/?q=ai:osorno.angelica-mSummary: We give an operadic definition of a \textit{genuine} symmetric monoidal \(G\)-category, and we prove that its classifying space is a genuine \(E_\infty G\)-space. We do this by developing some very general categorical coherence theory. We combine results of \textit{A. S. Corner} and \textit{N. Gurski} [``Operads with general groups of equivariance, and some 2-categorical aspects of operads in Cat'', Preprint, \url{arXiv:1312.5910}], \textit{A. J. Power} [J. Pure Appl. Algebra 57, No. 2, 165--173 (1989; Zbl 0668.18010)] and \textit{S. Lack} [J. Pure Appl. Algebra 175, No. 1--3, 223--241 (2002; Zbl 1142.18301)] to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal \(G\)-categories to genuine permutative \(G\)-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When \(G\) is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal \(G\)-categories as input to an equivariant infinite loop space machine that gives genuine \(\Omega\)-\(G\)-spectra as output.Descent in algebraic \(K\)-theory and a conjecture of Ausoni-Rognes.https://www.zbmath.org/1453.180112021-02-27T13:50:00+00:00"Clausen, Dustin"https://www.zbmath.org/authors/?q=ai:clausen.dustin"Mathew, Akhil"https://www.zbmath.org/authors/?q=ai:mathew.akhil"Naumann, Niko"https://www.zbmath.org/authors/?q=ai:naumann.niko"Noel, Justin"https://www.zbmath.org/authors/?q=ai:noel.justinSummary: Let \(A \to B\) be a \(G\)-Galois extension of rings, or more generally of \(\mathbb{E}_\infty \)-ring spectra in the sense of \textit{J. Rognes} [in: Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13--21, 2014. Vol. II: Invited lectures. Seoul: KM Kyung Moon Sa. 1259--1283 (2014; Zbl 1373.19002)]. A basic question in algebraic \(K\)-theory asks how close the map \(K(A) \to K(B)^{hG}\) is to being an equivalence, i.e., how close algebraic \(K\)-theory is to satisfying Galois descent. An elementary argument with the transfer shows that this equivalence is true rationally in most cases of interest. Motivated by the classical descent theorem of \textit{R. W. Thomason} [Ann. Sci. Éc. Norm. Supér. (4) 18, 437--552 (1985; Zbl 0596.14012)], one also expects such a result after periodic localization. We formulate and prove a general result which enables one to promote rational descent statements as above into descent statements after periodic localization. This reduces the localized descent problem to establishing an elementary condition on \(K_0(-)\otimes \mathbb{Q} \). As applications, we prove various descent results in the periodically localized \(K\)-theory, \(TC, THH\), etc. of structured ring spectra, and verify several cases of a conjecture of \textit{C. Ausoni} and \textit{J. Rognes} [Geom. Topol. 16, No. 4, 2037--2065 (2012; Zbl 1260.19004)].Reconstruction of tensor categories from their structure invariants.https://www.zbmath.org/1453.180162021-02-27T13:50:00+00:00"Chen, Hui-Xiang"https://www.zbmath.org/authors/?q=ai:chen.huixiang"Zhang, Yinhuo"https://www.zbmath.org/authors/?q=ai:zhang.yinhuoAuthors' abstract: We study tensor (or monoidal) categories of finite rank over an algebraically closed field \(\mathcal F\). Given a tensor category \(\mathcal C\), we have two structure invariants of \(\mathcal C\): the Green ring (or the representation ring) \(r(\mathcal C)\) and the Auslander algebra \(A(\mathcal C)\) of \(\mathcal C\). We show that a Krull-Schmit abelian tensor category \(\mathcal C\) of finite rank is uniquely determined (up to tensor equivalences) by its two structure invariants and the associated associator system of \(\mathcal C\). In fact, we can reconstruct the tensor category \(\mathcal C\) from its two invariants and the associator system. More general, we can construct a Krull-Schmidt and abelian tensor category \(\mathcal C\) over \(\mathcal F\) such that \(R\) is the Green ring of \(\mathcal C\) and \(A\) is the Auslander algebra of \(\mathcal C\). Moreover, we will give a necessary and sufficient condition for such two tensor categories to be tensor equivalent.
Reviewer: J. N. Alonso Alvarez (Vigo)The Farrell-Jones conjecture for hyperbolic and CAT(0)-groups revisited.https://www.zbmath.org/1453.200592021-02-27T13:50:00+00:00"Kasprowski, Daniel"https://www.zbmath.org/authors/?q=ai:kasprowski.daniel"Rüping, Henrik"https://www.zbmath.org/authors/?q=ai:ruping.henrikCotorsion pairs and a \(K\)-theory localization theorem.https://www.zbmath.org/1453.190042021-02-27T13:50:00+00:00"Sarazola, Maru"https://www.zbmath.org/authors/?q=ai:sarazola.maruSummary: We show that a complete hereditary cotorsion pair \((\mathcal{C}, \mathcal{C}^\bot)\) in an exact category \(\mathcal{E} \), together with a subcategory \(\mathcal{Z} \subseteq \mathcal{E}\) containing \(\mathcal{C}^\bot\), determines a Waldhausen category structure on the exact category \(\mathcal{C}\), in which \(\mathcal{Z}\) is the class of acyclic objects. This allows us to prove a new version of Quillen's Localization Theorem, relating the \(K\)-theory of exact categories \(\mathcal{A} \subseteq \mathcal{B}\) to that of a cofiber. The novel idea in our approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, we do not require \(\mathcal{A}\) to be a Serre subcategory, which produces new examples.
Due to the algebraic nature of our Waldhausen categories, we are able to recover a version of Quillen's Resolution Theorem, now in a more homotopical setting that allows for weak equivalences.Reductions between certain incidence problems and the continuum hypothesis.https://www.zbmath.org/1453.030562021-02-27T13:50:00+00:00"da Silva, Samuel G."https://www.zbmath.org/authors/?q=ai:gomes-da-silva.samuelThe present paper treats the set theory of the real line in the spirit of \textit{C. Freiling} [J. Symb. Log. 51, 190--200 (1986; Zbl 0619.03035)]. Informally speaking, two families \(\mathcal{P}_1\) and \(\mathcal{P}_2\) of incidence problems
dealing with real numbers and countable subsets of the real line are considered. A typical problem from \(\mathcal{P}_1\) is as follows: given a real number \(x\), pick a countable set \(A\) of reals at random, hoping that \(x\in A\). A typical problem from \(\mathcal{P}_2\) is slightly different: given a countable set \(A\) of reals, pick a real number \(x\) at random, hoping that
\(x\not\in A\). At least intuitively, one could defend the assertion that problems from \(\mathcal{P}_2\) are at least as easy
to solve as problems from \(\mathcal{P}_1\); and, after setting up some suitable formalization, the author proves (within \(\mathsf{ZFC}\)) that this is indeed the case. However, the assertion that problems from \(\mathcal{P}_1\) have the same level of difficulty as problems from \(\mathcal{P}_2\) is shown to be equivalent to the continuum hypothesis.
Reviewer: Paul Bankston (Milwaukee)Degrees of relatedness. A unified framework for parametricity, irrelevance, ad hoc polymorphism, intersections, unions and algebra in dependent type theory.https://www.zbmath.org/1453.030052021-02-27T13:50:00+00:00"Nuyts, Andreas"https://www.zbmath.org/authors/?q=ai:nuyts.andreas"Devriese, Dominique"https://www.zbmath.org/authors/?q=ai:devriese.dominiqueTilting bundles for weighted projective lines of weight type \((p,q)\).https://www.zbmath.org/1453.140552021-02-27T13:50:00+00:00"Zhang, Xiaofeng"https://www.zbmath.org/authors/?q=ai:zhang.xiaofengSummary: We give a combinatorial description of tilting bundles in the category of coherent sheaves over the weighted projective line of type \((p,q)\). We define a set of admissible paths in the rectangular area \(\mathbb{D}_{(p,q)}=[1,p-1]\times [1,q-1]\), and show that there is a bijection between the fundamental tilting bundles and the admissible paths in \(\mathbb{D}_{(p,q)}\).Functors and morphisms determined by subcategories.https://www.zbmath.org/1453.180012021-02-27T13:50:00+00:00"Zhu, Shijie"https://www.zbmath.org/authors/?q=ai:zhu.shijieSummary: We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a Hom-finite hereditary abelian category with enough projectives, we prove that the Auslander-Reiten-Smalø-Ringel formula of the minimal right determiner still holds. As an application, we give a formula of minimal right determiners in the category of finitely presented representations of strongly locally finite quivers.Lax orthogonal factorisations in monad-quantale-enriched categories.https://www.zbmath.org/1453.180082021-02-27T13:50:00+00:00"Clementino, Maria Manuel"https://www.zbmath.org/authors/?q=ai:clementino.maria-manuel"López Franco, Ignacio"https://www.zbmath.org/authors/?q=ai:lopez-franco.ignacio-lSummary: We show that, for a quantale \(V\) and a \(\mathsf{Set}\)-monad \(\mathbb{T}\) laxly extended to \(V\)-\(\mathsf{Rel}\), the presheaf monad on the category of \((\mathbb{T},V)\)-categories is simple, giving rise to a lax orthogonal factorisation system (lofs) whose corresponding weak factorisation system has embeddings as left part. In addition, we present presheaf submonads and study the LOFSs they define. This provides a method of constructing weak factorisation systems on some well-known examples of topological categories over \(\mathsf{Set}\).Bases for pseudovarieties closed under bideterministic product.https://www.zbmath.org/1453.200762021-02-27T13:50:00+00:00"Costa, Alfredo"https://www.zbmath.org/authors/?q=ai:costa.alfredo"Escada, Ana"https://www.zbmath.org/authors/?q=ai:escada.ana-pSummary: We show that if \(\mathsf{V}\) is a semigroup pseudovariety containing the finite semilattices and contained in \(\mathsf{DS}\), then it has a basis of pseudoidentities between finite products of regular pseudowords if, and only if, the corresponding variety of languages is closed under bideterministic product. The key to this equivalence is a weak generalization of the existence and uniqueness of \(\mathsf{J}\)-reduced factorizations. This equational approach is used to address the locality of some pseudovarieties. In particular, it is shown that \(\mathsf{DH}\cap \mathsf{ECom}\) is local, for any group pseudovariety \(\mathsf{H}\).On two notions of a gerbe over a stack.https://www.zbmath.org/1453.140032021-02-27T13:50:00+00:00"Chatterjee, Saikat"https://www.zbmath.org/authors/?q=ai:chatterjee.saikat"Koushik, Praphulla"https://www.zbmath.org/authors/?q=ai:koushik.praphullaThis paper is concerned with the relationship between two different notions of a \textit{differentiable gerbe over a differentiable stack}. One is a morphism of stacks abiding by some additional requirements [\textit{K. Behrend} and \textit{P. Xu}, J. Symplectic Geom. 9, No. 3, 285--341 (2011; Zbl 1227.14007); \url{https://www.uni-due.de/~hm0002/stacks.pdf]}; \url{https://arxiv.org/pdf/math/0605694.pdf}]. The other is a Morita equivalence class of a Lie groupoid extension [\textit{C. Laurent-Gengoux} et al., Adv. Math. 220, No. 5, 1357--1427 (2009; Zbl 1177.22001)]. On the one hand, given a Lie groupoid \(\mathcal{G}\), the category of principal \(\mathcal{G}\)-bundles, denoted by \(B\mathcal{G}\), is a differentiable stack [loc. cit.]. On the other hand, given a differentiable stack \(\mathcal{D}\), there exists a Lie groupoid \(\mathcal{H}\)\ such that \(\mathcal{D}\)\ is isomorphic to \(B\mathcal{H}\)\ [\textit{E. Lerman}, Enseign. Math. (2) 56, No. 3--4, 315--363 (2010; Zbl 1221.14003)]. The central idea in this paper is the correspondence between Lie groupoids and differentiable stacks.
A synopsis of the paper consisting of five sections goes as follows. \S 2 aims to introduce the differentiable gerbe over a stack. \S 3 aims to introduce the classifying stack of a Lie groupoid and the notion of \(\mathcal{G}\)-\(\mathcal{H}\) bibundle for Lie groupoids \(\mathcal{G}\)\ and \(\mathcal{H}\). \S 4 addresses the correspondence between the two definitions, exploring the possibility of finding a Morita equivalence class of a Lie groupoid extension from a gerbe over a stack \(F:\mathcal{D}\rightarrow\mathcal{C}\). \S 5 considers Morita equivalence class of a Lie groupoid extension, recovering a morphism of stacks obedient to the required properties of a gerbe over a stack on the lines of [loc. cit.].
Reviewer: Hirokazu Nishimura (Tsukuba)