Recent zbMATH articles in MSC 18https://zbmath.org/atom/cc/182022-11-17T18:59:28.764376ZWerkzeugStructural realism and category mistakeshttps://zbmath.org/1496.030262022-11-17T18:59:28.764376Z"Landry, Elaine"https://zbmath.org/authors/?q=ai:landry.elaineThe entire volume has been reviewed in [Zbl 1491.03006].
For the entire collection see [Zbl 1491.03006].Categorical structures for type theory in univalent foundationshttps://zbmath.org/1496.030532022-11-17T18:59:28.764376Z"Ahrens, Benedikt"https://zbmath.org/authors/?q=ai:ahrens.benedikt"Lumsdaine, Peter Lefanu"https://zbmath.org/authors/?q=ai:lumsdaine.peter-lefanu"Voevodsky, Vladimir"https://zbmath.org/authors/?q=ai:voevodsky.vladimir-aleksandrovichSummary: In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: \textit{categories with families}, \textit{split type-categories}, and \textit{representable maps of presheaves}. We study these in univalent type theory, where the comparisons between them can be given more elementarily than in set-theoretic foundations. Specifically, we construct maps between the various types of structures, and show that assuming the Univalence axiom, some of the comparisons are equivalences.
We then analyze how these structures transfer along (weak and strong) equivalences of categories, and, in particular, show how they descend from a category (not assumed univalent/saturated) to its Rezk completion. To this end, we introduce \textit{relative universes}, generalizing the preceding notions, and study the transfer of such relative universes along suitable structure.
We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the \texttt{UniMath} library.From term models to domainshttps://zbmath.org/1496.030632022-11-17T18:59:28.764376Z"Phoa, Wesley"https://zbmath.org/authors/?q=ai:phoa.wesleySummary: Let B be the closed term model of the \(\lambda\)-calculus in which terms with the same Böhm tree are identified. We investigate which partial equivalence relations (PERs) on B can be regarded as predomains or domains. Working inside the realizability topos on B, such PERs can be regarded simply as sets in a particular model of constructive set theory.
No well-behaved partial order has been identified for any class of PERs; but it is still possible to isolate those PERs which have `suprema of chains' in a certain sense, and all maps between such PERs in the model preserve such suprema of chains. One can also define what it means for such a PER to have a `bottom'; partial function spaces provide an example. For these PERs, fixed points of arbitrary endofunctions exist and are computed by the fixed point combinatory. There is also a notion of meet-closure for which all maps are stable.
The categories of predomains are closed under the formation of total and partial function spaces, polymorphic types and convex powerdomains. (Subtyping and bounded quantification can also be modelled.) They in fact form reflective subcategories of the realizability topos; and in this set-theoretic context, these constructions are very simple to describe.
For the entire collection see [Zbl 0875.00067].Coherence and valid isomorphism in closed categories. Applications of proof theory to category theory in a computer scientist perspective (notes for an invited lecture)https://zbmath.org/1496.031342022-11-17T18:59:28.764376Z"Longo, Guiseppe"https://zbmath.org/authors/?q=ai:longo.giuseppeFor the entire collection see [Zbl 0712.68006].A final coalgebra theoremhttps://zbmath.org/1496.032062022-11-17T18:59:28.764376Z"Aczel, Peter"https://zbmath.org/authors/?q=ai:aczel.peter"Mendler, Nax"https://zbmath.org/authors/?q=ai:mendler.nax-paulSummary: We prove that every set-based functor on the category of classes has a final coalgebra. This result strengthens the final coalgebra theorem announced in the book [Non-well-founded sets. Stanford, CA: Center for the Study of Language and Information (1988; Zbl 0668.04001)] by the first author.
For the entire collection see [Zbl 0712.68006].A Dialectica-like model of linear logichttps://zbmath.org/1496.032522022-11-17T18:59:28.764376Z"de Paiva, Valeria C. V."https://zbmath.org/authors/?q=ai:de-paiva.valeriaFor the entire collection see [Zbl 0712.68006].Categorical logic and model theoryhttps://zbmath.org/1496.032632022-11-17T18:59:28.764376Z"Bell, John L."https://zbmath.org/authors/?q=ai:bell.john-laneThe entire volume has been reviewed in [Zbl 1491.03006].
For the entire collection see [Zbl 1491.03006].Dictoseshttps://zbmath.org/1496.032642022-11-17T18:59:28.764376Z"Ehrhard, Thomas"https://zbmath.org/authors/?q=ai:ehrhard.thomasFor the entire collection see [Zbl 0712.68006].Coherent diagrammatic reasoning in compositional distributional semanticshttps://zbmath.org/1496.032672022-11-17T18:59:28.764376Z"Wijnholds, Gijs Jasper"https://zbmath.org/authors/?q=ai:wijnholds.gijs-jasperSummary: The framework of categorical compositional distributional models of meaning [\textit{B. Coecke} et al., ``Mathematical foundations for a compositional distributional model of meaning'', Preprint, \url{arXiv:1003.4394}], inspired by category theory, allows one to compute the meaning of natural language phrases, given basic meaning entities assigned to words. Composing word meanings is the result of a functorial passage from syntax to semantics. To keep one from drowning in technical details, diagrammatic reasoning is used to represent the information flow of sentences that exists independently of the concrete instantiation of the model. Not only does this serve the purpose of clarification, it moreover offers computational benefits as complex diagrams can be transformed into simpler ones, which under coherence can simplify computation on the semantic side. Until now, diagrams for compact closed categories and monoidal closed categories have been used (see [\textit{B. Coecke} et al., Ann. Pure Appl. Logic 164, No. 11, 1079--1100 (2013; Zbl 1280.03026); ``Mathematical foundations for a compositional distributional model of meaning'', Preprint, \url{arXiv:1003.4394}]). These correspond to the use of pregroup grammar [\textit{J. Lambek}, Lect. Notes Comput. Sci. 1582, 1--27 (1999; Zbl 0934.03043)] and the Lambek calculus [\textit{J. Lambek}, Am. Math. Mon. 65, 154--170 (1958; Zbl 0080.00702)] for syntactic structure, respectively. Unfortunately, the diagrammatic language of \textit{J. C. Baez} and \textit{M. Stay} [Lect. Notes Phys. 813, 95--172 (2011; Zbl 1218.81008)] has not been proven coherent. In this paper, we develop a graphical language for the (categorical formulation of) the nonassociative Lambek calculus [\textit{J. Lambek}, ``On the calculus of syntactic types'', in: Structure of language and its mathematical aspects. Providence, R.I.: American Mathematical Society (AMS). 166--178 (1961)]. This has the benefit of modularity where extension of the system are easily incorporated in the graphical language. Moreover, we show the language is coherent with monoidal closed categories without associativity, in the style of \textit{P. Selinger}'s survey paper [Lect. Notes Phys. 813, 289--355 (2011; Zbl 1217.18002)].
For the entire collection see [Zbl 1369.03021].Noncommutative geometry. A functorial approachhttps://zbmath.org/1496.140022022-11-17T18:59:28.764376Z"Nikolaev, Igor V."https://zbmath.org/authors/?q=ai:nikolaev.igor-vladimirovich|nikolaev.igor-vasilievichPublisher's description: Noncommutative geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. This book covers the key concepts of noncommutative geometry and its applications in topology, algebraic geometry, and number theory. Our presentation is accessible to the graduate students as well as nonexperts in the field. The second edition includes two new chapters on arithmetic topology and quantum arithmetic.
\begin{itemize}
\item An authoritative introductory treatment of noncommutative geometry now in its second edition.
\item A novel approach using functors is presented in detail.
\item Covers applications of the theory in topology, algebraic geometry, and number theory.
\end{itemize}
See the review of the first edition in [Zbl 1388.14001].On tensor products of matrix factorizationshttps://zbmath.org/1496.150102022-11-17T18:59:28.764376Z"Fomatati, Yves Baudelaire"https://zbmath.org/authors/?q=ai:fomatati.yves-baudelaireSummary: Let \(K\) be a field. Let \(f \in K [[x_1, \dots, x_r]]\) and \(g \in K [[ y_1,\dots, y_s]]\) be nonzero elements. If \(X\) (resp. \(Y)\) is a matrix factorization of \(f\) (resp. \(g)\), Yoshino had constructed a tensor product (of matrix factorizations) \(\widehat{\otimes}\) such that \(X \widehat{\otimes} Y\) is a matrix factorization of \(f + g \in K [[x_1, \dots, x_r, y_1, \dots, y_s]]\). In this paper, we propose a bifunctorial operation \(\widetilde{\otimes}\) and its variant \(\widetilde{\otimes}^\prime\) such that \(X \widetilde{\otimes} Y\) and \(X \widetilde{\otimes}^\prime Y\) are two different matrix factorizations of \(f g \in K [[x_1, \dots, x_r, y_1, \dots, y_s]]\). We call \(\widetilde{\otimes}\)\textit{the multiplicative tensor product} of \(X\) and \(Y\). Several properties of \(\widetilde{\otimes}\) are proved. Moreover, we find three functorial variants of Yoshino's tensor product \(\widehat{\otimes}\). Then, \(\widetilde{\otimes}\) (or its variant) is used in conjunction with \(\widehat{\otimes}\) (or any of its variants) to give an improved version of the standard algorithm for factoring polynomials using matrices on the class of \textit{summand-reducible polynomials} defined in this paper. Our algorithm produces matrix factors whose size is at most one half the size one obtains using the standard method.Reduction techniques for the finitistic dimensionhttps://zbmath.org/1496.160092022-11-17T18:59:28.764376Z"Green, Edward L."https://zbmath.org/authors/?q=ai:green.edward-lee"Psaroudakis, Chrysostomos"https://zbmath.org/authors/?q=ai:psaroudakis.chrysostomos"Solberg, Øyvind"https://zbmath.org/authors/?q=ai:solberg.oyvindSummary: In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples.Resolving resolution dimension of recollements of abelian categorieshttps://zbmath.org/1496.160102022-11-17T18:59:28.764376Z"Zhang, Houjun"https://zbmath.org/authors/?q=ai:zhang.houjun"Zhu, Xiaosheng"https://zbmath.org/authors/?q=ai:zhu.xiaoshengTilting modules and dominant dimension with respect to injective moduleshttps://zbmath.org/1496.160142022-11-17T18:59:28.764376Z"Adachi, Takahide"https://zbmath.org/authors/?q=ai:adachi.takahide"Tsukamoto, Mayu"https://zbmath.org/authors/?q=ai:tsukamoto.mayuSummary: In this paper, we study a relationship between tilting modules with finite projective dimension and dominant dimension with respect to injective modules as a generalization of results of Crawley-Boevey-Sauter, Nguyen-Reiten-Todorov-Zhu and Pressland-Sauter. Moreover, we give characterizations of almost \(n\)-Auslander-Gorenstein algebras and almost \(n\)-Auslander algebras by the existence of tilting modules. As an application, we describe a sufficient condition for almost 1-Auslander algebras to be strongly quasi-hereditary by comparing such tilting modules and characteristic tilting modules.Derived Picard groups of preprojective algebras of Dynkin typehttps://zbmath.org/1496.160182022-11-17T18:59:28.764376Z"Mizuno, Yuya"https://zbmath.org/authors/?q=ai:mizuno.yuyaSummary: In this paper, we study two-sided tilting complexes of preprojective algebras of Dynkin type. We construct the most fundamental class of two-sided tilting complexes, which has a group structure by derived tensor products and induces a group of auto-equivalences of the derived category. We show that the group structure of the two-sided tilting complexes is isomorphic to the braid group of the corresponding folded graph. Moreover, we show that these two-sided tilting complexes induce tilting mutation and any tilting complex is given as the derived tensor products of them. Using these results, we determine the derived Picard group of preprojective algebras for type \(A\) and \(D\).On cohomologies and algebraic \(K\)-theory of Lie \(p\)-superalgebrashttps://zbmath.org/1496.170182022-11-17T18:59:28.764376Z"Rakviashvili, Giorgi"https://zbmath.org/authors/?q=ai:rakviashvili.giorgiSummary: An enveloping associative superalgebra \(\Lambda [L,\alpha ,\beta ]\) and its groups of cohomologies are defined and it is proved that there exists Frobenius multiplication of the Quillen algebraic \(K\)-functors of \(\Lambda [L,\alpha ,\beta ]\). These results generalize corresponding results for Lie \(p\)-algebras which were proved by the author earlier.A synthetic perspective on \((\infty,1)\)-category theory. Fibrational and semantic aspectshttps://zbmath.org/1496.180012022-11-17T18:59:28.764376Z"Weinberger, Jonathan Maximilian Lajos"https://zbmath.org/authors/?q=ai:weinberger.jonathan-maximilian-lajos(no abstract)Structuralism, invariance, and univalencehttps://zbmath.org/1496.180022022-11-17T18:59:28.764376Z"Awodey, Steve"https://zbmath.org/authors/?q=ai:awodey.steveThe entire volume has been reviewed in [Zbl 1491.03006].
For the entire collection see [Zbl 1491.03006].Category theory and foundationshttps://zbmath.org/1496.180032022-11-17T18:59:28.764376Z"Ernst, Michael"https://zbmath.org/authors/?q=ai:ernst.michael-dThe entire volume has been reviewed in [Zbl 1491.03006].
For the entire collection see [Zbl 1491.03006].Categories and modalitieshttps://zbmath.org/1496.180042022-11-17T18:59:28.764376Z"Kishida, Kohei"https://zbmath.org/authors/?q=ai:kishida.koheiThe entire volume has been reviewed in [Zbl 1491.03006].
For the entire collection see [Zbl 1491.03006].Canonical mapshttps://zbmath.org/1496.180052022-11-17T18:59:28.764376Z"Marquis, Jean-Pierre"https://zbmath.org/authors/?q=ai:marquis.jean-pierreThe entire volume has been reviewed in [Zbl 1491.03006].
For the entire collection see [Zbl 1491.03006].Unfolding FOLDS: a foundational framework for abstract mathematical conceptshttps://zbmath.org/1496.180062022-11-17T18:59:28.764376Z"Marquis, Jean-Pierre"https://zbmath.org/authors/?q=ai:marquis.jean-pierreThe entire volume has been reviewed in [Zbl 1491.03006].
For the entire collection see [Zbl 1491.03006].Homotopy type theory: a synthetic approach to higher equalitieshttps://zbmath.org/1496.180072022-11-17T18:59:28.764376Z"Shulman, Michael"https://zbmath.org/authors/?q=ai:shulman.michael-aThe entire volume has been reviewed in [Zbl 1491.03006].
For the entire collection see [Zbl 1491.03006].Categories as mathematical modelshttps://zbmath.org/1496.180082022-11-17T18:59:28.764376Z"Spivak, David I."https://zbmath.org/authors/?q=ai:spivak.david-iThe entire volume has been reviewed in [Zbl 1491.03006].
For the entire collection see [Zbl 1491.03006].Category theory and the foundations of classical space-time theorieshttps://zbmath.org/1496.180092022-11-17T18:59:28.764376Z"Weatherall, James Owen"https://zbmath.org/authors/?q=ai:weatherall.james-owenThe entire volume has been reviewed in [Zbl 1491.03006].
For the entire collection see [Zbl 1491.03006].Algebraic models of cubical weak \(\infty\)-categories with connectionshttps://zbmath.org/1496.180102022-11-17T18:59:28.764376Z"Kachour, Camell"https://zbmath.org/authors/?q=ai:kachour.camellThe author defines the category of cubical categorical stretchings, which is the cubical analogue of the category of globular categorical stretchings in [\textit{J. Penon}, Cah. Topologie Géom. Différ. Catégoriques 40, No. 1, 31--80 (1999; Zbl 0918.18006)]. The key ingredient is a cubical analogue of the globular contractions built there. A monad \(\mathbb{W}\) on the category of cubical sets whose algebras are the models of cubical weak \(\infty\)-categories \ with connections is given. The monad is the cubical analogue of the monad \(\mathbb{P}_{C}^{0}\) there, whose \(\mathbb{P}_{C}^{0}\)-algebras are the globular \(\infty\)-categories of Penon. Main proofs use sketch theory initiated by Charles Ehresmann and his students, especially [\textit{L. Coppey} and \textit{C. Lair}, Diagrammes 13, 112 p. (1985; Zbl 0594.18006)].
In [\textit{K. Kachour}, Cah. Topol. Géom. Différ. Catég. 49, No. 1, 1--68 (2008; Zbl 1202.18004); Theory Appl. Categ. 30, 775--807 (2015; Zbl 1320.18004); \textit{J. Penon}, Cah. Topologie Géom. Différ. Catégoriques 40, No. 1, 31--80 (1999; Zbl 0918.18006)] some computations were described for globular higher structures born with globular stretcchings. It was proved in [loc. cit.] that in dimension \(2\), the globular weak \(\infty\)-categories of Penon are bicategories. The author gives in \S 4.3 a precise definition of the dimension for algebras of models of cubical weak \(\infty\)-categories. Computations in dimension \(2\) lead to long computations and go beyond the scope of this paper, but the reader can verify that the models of dimension \(2\) are weak double categories in the sense of [\textit{D. Verity}, Repr. Theory Appl. Categ. 2011, No. 20, 266 p. (2011; Zbl 1254.18001)]. The author exhibits an example of cubical coherence cell in dimension \(2\) in \S 4.1.
Reviewer: Hirokazu Nishimura (Tsukuba)Monadic forgetful functors and (non-)presentability for \(C^\ast\)- and \(W^\ast\)-algebrashttps://zbmath.org/1496.180112022-11-17T18:59:28.764376Z"Chirvasitu, Alexandru"https://zbmath.org/authors/?q=ai:chirvasitu.alexandru"Ko, Joanna"https://zbmath.org/authors/?q=ai:ko.joannaThe original impetus for this paper was [\textit{J. Rosický}, Commun. Algebra 50, No. 1, 268--274 (2022; Zbl 1483.18006)] asking whether the forgetful functors
\[
G:\mathcal{C}_{1}^{\ast}\rightarrow\mathrm{Ban}
\]
and
\[
G_{c}:\mathcal{C}_{c,1}^{\ast}\rightarrow\mathrm{Ban}
\]
are monadic, where \(\mathcal{C}_{1}^{\ast}\)\ and \(\mathcal{C}_{c,1}^{\ast} \)\ are the categories of unital \(C^{\ast}\)- and unitial commutative \(C^{\ast} \)-algebras respectively, while \(\mathrm{Ban}\)\ is the category of Banach spaces and linear maps of norm \(\leq1\)\ as morphisms. This paper gives the affirmative answer to the above question (Theorem 2.4 and Corollary 2.6).
Theorem. The forgetful functors from the category \(\mathcal{C}_{1}^{\ast}\) to the categories of unital Banach *-algebras, unital Banach algebras and Banach spaces are all monadic. The same holds for commutative (\(C^{\ast}\)- and Banach) algebras.
The obvious modification of the previous result goes through for von Neumann or \(W^{\ast}\)-algebras (Theorem 4.11 and Corollary 4.13).
Theorem. The forgetful functors from the category \(\mathcal{W}_{1}^{\ast}\)\ of \(W^{\ast}\)-algebras to the categories of \(C^{\ast}\)-algebras, unital Banach *-algebras, unital Banach algebras and Banach spaces are all monadic.
It has been known for some time that the categories \(\mathcal{C}_{1}^{\ast} \)\ and \(\mathcal{C}_{c,1}^{\ast}\)\ are locally \(\aleph_{1}\)-presentable [\textit{J. Adámek} and \textit{J. Rosický}, Locally presentable and accessible categories. Cambridge: Cambridge University Press (1994; Zbl 0795.18007), Theorem 3.28; \textit{J. W. Pelletier} and \textit{J. Rosický}, Algebra Univers. 30, No. 2, 275--284 (1993; Zbl 0817.46057), Theorem 2.4]. A strong negation of local presentability for \(\mathcal{W}_{1}^{\ast}\)\ is demonstrated (Theorem 4.2 and Proposition 4.10).
Theorem. The only presentable objects in the category \(\mathcal{W}_{1}^{\ast}\)\ of von Neumann algebras are \(\left\{ 0\right\} \)\ and \(\mathbb{C}\).
The following speculation is established (Proposition 3.1, Corollary 3.2, Proposition 3.3 and Corollary 3.4).
Theorem. Let \(A\)\ be a commutative unital \(C^{\ast}\)-algebra and \(\mathcal{M}\)\ the class of isometric \(C^{\ast}\) morphisms.
\begin{itemize}
\item \(A\) is \(\aleph_{0}\)-generated with respect to \(\mathcal{M}\)\ in the (plain or enriched) category \(\mathcal{C}_{c,1}^{\ast}\)\ iff it is finite-dimensional.
\item \(A\) is \(\aleph_{0}\)-generated with respect to \(\mathcal{M}\)\ in the ordinary category \(\mathcal{C}_{1}^{\ast}\)\ iff it has dimension \(\leq1\).
\item \(A\) is \(\aleph_{0}\)-generated with respect to \(\mathcal{M}\)\ in the \(\mathrm{CMet}\)-enriched category \(\mathcal{C}_{1}^{\ast}\)\ iff it is finite-dimensional, where \(\mathrm{CMet}\)\ is the category of complete generalized metric spaces.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Six model categories for directed homotopyhttps://zbmath.org/1496.180122022-11-17T18:59:28.764376Z"Gaucher, Philippe"https://zbmath.org/authors/?q=ai:gaucher.philippeTwo categories are considered for modeling directed homotopy. The first of them, the category of multipointed \(d\)-spaces, is described in [\textit{P. Gaucher}, Theory Appl. Categ. 22, 588--621 (2009; Zbl 1191.55013)]. The second, the category of flows introduced in [\textit{P. Gaucher}, Homology Homotopy Appl. 5, No. 1, 549--599 (2003; Zbl 1069.55008)]. In the paper under review, three model structures are constructed for each of these categories and a comparison of the six model categories obtained is carried out.
The model category \(\mathcal{K}\) is given by the model structure \((\mathcal{C}, \mathcal{W}, \mathcal{F})\), which consists of cofibrations \(\mathcal{C}\), weak equivalences \(\mathcal{W}\), and fibrations \(\mathcal{F}\) satisfying the axioms described in [\textit{M. Hovey}, Model categories. Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0909.55001), Definition 1.1.4].
Let \(Top\) be the category of \(\Delta\)-generated spaces. The category of general topological spaces is denoted by \(\mathcal{TOP}\). There are the following three model structures on the category \(Top\).
\begin{itemize}
\item The \(q\)-model structure \((\mathcal{C}_q, \mathcal{W}_q, \mathcal{F}_q)\): the cofibrations are the retracts of the transfinite compositions of the inclusions \(S^{n-1}\subset D^n\) for \(n\geq 0\), the weak equivalences are the weak homotopy equivalence, and the fibrations are the maps satisfying the RLP with respect to inclusions \(D^n \subset D^{n+1}\) for \(n\geq 0\). The existence of this model goes back to [\textit{D. G. Quillen}, Homotopical algebra. Lecture Notes in Mathematics. 43. Berlin-Heidelberg-New York: Springer-Verlag. (1967; Zbl 0168.20903)].
\item The \(h\)-model structure \((\mathcal{C}_{\bar h}, \mathcal{W}_h, \mathcal{F}_h)\): the fibrations are the maps satisfying the RLP with respect to inclusions \(X\times \{0\} \subset X\times [0,1]\) for all topological spaces \(X\), and the weak equivalences are the homotopy equivalences. The \(h\)-model described in [\textit{A. Strøm}, Arch. Math. 23, 435--441 (1972; Zbl 0261.18015)] and [\textit{T. Barthel} and \textit{E. Riehl}, Algebr. Geom. Topol. 13, No. 2, 1089--1124 (2013; Zbl 1268.18001)].
\item The \(m\)-model structure \((\mathcal{C}_m, \mathcal{W}_m, \mathcal{F}_m)\): \((\mathcal{C}_m, \mathcal{W}_m, \mathcal{F}_m) = (\mathcal{C}_m, \mathcal{W}_q, \mathcal{F}_h)\): the cofibrations are constructed using the LLP condition with respect to \(\mathcal{W}_q\cap \mathcal{F}_h\). Its existence is a consequence of [\textit{M. Cole}, Topology Appl. 153, No. 7, 1016--1032 (2006; Zbl 1094.55015), Theorem 2.1].
\end{itemize}
Section 6 is devoted to the category of multipointed \(d\)-spaces \({\mathcal{G}}dTop\)
A multipointed \(d\)-space \(X\) is a pair \((|X|, X^0)\) where \(|X|\) is a topological space and \(X^0\) is a subset of \(|X|\). A morphism \(f: X=(|X|, X^0)\to Y=(|Y|, Y^0)\) is a continuous map \(|f|: |X|\to |Y|\) and a map \(f^0: X^0\to Y^0\) such that \((\forall s\in X^0)f^0(s)= |f|(s)\).
Let \(\mathcal{G}\) be the topological group of nondecreasing homeomorphisms of \([0, 1]\).
A multipoinded \(d\)-space \(X\) (Definition 6.3) is a triple \((|X|, X^0, \mathbb{P}^{\mathcal{G}}X)\) where \((|X|, X^0)\) is a multipointed space and \(\mathbb{P}^{\mathcal{G}}X\) is a set of continuous maps from \([0,1]\) to \(|X|\) called the execution paths, satisfying the following axioms:
\begin{itemize}
\item For any execution path \(\gamma\), one has \(\gamma(0), \gamma(1)\in X^0\).
\item For any execution path \(\gamma\) of \(X\), any composite \(\gamma.\phi\) with \(\phi\in \mathcal{G}\) is an execution path of \(X\).
\item If \(\gamma_1\) and \(\gamma_2\) are composable execution paths of \(X\), then the normalized composition \(\gamma_1*_N\gamma_2\) is an execution path of \(X\).
\end{itemize}
A morphism \(f: X\to Y\) of multipointed \(d\)-spaces is a map of multipointed spaces from \((|X|,X^0)\) to \((|Y|, Y^0)\) such that for any execution path \(\gamma\) of \(X\), the map \(f.\gamma\) is an execution path of \(Y\). The subset of execution paths from \(\alpha\) to \(\beta\) is the set of \(\gamma\in \mathbb{P}^{\mathcal{G}}X\) such that \(\gamma(0)=\alpha\) and \(\gamma(1)=\beta\) and is denoted by \(\mathbb{P}^{\mathcal{G}}_{\alpha, \beta}X\). It is equipped with the kelleyfication of the initial topology making the inclusion \(\mathbb{P}^{\mathcal{G}}_{\alpha,\beta}X \subset \mathcal{TOP}([0,1], |X|)\) is continuous.
Theorem 6.14. Let \((\mathcal{C}, \mathcal{W}, \mathcal{F})\) be one of the three model structures \[ (\mathcal{C}_q, \mathcal{W}_q, \mathcal{F}_q), (\mathcal{C}_{\bar h}, \mathcal{W}_h, \mathcal{F}_h), (\mathcal{C}_m, \mathcal{W}_m, \mathcal{F}_m) \] of \(Top\). Then there exists a unique model structure on \({\mathcal{G}}dTop\) such that:
\begin{itemize}
\item A map of multipointed \(d\)-spaces \(f: X \to Y\) is a weak equivalence if and only if \(f^0: X^0\to Y^0\) is a bijection and for all \((\alpha,\beta)\in X^0\times X^0\), the continuous map \(\mathbb{P}^{\mathcal{G}}_{\alpha, \beta}X \to \mathbb{P}^{\mathcal{G}}_{f(\alpha), f(\beta)}Y\) belongs to \(\mathcal{W}\).
\item A map of multipointed \(d\)-spaces \(f: X \to Y\) is a fibration if and only if for all \((\alpha,\beta)\in X^0\times X^0\), the continuous map \(\mathbb{P}^{\mathcal{G}}_{\alpha, \beta}X \to \mathbb{P}^{\mathcal{G}}_{f(\alpha), f(\beta)}Y\) belongs to \(\mathcal{F}\).
\end{itemize}
Moreover, this model structure is accessible and all objects are fibrant.
Section 7 is devoted to the category \(Flow\).
Definition 7.1. [\textit{P. Gaucher}, Homology Homotopy Appl. 5, No. 1, 549--599 (2003; Zbl 1069.55008)] A flow \(X\) consists of a topological space \(\mathbb{P} X\) of execution paths, a discrete space \(X^0\) of states, two continuous maps \(s\) and \(t\) from \(\mathbb{P} X\) to \(X^0\) called the source and target map, respectively, and a continuous and associative map
\[
*: \{(x,y)\in \mathbb{P} X \times \mathbb{P} X; t(x)= s(y)\}\to \mathbb{P} X
\]
such that \(s(x*y)=s(x)\) and \(t(x*y)= t(y)\). A morphism of flows \(f: X\to Y\) consists of a set map \(f^0: X^0\to Y^0\) together with a continuous map \(\mathbb{P} f: \mathbb{P} X \to \mathbb{P} Y\) such that \(f(s(x))= s(f(x))\), \(f(t(x))= t(f(x))\) and \(f(x*y)= f(x)*f(y)\). The corresponding category is denoted by \(Flow\). Let \(\mathbb{P}_{\alpha,\beta}X= \{x\in \mathbb{P} X | s(x)= \alpha \text{ and } t(x)=\beta\}\).
Theorem 7.4. Let \((\mathcal{C}, \mathcal{W}, \mathcal{F})\) be one of the three model structures
\[
(\mathcal{C}_q, \mathcal{W}_q, \mathcal{F}_q), (\mathcal{C}_{\bar h}, \mathcal{W}_h, \mathcal{F}_h), (\mathcal{C}_m, \mathcal{W}_m, \mathcal{F}_m)
\]
of \(Top\). Then there exists a unique model structure on \(Flow\) such that:
\begin{itemize}
\item A map of flows \(f: X \to Y\) is a weak equivalence if and only if \(f^0: X^0\to Y^0\) is a bijection and for all \((\alpha,\beta)\in X^0\times X^0\), the continuous map \(\mathbb{P}^{\mathcal{G}}_{\alpha, \beta}X \to \mathbb{P}^{\mathcal{G}}_{f(\alpha), f(\beta)}Y\) belongs to \(\mathcal{W}\).
\item A map of flows \(f: X \to Y\) is a fibration if and only if for all \((\alpha,\beta)\in X^0\times X^0\), the continuous map \(\mathbb{P}^{\mathcal{G}}_{\alpha, \beta}X \to \mathbb{P}^{\mathcal{G}}_{f(\alpha), f(\beta)}Y\) belongs to \(\mathcal{F}\).
\end{itemize}
Moreover, this model structure is accessible and all objects are fibrant.
From the text: ``We obtain the following results:
\begin{itemize}
\item a \(q\)-model structure, an \(h\)-model structure and an \(m\)-model structure on multipointed \(d\)-spaces and on flows in one step (!)
\item the identity functor induces a Quillen equivalence between the \(q\)-model structure and the \(m\)-model structure on multipointed \(d\)-spaces (on flows, respectively)
\item the two \(q\)-model structures are combinatorial and left determined and they coincide with that of \textit{P. Gaucher} [Homology Homotopy Appl. 5, No. 1, 549--599 (2003; Zbl 1069.55008); Theory Appl. Categ. 22, 588--621 (2009; Zbl 1191.55013); Cah. Topol. Géom. Différ. Catég. 61, No. 2, 208--226 (2020; Zbl 1452.18010)], respectively
\item the four other model structures (the two \(m\)-model structures and the two \(h\)-model structures) are accessible
\item all objects are fibrant in these six model structures
\item there are the implications \(q\)-cofibrant \(\Rightarrow\) \(m\)-cofibrant \(\Rightarrow\) \(h\)-cofibrant for multipointed \(d\)-spaces and flows
\item there exist multipointed \(d\)-spaces and flows which are not \(q\)-cofibrant, not \(h\)-cofibrant and not \(m\)-cofibrant.''
\end{itemize}
Reviewer: Ahmet A. Khusainov (Komsomolsk-om-Amur)Bilimits are bifinal objectshttps://zbmath.org/1496.180132022-11-17T18:59:28.764376Z"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.edoardoThe theory of limits and colimits lies at the center of category theory. The notions of \(2\)-limits and \(2\)-colimits were first introduced independently in [\textit{C. Auderset}, Cah. Topologie Géom. Différ. Catégoriques 15, 3--20 (1974; Zbl 0364.18007)], where the Eilenberg-Moore and the Kleisli category of a monad are recovered as a \(2\)-limit and \(2\)-colimit, and in [\textit{F. Borceux} and \textit{G. M. Kelly}, Bull. Aust. Math. Soc. 12, 49--72 (1975; Zbl 0329.18011)], where the notion of enriched limits and colimit were introduced. These notions were further investigated and developed by \textit{R. Street} [J. Pure Appl. Algebra 8, 149--181 (1976; Zbl 0335.18005)], \textit{G. M. Kelly} [Bull. Aust. Math. Soc. 39, No. 2, 301--317 (1989; Zbl 0657.18004); The basic concepts of enriched category theory. Seminarber. Fachber. Math., Fernuniv. Hagen 9 (1981; Zbl 0709.18501); Basic concepts of enriched category theory. Cambridge etc.: Cambridge University Press (1982; Zbl 0478.18005); Repr. Theory Appl. Categ. 2005, No. 10, 1--136 (2005; Zbl 1086.18001)] and \textit{S. Lack} [IMA Vol. Math. Appl. 152, 105--191 (2010; Zbl 1223.18003)], who also introduced and investigted the lax and weighted versions. Two papers by \textit{T. Clingman} and \textit{L. Moser} [``2-limits and 2-terminal objects are too different'', Preprint, \url{arXiv:2001.01313}; ``Bi-initial objects and bi-representations are not so different'', Preprint, \url{arXiv:2009.05545}] have investigated whether the well-known result that limits are terminal cones extends to the \(2\)-dimensional framework, the first establishing the negative answer while the second leveraging on results from double-category theory on representability of \(\mathcal{C}at\)-valued functors to show that being terminal still captures the notion of limit, provided one is willing to work with an alternative \(2\)-category than that of cones.
This paper aims to clarify, with its main result (Theorem 4.10), that a natural characterization of lax bilimits in terms of cones is still possible. The synopsis of the paper goes as follows. The use of a marking on the domain of the \(2\)-functor of which the authors want to study the bilimits addresses in a fundamental way all the possible levels of laxity of the bilimit (pseudo, lax or anything in between). It is remarked that the further level of generality is necessary from a technical viewpoint to coherently interpolate laxity from pseudo to lax in all the \(2\)-categorical constructions. The authors aim also to fill the gap in the literature as concerns final objects in \(2\)-categorical theory, though some results on final \(2\)-functors can be seen in [\textit{F. Abellán García} and \textit{W. H. Stern}, J. Pure Appl. Algebra 226, No. 9, Article ID 107040, 43 p. (2022; Zbl 1495.18026)].
\begin{itemize}
\item[\S 1] recalls the necessary background on \(2\)-categories and relevant constructions, namely joins, slices and the Grothendieck construction for fibrations of \(2\)-categories.
\item[\S 2] introduces lax marked bilimits and contractions on the lines of [\textit{M. E. Descotte} et al., Adv. Math. 333, 266--313 (2018; Zbl 1401.18004)]. It is established (Proposition 2.2.11) that final objects can be characterized in several ways, one of which involves contractions.
\item[\S 3] further studies the slice fibration
\[
p:\mathcal{C}^{/F}\rightarrow\mathcal{C}
\]
to a marked \(2\)-category \(\overline{\mathcal{J}}\mathcal{=}\left( \mathcal{J},E\right) \) and a \(2\)-functor \(F:\mathcal{J}\rightarrow \mathcal{C}\). \S 3.1 is an investigation on representable fibrations and the properties of the corresponding representing objects. \S 3.2 focuses on the particular case with \(\mathcal{J}=\mathrm{D}_{0}\), so that \(\mathcal{C} ^{/F}=\mathcal{C}_{/F\left( \ast\right) }\) is a representable fibration, showing that in this case the object \(1_{F\left( \ast\right) }\) is the center of a contraction to the collection of \(p\)-cartesian edges. \S 3.3 constructs a modified \(2\)-category of cones which projects to \(\mathcal{C} ^{/F}\), showing that this projection is a biequivalence iff \(F\)\ admits an \(E\)-bilimits.
\item[\S 4] blends all together, culminating in the main result (Theorem 4.10), which characterizes (lax, maraked) bilimits as limiting bifinal cones. It is established that such bilimits are also terminal in the appropriate subcategory of cones obtained by restricting to cartesian morphisms between them.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)The limit and tensor product in the category of Q-P quantale moduleshttps://zbmath.org/1496.180142022-11-17T18:59:28.764376Z"Liang, Shaohui"https://zbmath.org/authors/?q=ai:liang.shaohuiSummary: In this paper, firstly, the definition of Q-P quantale modules and some relative concepts were introduced. we prove that the category of Q-P quantale modules is a pointed and connected category. Secondly, we give the structure of the limit of this category, so it is complete. At last, The definition of bimorphism of Q-P quantale modules is given. The tensor product of Q-P quantale modules is obtained, and some of their properties are discussed.Morita theorem for hereditary Calabi-Yau categorieshttps://zbmath.org/1496.180152022-11-17T18:59:28.764376Z"Hanihara, Norihiro"https://zbmath.org/authors/?q=ai:hanihara.norihiroThe paper characterizes the structure of Calabi-Yau triangulated category with a hereditary cluster tilting object. Recall that a triangulated category \(\mathcal T\) is \(d\)-Calabi-Yau if it has finite dimensional morphism spaces over a field \(k\), and satisfies the functorial isomorphism \(\mathcal T(A,B)\cong \Hom_k(\mathcal T(B,A[d]),k)\) for all \(A,B\in \mathcal T\). An example of \(d\)-Calabi-Yau category is the \(d\)-cluster category of a hereditary algebra \(H\), which is the orbit category of derived category \(D^b(H)\) under the functor \(\tau^{-1}[d-1]\). The cluster category has a \(d\)-cluster tilting object.
Morita theory suggests that module categories of two rings are equivalent if and only there exists progengentors inducing the equivalence. The author proves a version of Morita theory for Calabi-Yau categories. The \(d\)-cluster tilting objects play the role of progenerators. More precisely, an algebraic \(d\)-Calabi-Yau category with a hereditary \(n\)-cluster tilting object \(T\) is equivalent the \(d\)-cluster category of the hereditary endomorphism algebra of \(T\oplus T[-1]\oplus \dots\oplus T[2-d]\).
As a result, Morita theorems of Keller-Reiten and Keller-Murfet-Van den Berg are followed by specified \(d\) and \(T\) [\textit{B. Keller} and \textit{I. Reiten}, Compos. Math. 144, No. 5, 1332--1348 (2008; Zbl 1171.18008); \textit{B. Keller} et al., Compos. Math. 147, No. 2, 591--612 (2011; Zbl 1264.13016)].
Reviewer: Zhe Han (Kaifeng)Microthesis. Sheaves over the spectrum of a tensor triangulated categoryhttps://zbmath.org/1496.180162022-11-17T18:59:28.764376Z"Rowe, James"https://zbmath.org/authors/?q=ai:rowe.jamesSummary: The spectrum of a tensor triangulated category is a topological space which is vital in classifying different types of subcategories. We investigate sheaves built from data contained in the original tensor triangulated category defined over this spectrum and see how their behaviour relates to various categorical or algebraic properties.Loday constructions on twisted products and on torihttps://zbmath.org/1496.180172022-11-17T18:59:28.764376Z"Hedenlund, Alice"https://zbmath.org/authors/?q=ai:hedenlund.alice"Klanderman, Sarah"https://zbmath.org/authors/?q=ai:klanderman.sarah"Lindenstrauss, Ayelet"https://zbmath.org/authors/?q=ai:lindenstrauss.ayelet"Richter, Birgit"https://zbmath.org/authors/?q=ai:richter.birgit"Zou, Foling"https://zbmath.org/authors/?q=ai:zou.foling``We develop a spectral sequence for the homotopy groups of Loday constructions with respect to twisted Cartesian products in the case where the group involved is discrete. We show that for commutative Hopf algebra spectra Loday constructions are stable, generalizing a result by \textit{Y. Berest} et al. [Int. Math. Res. Not. 2022, No. 6, 4093--4180 (2022; Zbl 1494.57050)], but prove that several Loday constructions of truncated polynomial rings with reduced coefficients are not stable by investigating their torus homology.''
This extensive work has four sections. The first section is entitled ``The Loday construction: basic features''. The authors recall some definition concerning Loday construction [\textit{J.-L. Loday}, Cyclic homology. 2nd ed. Berlin: Springer (1998; Zbl 0885.18007)] and fix notation. The authors write instead ``For most of our work we can use any good symmetric monoidal category of spectra whose category of commutative monoids is Quillen equivalent to the category of \(E_\infty\)-ring spectra, such as \textbf{symmetric spectra} [\textit{M. Hovey} et al., J. Am. Math. Soc. 13, No. 1, 149--208 (2000; Zbl 0931.55006)], \textbf{orthogonal spectra} [\textit{M. A. Mandell} and \textit{J. P. May}, Equivariant orthogonal spectra and \(S\)-modules. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1025.55002)] or \(\textsl{\textbf{S}}\)-\textbf{modules} [\textit{A. D. Elmendorf} et al., Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole. Providence, RI: American Mathematical Society (1997; Zbl 0894.55001)]. As parts of the paper require us to work with a specific model category we chose to work with the category of \(S\)-modules everywhere except in Section 3, where we will work in the \(\infty\)-\textbf{category of spectra in the sense of Luri} [\textit{J. Lurie}, Higher algebra. (2017), \url{http://www.math.harvard.edu/-lurie/papers/HA.pdf}].''
Let \(X\) be a finite pointed simplicial set and \(R\rightarrow A\rightarrow C\) be a sequence of maps of commutative ring spectra.
{Definition 1.1}. The \textsl{Loday construction with respect to} \(X\) \textsl{of} \(A\) \textsl{over} \(R\) \textsl{with} \textsl{coefficients in} \(C\) is the simplicial commutative augumented \(C\) -algebra spectrum \(\mathfrak{L}^R_X(A;C)\) given by \(\mathfrak{L}^R_X(A;C)_n=C\wedge \bigwedge_{x\in X_n\backslash \ast} A\), where the smash products are taken over \(R\). Here, \(\ast\) denotes the basepoint of \(X\) and we place a copy of \(C\) at the basepoint.
The authors assume in addition that \(R\) is a cofibrant commutative \(S\)-algebra, \(A\) is a cofibrant commutative \(R\)-algebra and \(C\) is a cofibrant commutative \(A\)-algebra. This ensures that the homotopy type of \(\mathfrak{L}^R_X(A;C)\) is well-defined and depends only on the homotopy type of \(X\). Under these conditions, new notations are adopted for some used in [\textit{J.-L. Loday}, Cyclic homology. 2nd ed. Berlin: Springer (1998; Zbl 0885.18007); \textit{A. D. Elmendorf} et al., Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole. Providence, RI: American Mathematical Society (1997; Zbl 0894.55001); \textit{T. Pirashvili}, Ann. Sci. Éc. Norm. Supér. (4) 33, No. 2, 151--179 (2000; Zbl 0957.18004)]. Thus \(\mathfrak{L}^R_X(A;C)\) instead of \(C\otimes\bigotimes_{x\in {X_n}\setminus\ast}A\), in [\textit{J.-L. Loday}, Cyclic homology. 2nd ed. Berlin: Springer (1998; Zbl 0885.18007)], for \(R\rightarrow A\rightarrow C\) a sequence of commutative rings. And, for \(X=S^n\), \(\mathfrak{L}^R_{S^n}(A;C)\), instead of \(THH^{[n],R}(A;C)\), in [\textit{A. D. Elmendorf} et al., Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole. Providence, RI: American Mathematical Society (1997; Zbl 0894.55001)] and [\textit{T. Pirashvili}, Ann. Sci. Éc. Norm. Supér. (4) 33, No. 2, 151--179 (2000; Zbl 0957.18004)].
Section 2, entitled ``A spectral sequence for twisted Cartesian products'', contains the construction of a spectral sequence \(E^2_{p,q}=\pi_p((\mathfrak{L}^R_B(\pi_\ast \mathfrak{L}^R_F(A)^\tau))_q)\Rightarrow \pi_{p+q}(\mathfrak{L}^R_{E(\tau)}(A))\) (Theorem 2.10) for the homotopy groups \(\pi_\ast(\mathfrak{L}^R_F(A))\) of Loday constructions with respect to a twisted Cartesian products (TCP), \(E(\tau)=F\times_\tau B, \) [\textit{J. P. May}, Simplicial objects in algebraic topology. Chicago: The University of Chicago Press (1992; Zbl 0769.55001)]. Two examples are given.One starting from a TCP constructed by the connected \(n\)-fold of \(S^1\) given by degree \(n\) map. And the second,conversely, for the Klein bottle, \(K\ell\), is recomputed the homotopy groups of the Loday construction of the polynomial algebra \(k[x]\) for a field \(k\), such that 2 is invertible in \(k\), using the above TCP spectral sequence.
Section 3, ``Hopf algebras in spectra'', the authors prove that the Loday construction is stable for commutative Hopf algebra spectra, generalizing a result of \textit{Y. Berest} et al. [Int. Math. Res. Not. 2022, No. 6, 4093--4180 (2022; Zbl 1494.57050)]. If CAlg denote the \(\infty\)-category of \(E_\infty\)-ring spectra, then a commutative Hopf algebra spectrum is a cogroup in CAlg. In the introduction of this section the authors give some examples of Hopf algebra spectra. A first example starts from a topological abelian group \(G\), with the spherical group ring \(S[G]=\sum_{+}^\infty G\), equipped with the product induced by the product in \(G\), the coproduct induced by the diagonal map \(G\rightarrow G\times G\), and the antipodal map induced by the inverse map from \(G\) to \(G\) is a commutative Hopf algebra spectrum. Another example starts from an ordinary commutative Hopf algebra \(A\) over a commutative ring \(k\) and A is flat as a \(k\)-module. Then the Eilenberg -Mac Lane spectrum \(HA\) is a commutative Hopf algebra spectrum over \(Hk\). Other examples are also given and then is proved the following theorem.
{Theorem 3.6}. \textsl{If} \(\mathcal{H}\) \textsl{is a commutative Hopf algebra spectrum and if} \(\sum(X_+)\simeq \sum(X_+)\) \textsl{is an equivalence in} \(\mathcal{S}_\ast\) , \textsl{then there is an equivalence} \(X\otimes \mathcal{H}\simeq Y\otimes \mathcal{H}\) \textsl{in} CAlg. (Where \(\mathcal{S}_\ast\) denotes the \(\infty\)-category of based spaces).
Section 4 is entitled ``Truncated polynomial algbras''. But the notations in this part of the article are too complicated to be transcribed or summarized by the reviewer even only in the two big theorems (4.10 and 4.23) in this section. That is why the reviewer is limited to what the authors wrote in relation to this section. ``In Section 4 we prove that truncated polynomial algebras of the form \(\mathbb{Q}[t]/t^m\) and \(\mathbb{Z}[t]/t^m\) for \(m\geq 2\) are not muliplicatively stable by comparing the Loday construction of tori to the Loday construction of a bouquet of spheres corresponding to the cells of the tory. We also show that for \(2\leq m< p\) the pairs \((\mathbb{F}_p[t]/t^m;\mathbb{F}_p)\) are not stable''.
Reviewer: Ioan Pop (Iaşi)Two-dimensional topological theories, rational functions and their tensor envelopeshttps://zbmath.org/1496.180182022-11-17T18:59:28.764376Z"Khovanov, Mikhail"https://zbmath.org/authors/?q=ai:khovanov.mikhail-g"Ostrik, Victor"https://zbmath.org/authors/?q=ai:ostrik.victor"Kononov, Yakov"https://zbmath.org/authors/?q=ai:kononov.yakovThis paper works over a field \(\boldsymbol{k}\), occasionally specializing to a characteristic \(0\) field. The authors consider \(\boldsymbol{k}\)-linear symmetric monoidal categories called \textit{tensor categories}.
This paper is concerned with the following nine categories.
\begin{itemize}
\item \(\mathrm{Cob}_{2}\): Oriented \(2\)D cobordisms between one-manifolds
\item \(\mathrm{VCob}_{\alpha}\): Viewable cobordisms
\item \(\mathrm{SCob}_{\alpha}\): Skein category, denoted \(\mathrm{PCob}_{\alpha }\)
\item \(\mathrm{Cob}_{\alpha}\): Gligible quotient of \(\mathrm{SCob}_{\alpha} \) by the kernels of trace forms
\item \(\mathrm{DCob}_{\alpha}\): Deligne category
\item \(\underline{\mathrm{DCob}}_{\alpha}\): Gligible quotient of the Deligne category
\item \(\mathrm{SCob}_{\alpha}^{\oplus}\): The finite additive closure of \(\mathrm{SCob}_{\alpha}^{{}}\)
\item \(\mathrm{Cob}_{\alpha}^{\oplus}\): The finite additive closure of \(\mathrm{Cob}_{\alpha}\)
\end{itemize}
The nine categories are connected by functors as follows.
\[
\begin{array} [c]{ccccccccc} \mathrm{Cob}_{2} & \rightarrow & \mathrm{VCob}_{\alpha} & \rightarrow & \mathrm{SCob}_{\alpha}^{{}} & \rightarrow & \mathrm{SCob}_{\alpha}^{\oplus} & \rightarrow & \mathrm{DCob}_{\alpha}\\
& & & & \downarrow & & \downarrow & & \downarrow\\
& & & & \mathrm{Cob}_{\alpha} & \rightarrow & \mathrm{Cob}_{\alpha} ^{\oplus} & \rightarrow & \underline{\mathrm{DCob}}_{\alpha} \end{array}
\]
The four rightmost categories are additive and the three categories to the left of them are \(\boldsymbol{k}\)-linear and pre-additive, while \(\mathrm{Cob}_{2}\) is neither pre-additive nor \(\boldsymbol{k}\)-linear. All eight categories are rigid symmetic monoidal. The bottom three categories are gligible quotients of the respective categories above them, and their hom spaces carry non-degenerate bilinear forms. Category \(\mathrm{DCob}_{\alpha} \) is the analogue of the Deligne category \(\mathrm{Rep}(S_{t})\) and specializes to it when the sequence \(\alpha\) is constant,
\[
\alpha(t)=(t,t,\dots),\quad Z_{\alpha(t)}=\frac{t}{1-T},\quad t\in\boldsymbol{k}
\]
Category \(\underline{\mathrm{DCob}}_{\alpha}\) is the analogue of the quotient \underline{\(\mathrm{Rep}\)}\((S_{t})\) of \(\mathrm{Rep} (S_{t})\) by negligible morphisms, and specializes to it when the sequence \(\alpha\) is constant.
This paper investigates generalized Deligne categories \(\mathrm{DCob}_{\alpha }\), their quotients \(\underline{\mathrm{DCob}}_{\alpha}\) as well as categories \(\mathrm{SCob}_{\alpha}^{{}}\) and \(\mathrm{Cob}_{\alpha}\) for other rational series \(\alpha\), which are referred to as \textit{tensor envelopes} of \(\alpha\).
The synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 2] discusses basic properties of tensor envelopes. \S 2.1 points out the the scaling
\[
Z(T)\mapsto\lambda^{-1}Z(\lambda T)
\]
for an invertible \(\lambda=\mu^{2}\) does not change the categories considered. \S 2.2 explains that any commutative Frobenius algebra object in a pre-additive tensor category gives rise to a power series \(\alpha\) with coefficients in the commutative ring \(\mathrm{End}(\boldsymbol{1} )\) of endomorphisms of the unit object. \S 2.3 recalls the universal property of \(\mathrm{Cob}_{\alpha}\). \S 2.4 investigates direct sum decompositions of commutative Frobenius algebra objects which mirror partial decompositions of their rational generating series.
\item[\S 3] contains key semisimplicity and abelian realization criteria for the tensor envelopes of \(\alpha\), including Theorem 3.2 and Theorem 3.7, both of which characterize admitting an abelian realization. In particular, the authors classify series \(\alpha\) with the semisimple category \(\underline {\mathrm{DCob}}_{\alpha}\).
\item[\S 4] reviews properties of the endomorphism ring of the one-circle object in categories \(\mathrm{SCob}_{\alpha}\) and \(\mathrm{Cob}_{\alpha}\).
\item[\S 5] describes the structure of the gligible category \(\mathrm{Cob} _{\alpha}\) for the constant function (series \(\alpha=(\beta ,0,0,\dots)\)). Theorem 5.1 claims that the dimension of the state space \(A(n)\) of \(n\) circles for this function [\textit{M. Khovanov}, ``Universal construction of topological theories in two dimensions'', Preprint, \url{arXiv:2007.03361}] is the Catalin number, for \(\boldsymbol{k}\) of characteristic \(0\). A monoidal equivalence between the Karoubi envelope \(\underline{\mathrm{DCob}}_{\alpha} \) of \(\mathrm{Cob}_{\alpha}\), and a suitable category of finite-dimensional representations of the Lie superalgebra \(osp(1\mid2)\) is established in \S 5.5.
\item[\S 6] consists of two sections. \S 6.1 investigates Gram determinants of a natural spanning set of surfaces for the function \(\beta/(1-\gamma T)\), where tensor envelopes correspond to the Deligne category [\textit{M. Khovanov} and \textit{R. Sazdanovic}, ``Bilinear pairings on two-dimensional cobordisms and generalizations of the Deligne category'', Preprint, \url{arXiv:2007.11640}]. These are rank one theories. \S 6.2 gives determinant computations for various rank two theories.
\item[\S 7] considers the case of a polynomial generating function, beyond the constant function case studied in \S 5. When the function is linear, associated tensor envelopes can be expressed via the unoriented Brauer category and its gligible quotient, due to the presence of a commutative Frobenius object in the Brauer category with a linear generating function, as can be seen in \S 7.1. \S 7.2 provides numerical data for the Gram determinants in categories when the generating function is a polynomial of degree two or three. \S 7.3 considers arbitrary degree polynomials, where a conjectural basis in the state space of \(n\) circles for the theory is established, and some properties of the Gram determinant for the set of vectors is established.
\item[\S 8] explains how to enrich category \(\mathrm{Cob}_{2}\) of two-dimensional oriented cobordisms by adding codimension two defects (dots). Presence of the handle cobordism allows of adding relations intertwining the handle cobordism with dot decorations. Going from less general to more general examples, dots may be viewed as fractional handles, elements of a commutative monoid, or elements of a commutative algebra.
\end{itemize}
The works [\textit{J. Flake} et al., ``Indecomposable objects in Khovanov-Sazdanovic's generalizations of Deligne's interpolation categories'', Preprint, \url{arXiv:2106.05798}; \textit{E. Meir}, ``Interpolations of monoidal categories and algebraic structures by invariant theory'', Preprint, \url{arXiv:2105.04622}] are related to this one.
Reviewer: Hirokazu Nishimura (Tsukuba)Higher central charges and Witt groupshttps://zbmath.org/1496.180192022-11-17T18:59:28.764376Z"Ng, Siu-Hung"https://zbmath.org/authors/?q=ai:ng.siu-hung"Rowell, Eric C."https://zbmath.org/authors/?q=ai:rowell.eric-c"Wang, Yilong"https://zbmath.org/authors/?q=ai:wang.yilong"Zhang, Qing"https://zbmath.org/authors/?q=ai:zhang.qing.4|zhang.qing|zhang.qing.1|zhang.qing.3|zhang.qing.2Let \(\mathcal{C}\) be a fusion category. Using the number theoretic properties of the categorical dimension and the Frobenius-Perron dimension of \(\mathcal{C}\) respectively, the paper under review defines two notions of signature of \(\mathcal{C}\), defining homomorphisms from the (super-)Witt group \(\mathcal{W}\) of non-degenerate braided fusion categories to the group of maps from the absolute Galois group \(\operatorname{Gal}(\overline{\mathbb{Q}})\) to \(\left\{ 1,-1 \right\}\). In the case of Frobenius-Perron dimension, the signature homomorphism generalizes to the setting of fusion categories over a symmetric fusion category studied in [\textit{A. Davydov} et al., Sel. Math., New Ser. 19, No. 1, 237--269 (2013; Zbl 1345.18005)].
If \(\mathcal{C}\) is modular, the paper gives a description of higher charges of \(\mathcal{C}\), introduced in [\textit{S. Ng} et al., Sel. Math., New Ser. 25, No. 4, Paper No. 53, 32 p. (2019; Zbl 1430.18015)], in terms of the signature of \(\mathcal{C}\). Viewing the central charge construction for \(\mathcal{C}\) as a function \(\Psi_{\mathcal{C}}\) from \(\operatorname{Gal}(\overline{\mathbb{Q}})\), the signature description is used to show that the assignment \(\mathcal{C} \mapsto \Psi_{\mathcal{C}}\) gives a group homomorphism from the Witt group of pseudounitary modular categories to the group of functions from \(\operatorname{Gal}(\overline{\mathbb{Q}})\) to \(\bigcup_{n=1}^{\infty} \mu_{n}\), where \(\mu_{n}\) is the group of \(n\)th roots of unity.
Further, the paper determines the signatures of certain infinite families of categories \(\mathcal{C}_{r} := \mathfrak{so}(2r+1)_{2r+1}\) of integrable highest weight modules of level \(2r+1\) over the affinization of \(\mathfrak{so}(2r+1)\). Using these results, it is shown that, for an Ising modular category \(\mathcal{I}\), the equation \(x^{2} = [I]\) has infinitely many solutions in the quotient \(\mathcal{W}/\mathcal{W}_{\operatorname{pt}}\) of \(\mathcal{W}\) by the pointed part of \(\mathcal{W}\). This is shown to imply that for the super-Witt group \(s\mathcal{W}\), the torsion subgroup \(s\mathcal{W}_{2}\) generated by the completely anisotropic \(s\)-simple braided fusion categories is of infinite rank. This confirms a conjecture made in [\textit{A. Davydov} et al., Sel. Math., New Ser. 19, No. 1, 237--269 (2013; Zbl 1345.18005)].
Reviewer: Mateusz Stroiński (Uppsala)On webs in quantum type \(C\)https://zbmath.org/1496.180202022-11-17T18:59:28.764376Z"Rose, David E. V."https://zbmath.org/authors/?q=ai:rose.david-e-v"Tatham, Logan C."https://zbmath.org/authors/?q=ai:tatham.logan-cThe authors give a linear pivotal category \(\mathbf{Web}(\mathfrak{sp}_{6}) \) defined by diagrams and relations, and conjecture its equivalence to the full subcategory \(\mathbf{FundRep}(U_q(\mathfrak{sp}_{6})) \) of finite-dimensional representations of \( \mathfrak{sp}_{6} \) tensor-generated by fundamental representations. This is a step towards generalizing to type \(C\) case of the main open problem from Kuperberg's Spider for rank 2 Lie algebras [\textit{G. Kuperberg} Commun. Math. Phys. 180, No. 1, 109--151 (1996; Zbl 0870.17005)]. They prove a number of results that support the conjecture. Namely, they construct a functor from \(\mathbf{Web}(\mathfrak{sp}_{6}) \) to \(\mathbf{FundRep}(U_q(\mathfrak{sp}_{6})) \) that is full and essentially surjective (they prove that the well-known surjection from the BMW algebra to the representation category factors through the web category). Moreover they prove that all \(\Hom\)-spaces in \(\mathbf{Web}(\mathfrak{sp}_{6}) \) are finite-dimensional and that the endomorphism algebra of the tensor unit in \(\mathbf{Web}(\mathfrak{sp}_{6}) \) is one-dimensional. As a consequence, the authors give a new approach to the quantum \( \mathfrak{sp}_{6} \) link invariants in the same lines as the Kauffman bracket description of the Jones polynomial. In this paper it is also given a thickening of \(\mathbf{Web}(\mathfrak{sp}_{6}) \) to a category constructed using ladders that is one of the ingredients is proving the results above.
Reviewer's remark: The conjecture that is the subject of this paper is a consequence of the results in \url{arXiv:2103.14997} for \(sp_{2n}\) by the authors together with \textit{E. Bodish} and \textit{B. Elias}.
Reviewer: Pedro Vaz (Louvain-la-Neuve)Cyclic Gerstenhaber-Schack cohomologyhttps://zbmath.org/1496.180212022-11-17T18:59:28.764376Z"Fiorenza, Domenico"https://zbmath.org/authors/?q=ai:fiorenza.domenico"Kowalzig, Niels"https://zbmath.org/authors/?q=ai:kowalzig.nielsWe know that deformations of an associative algebra is controlled by its Hochschild cohomology [\textit{M. Gerstenhaber}, Ann. Math. (2) 78, 267--288 (1963; Zbl 0131.27302)]. The same way, deformations of a bialgebra is controlled by Gerstenhaber-Schack cohomology [\textit{M. Gerstenhaber} and \textit{S. D. Schack}, Contemp. Math. 134, 51--92 (1992; Zbl 0788.17009)]. The main problem the authors tackle arises from the question whether Gerstenhaber-Schack cohomology carries a Gerstenhaber bracket analogous to the bracket structure on Hochschild cohomology. The authors show that when the underlying bialgebra is a Hopf algebra the diagonal of the Gerstenhaber-Schack complex carries an operad structure with multiplication (Theorem B). Since bisimplicial objects are homotopic to their diagonals, the Gerstenhaber-Schack cohomology carries a natural Gerstenhaber bracket due to [\textit{M. Gerstenhaber} and \textit{S. D. Schack}, Contemp. Math. 134, 51--92 (1992; Zbl 0788.17009); \textit{J. E. McClure} and \textit{J. H. Smith}, Contemp. Math. 293, 153--193 (2002; Zbl 1009.18009)]. Moreover, they also show that (Theorem C) when the antipode of the underlying Hopf algebra is involutive, or when the Hopf algebra has a modular pair in involution, then operad is cyclic, and therefore, the Gerstenhaber-Schack cohomology supports a Batalin-Vilkovisky algebra structure by [\textit{L. Menichi}, \(K\)-Theory 32, No. 3, 231--251 (2004; Zbl 1101.19003)]. Interestingly, the bracket is trivial when the Hopf algebra is finite dimensional (Theorem D), and therefore, Gerstenhaber-Schack cohomology has a \(e_3\)-algebra structure due to [\textit{D. Fiorenza} and \textit{N. Kowalzig}, Int. Math. Res. Not. 2020, No. 23, 9148--9209 (2020; Zbl 1468.55008)].
Reviewer: Atabey Kaygun (İstanbul)Resolutions of operads via Koszul (bi)algebrashttps://zbmath.org/1496.180222022-11-17T18:59:28.764376Z"Tamaroff, Pedro"https://zbmath.org/authors/?q=ai:tamaroff.pedroFor any bialgebra \(H\), the category of associative algebras in the category of \(H\)-modules is isomorphic to the category of algebras over an operad \(\mathrm{Ass}_H\) which is described in this paper. This operad is generally not quadratic. When \(H\) is Koszul (as an algebra), then a minimal model of \(\mathrm{Ass}_H\) is described, involving only the coproduct of \(H\) and the Koszul model of \(H\). It is also shown how to construct a Gröbner basis of \(\mathrm{Ass}_H\) from a Gröbner basis of \(H\). Links are made with homotopy theory when \(H\) is the mod-2 Steenrod algebra.
Reviewer: Loïc Foissy (Calais)On lax epimorphisms and the associated factorizationhttps://zbmath.org/1496.180232022-11-17T18:59:28.764376Z"Lucatelli Nunes, Fernando"https://zbmath.org/authors/?q=ai:lucatelli-nunes.fernando"Sousa, Lurdes"https://zbmath.org/authors/?q=ai:sousa.lurdesLax epimorphisms or co-fully-faithful morphisms in a 2-category are 1-cells \(f:A\to B\) which induce fully faithful contravariant Hom-functors \(\mathrm{Hom}(f,C)\) for all \(C\). The authors give a concrete description of an \((\mathcal E,\mathcal M)\)-factorization system in \(\mathsf{Cat}\), where \(\mathcal E\) is the class of all lax epimorphisms. After proving that lax epimorphisms are closed under 2-colimits, they further show that under suitable conditions if a 2-category has all 2-colimits then it admits an orthogonal factorization system \((\mathit{LaxEpi},\mathit{LaxStrongMono})\). Finally they study several characterizations of the class of lax epimorphisms in enriched 2-categories \(\mathcal V\text{-}\mathsf{Cat}\) under suitable properties on the enriching symmetric monoidal category \(\mathcal V\).
Reviewer: Amit Kuber (Kanpur)Regularity of spectral stacks and discreteness of weight-heartshttps://zbmath.org/1496.180242022-11-17T18:59:28.764376Z"Sosnilo, Vladimir"https://zbmath.org/authors/?q=ai:sosnilo.vladimir-aThe main motivation of this paper is the study of bounded \(t\)-structures, after the key contributions of Antieau, Gepner and Heller who proved that obstructions to the existence of bounded \(t\)-structures on a stable \(\infty\)-category \(\mathscr{C}\) are controlled by the first negative \(K\)-group \(K_{-1}(\mathscr{C})\) [\textit{B. Antieau} et al., Invent. Math. 216, No. 1, 241--300 (2019; Zbl 1430.18009)]. This work tackles the problem assuming that \(\mathscr{C}\) is endowed with a \textit{weight structure}, a structure introduced in [\textit{M. V. Bondarko}, J. \(K\)-Theory 6, No. 3, 387--504 (2010; Zbl 1303.18019)], similar (but not dual) to a \(t\)-structure, which axiomatizes the properties of naive truncations of chain complexes. This notion is closely related to the concept of \textit{regularity} of \(\mathbb{E}_1\)-ring spectra, introduced in [\textit{C. Barwick} and \textit{T. Lawson}, ``Regularity of structured ring spectra and localization in \(K\)-theory'', Preprint, \url{arXiv:1402.6038}].
In the first part, the author contributes to the topic of regular \(\mathbb{E}_1\)-ring spectra by proving the stability of regularity spectra under localizations (Proposition \(2.8\)) and the discreteness of bounded regular \(\mathbb{E}_1\)-ring spectra which are \textit{quasicommutative} (Theorem \(2.11\)). The paper also provides a counterexample to the latter statement in the non-quasicommutative case (Construction \(2.12\)). The bulk of this section, however, is the (twofold) generalization of the concept of regularity to spectral stacks (Definition \(2.15\)): a spectral stack \(X\) is \textit{regular} if there exists a regular atlas \(\operatorname{Spec}(R) \to X\), while it is \textit{homological regular} if the standard \(t\)-structure on \(\operatorname{QCoh}(X)\) restricts to the subcategory of compact objects. While in general not equivalent, the two definitions agree if \(X\) is an affine spectral scheme or, with some additional hypothesis, if \(X\) is a quotient of a Noetherian connective \(\mathbb{E}_{\infty}\)-\(k\)-algebra under the action of a smooth affine group scheme (Theorem \(2.16\)).
In the second part, the author recalls the main definitions, properties and examples of weight structures on stable \(\infty\)-categories, and introduces the notion of \textit{adjacent structures} (Definition \(3.9\)) - i.e., a weight structure \(\left(\mathscr{C}_{w\geqslant 0},\mathscr{C}_{w\leqslant 0}\right)\) and a \(t\)-structure \(\left(\mathscr{C}_{t\geqslant 0},\mathscr{C}_{t\leqslant 0}\right)\) such that \(\mathscr{C}_{w\geqslant 0}=\mathscr{C}_{t\geqslant 0}\). Arguing that the standard \(t\)-structure and the standard weight structure on \(\operatorname{Mod}^{\operatorname{perf}}_R\) are adjacent if and only if \(R\) is regular, the author conjectures that the existence of adjacent structures on a stable \(\infty\)-category \(\mathscr{C}\) should be a noncommutative analogue of regularity. Hence, it is proposed that if all the mapping spaces in the heart of the weight structure \(\operatorname{Hw}(\mathscr{C})\) are \(N\)-truncated for some fixed \(N\), the \(\infty\)-category \(\operatorname{Hw}(\mathscr{C})\) is actually discrete (Conjecture \(3.12\)). Finally, the author proves that if \(X\) is a quotient spectral stack satisfying the assumptions of Theorem \(2.16\), then Conjecture \(3.12\) holds for \(\operatorname{QCoh}(X)\).
Reviewer: Emanuele Pavia (Trieste)\((b, c)\)-inverse, inverse along an element, and the Schützenberger category of a semigrouphttps://zbmath.org/1496.201252022-11-17T18:59:28.764376Z"Mary, Xavier"https://zbmath.org/authors/?q=ai:mary.xavierSummary: We prove that the \((b, c)\)-inverse and the inverse along an element
in a semigroup are actually genuine inverse when considered as morphisms
in the Schützenberger category of a semigroup. Applications to the Reverse
Order Law are given.\(\mathcal{L}\)-semireflexive subcategorieshttps://zbmath.org/1496.460762022-11-17T18:59:28.764376Z"Botnaru, Dumitru"https://zbmath.org/authors/?q=ai:botnaru.dumitruRésumée: Si \((\mathcal{K,L})\) est un couple de sous-catégories conjuguées de la catégorie des espaces localement convexes topologiques vectoriels Hausdorff, alors les catégories \(\mathcal{K}\) et \(\mathcal{L}\) sont isomorphes. Ainsi:
\begin{itemize}
\item[1.] Les lattices \(\mathbb{R}(\mathcal{K})\) et \(\mathbb{R}(\mathcal{L})\) des sous-catégories réflectives des catégories \(\mathcal{K}\)
et \(\mathcal{L}\) sont isomorphes.
\item[2.] Les lattices \(\mathbb{K}(\mathcal{K})\) et \(\mathbb{K}(\mathcal{L})\) des sous-catégories coréflectives des catégories
\(\mathcal{K}\) et \(\mathcal{L}\) sont isomorphes.
\end{itemize}
En construisant l'isomorphisme des lattices \(\mathbb{R}(\mathcal{K})\) et \(\mathbb{R}(\mathcal{L})\), on constate que ces lattices sont isomorphes avec la lattice \(\mathbb{R}^s_f(\varepsilon\mathcal{L})\) des sous-catégories \(\mathcal{L}\)-semi-réflexives de la catégorie \(\mathcal{C}_2\mathcal{V}\), et l'isomorphisme des lattices \(\mathbb{K}(\mathcal{K})\) et \(\mathbb{K}(\mathcal{L})\) nous mène à leur isomorphisme avec la lattice \(\mathbb{K}^s_f(\mu\mathcal{K})\) des sous-catégories \(\mathcal{K}\)-semi-coréflexives.Real homotopy of configuration spaces. Peccot lecture, Collège de France, March \& May 2020https://zbmath.org/1496.550012022-11-17T18:59:28.764376Z"Idrissi, Najib"https://zbmath.org/authors/?q=ai:idrissi.najibThe notion of the configuration space \(\operatorname{Conf}_{r}(M)\) of ordered \(r\)-tuples of distinct points in a manifold \(M\) has played an important role in algebraic geometry and topology for over a century, though their homotopy theory was first studied in an orderly way by \textit{E. Fadell} and \textit{L. Neuwirth} [Math. Scand. 10, 111--118 (1962; Zbl 0136.44104)]. In particular, these spaces provide important invariants of manifolds (see \textit{E. R. Fadell} and \textit{S. Y. Husseini}'s survey [Geometry and topology of configuration spaces. Berlin: Springer (2001; Zbl 0962.55001)]). However, an example due to \textit{R. Longoni} and \textit{P. Salvatore} [Topology 44, No. 2, 375--380 (2005; Zbl 1063.55015)] shows that configuration spaces are not a homotopy invariant, even for closed manifolds.
The present monograph, on the real homotopy type of configuration spaces, is based on the Peccot Lectures given by the author at the Collège de France in the spring of 2020. \textit{P. Lambrechts} and \textit{D. Stanley} [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 497--511 (2008; Zbl 1172.13009)] constructed a certain Poincaré duality commutative differential graded algebra (CDGA) \(G_{A}(r)\) from a CDGA model \(A\) for a simply-connected closed manifold \(M\), and in his thesis, published in [Invent. Math. 216, No. 1, 1--68 (2019; Zbl 1422.55031)], the author showed that this \(G_{A}(r)\) is indeed a real CDGA model for \(\operatorname{Conf}_{r}(M)\), as they had conjectured. In [``A model for configuration spaces of points'', Preprint, \url{arXiv:1604.02043}], \textit{R. Campos} and \textit{T. Willwacher} constructed an alternative real CDGA model for \(\operatorname{Conf}_{r}(M)\), using certain graph complexes based on Kontsevich's graph cooperad, and used this for an alternative proof of the real homotopy invariance (described in detail in Chapter 3 of the present book). These ideas were then used, in collaboration with Idrissi and Lambrechts, to prove a version of this statement for manifolds with boundary (described in Chapter 4). Finally, Chapter 5 elucidates the relationship between configuration spaces, factorization homology, operads, and formality.
Reviewer: David Blanc (Haifa)From deformation theory of wheeled props to classification of Kontsevich formality mapshttps://zbmath.org/1496.550122022-11-17T18:59:28.764376Z"Andersson, Assar"https://zbmath.org/authors/?q=ai:andersson.assar"Merkulov, Sergei"https://zbmath.org/authors/?q=ai:merkulov.sergei-aAuthors' abstract: We study the homotopy theory of the wheeled prop controlling Poisson structures on formal graded finite-dimensional manifolds and prove, in particular, that the Grothendieck-Teichmüller group acts on that wheeled prop faithfully and homotopy nontrivially. Next, we apply this homotopy theory to the study of the deformation complex of an arbitrary Kontsevich formality map and compute the full cohomology group of that deformation complex in terms of the cohomology of a certain graph complex introduced earlier by \textit{M. Kontsevich} [Math. Phys. Stud. 20, 139--156 (1997; Zbl 1149.53325)] and studied by \textit{T. Willwacher} [Invent. Math. 200, No. 3, 671--760 (2015; Zbl 1394.17044)].
Reviewer: Iakovos Androulidakis (Athína)Homotopy invariance of the cyclic homology of \(A_{\infty}\)-algebras under homotopy equivalences of \(A_{\infty}\)-algebrashttps://zbmath.org/1496.550192022-11-17T18:59:28.764376Z"Lapin, Sergey V."https://zbmath.org/authors/?q=ai:lapin.sergey-vIn [\textit{S. V. Lapin}, Math. Notes 102, No. 6, 806--823 (2017; Zbl 1381.16009); translation from Mat. Zametki 102, No. 6, 874--895 (2017)], Lapin constructed the cyclic bicomplex of an \(A_{\infty}\)-algebra over a commutative unital ring. This generalized the famous cyclic bicomplex for an associative algebra due to \textit{B. L. Tsygan} [Russ. Math. Surv. 38, No. 2, 198--199 (1983; Zbl 0526.17006)] and \textit{J.-L. Loday} and \textit{D. Quillen} [Comment. Math. Helv. 59, 565--591 (1984; Zbl 0565.17006)]. The cyclic homology of an \(A_{\infty}\)-algebra is defined as the homology of the chain complex associated to Lapin's bicomplex.
In the paper under review Lapin proves that the cyclic homology of \(A_{\infty}\)-algebras is homotopy invariant under homotopy equivalences of \(A_{\infty}\)-algebras.
The paper is organized as follows. Section 2 recalls the definition of a cyclic module with \(\infty\)-simplicial faces, or \(CF_{\infty}\)-module, as introduced in [\textit{S. V. Lapin}, loc. cit.]. In particular, as recalled in Section 4 of the paper under review, any \(A_{\infty}\)-algebra gives rise to a \(CF_{\infty}\)-module. The notion of homotopy equivalence of \(CF_{\infty}\)-modules is introduced.
Section 3 recalls the definition of cyclic homology of \(CF_{\infty}\)-modules. It is shown that there is a cyclic homology functor from the category of \(CF_{\infty}\)-modules to the category of graded modules over the ground ring. Furthermore, it is shown that this functor sends homotopy equivalences of \(CF_{\infty}\)-modules to isomorphisms of graded modules.
In Section 4, the author recalls the definition of \(A_{\infty}\)-algebra and how to construct a \(CF_{\infty}\)-module from an \(A_{\infty}\)-algebra. The results of Section 3 are then applied to give a cyclic homology functor from the category of \(A_{\infty}\)-algebras to the category of graded modules, sending homotopy equivalences of \(A_{\infty}\)-algebras to isomorphisms of graded modules.
Reviewer: Daniel Graves (Leeds)HOMFLYPT homology for links in handlebodies via type \textsf{A} Soergel bimoduleshttps://zbmath.org/1496.570132022-11-17T18:59:28.764376Z"Rose, David E. V."https://zbmath.org/authors/?q=ai:rose.david-e-v"Tubbenhauer, Daniel"https://zbmath.org/authors/?q=ai:tubbenhauer.danielSummary: We define a triply-graded invariant of links in a genus \(g\) handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-sphere. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family of categorical actions built from complexes of (singular) Soergel bimodules.Mongruences and cofree coalgebrashttps://zbmath.org/1496.681102022-11-17T18:59:28.764376Z"Jacobs, Bart"https://zbmath.org/authors/?q=ai:jacobs.bartSummary: A coalgebra is introduced here as a model of a certain signature consisting of a type \(X\) with various ``destructor'' function symbols, satisfying certain equations. These destructor function symbols are like methods and attributes in object-oriented programming: they provide access to the type (or state) \(X\). We show that the category of such coalgebras and structure preserving functions is comonadic over sets. Therefore we introduce the notion of a `mongruence' (predicate) on a coalgebra. It plays the dual role of a congrence (relation) on an algebra.
For the entire collection see [Zbl 1492.68008].Weighted models for higher-order computationhttps://zbmath.org/1496.681462022-11-17T18:59:28.764376Z"Laird, James"https://zbmath.org/authors/?q=ai:laird.james-dSummary: We study a class of quantitative models for higher-order computation: Lafont categories with (infinite) biproducts. Each of these has a complete ``internal semiring'' and can be enriched over its modules. We describe a semantics of nondeterministic PCF weighted over this semiring in which fixed points are obtained from the bifree algebra over its exponential structure. By characterizing them concretely as infinite sums of approximants indexed over nested finite multisets, we prove computational adequacy.
We can construct examples of our semantics by weighting existing models such as categories of games over a complete semiring. This transition from qualitative to quantitative semantics is characterized as a ``change of base'' of enriched categories arising from a monoidal functor from coherence spaces to modules over a complete semiring. For example, the game semantics of Idealized Algol is coherence space enriched and thus gives rise to to a weighted model, which is fully abstract.CPO models for infinite term rewritinghttps://zbmath.org/1496.681632022-11-17T18:59:28.764376Z"Corradini, Andrea"https://zbmath.org/authors/?q=ai:corradini.andrea"Gadducci, Fabio"https://zbmath.org/authors/?q=ai:gadducci.fabioSummary: Infinite terms in universal algebras are a well-known topic since the seminal work of the ADJ group [\textit{J. A. Goguen} et al., J. Assoc. Comput. Mach. 24, 68--95 (1977; Zbl 0359.68018)]. The recent interest in the field of \textit{term rewriting} (tr) for infinite terms is due to the use of \textit{term graph rewriting} to implement tr, where terms are represented by graphs: so, a cyclic graph is a finitary description of a possibly infinite term. In this paper we introduce \textit{infinite rewriting logic}, working on the framework of \textit{rewriting logic} proposed by \textit{J. Meseguer} [Functorial semantics of rewrite systems. Techn. Rep. CSL-93-02R, Computer Science Laboratory (1990); Theor. Comput. Sci. 96, No. 1, 73--155 (1992; Zbl 0758.68043)]. We provide a simple algebraic presentation of infinite computations, recovering the \textit{infinite parallel term rewriting}, originally presented by the first author [Lect. Notes Comput. Sci. 668, 468--484 (1993; \url{doi:10.1007/3-540-56610-4_83})] to extend the classical, set-theoretical approach to tr with infinite terms. Moreover, we put all the formalism on firm theoretical bases, providing (for the first time, to the best of our knowledge, for infinitary rewriting systems) a clean algebraic semantics by means of (internal) 2-categories.
For the entire collection see [Zbl 1492.68008].ESM systems and the composition of their computationshttps://zbmath.org/1496.681642022-11-17T18:59:28.764376Z"Janssens, D."https://zbmath.org/authors/?q=ai:janssens.dirk|janssens.davySummary: ESM systems are graph rewriting systems where productions are morphisms in a suitable category, ESM. The way graphs are transformed in ESM systems is essentially the same as in actor grammars, which were introduced in
[the author and \textit{G. Rozenberg}, Math. Syst. Theory 22, No. 2, 75--107 (1989; Zbl 0677.68082)].
It is demonstrated that a rewriting step corresponds to a (single) pushout construction, as in the approach from
[\textit{M. Löwe}, Theor. Comput. Sci. 109, No. 1--2, 181--224 (1993; Zbl 0787.18001)].
Rewriting processes in ESM systems are represented by computation structures, and it is shown that communication of rewriting processes corresponds to a gluing operation on computation structures. In the last section we briefly sketch how one may develop a semantics for ESM systems, based on computation structures, that is compositional w.r.t. the union of ESM systems.
For the entire collection see [Zbl 0825.00054].Canonical selection of colimitshttps://zbmath.org/1496.682032022-11-17T18:59:28.764376Z"Mossakowski, Till"https://zbmath.org/authors/?q=ai:mossakowski.till"Rabe, Florian"https://zbmath.org/authors/?q=ai:rabe.florian"Codescu, Mihai"https://zbmath.org/authors/?q=ai:codescu.mihaiSummary: Colimits are a powerful tool for the combination of objects in a category. In the context of modeling and specification, they are used in the institution-independent semantics (1) of instantiations of parameterised specifications (e.g. in the specification language CASL), and (2) of combinations of networks of specifications (in the OMG standardised language DOL).
The problem of using colimits as the semantics of certain language constructs is that they are defined only up to isomorphism. However, the semantics of a complex specification in these languages is given by a signature and a class of models over that signature -- not by an isomorphism class of signatures. This is particularly relevant when a specification with colimit semantics is further translated or refined. The user needs to know the symbols of a signature for writing a correct refinement.
Therefore, we study how to usefully choose one representative of the isomorphism class of all colimits of a given diagram. We develop criteria that colimit selections should meet. We work over arbitrary inclusive categories, but start the study how the criteria can be met with \(\mathbb Set\)-like categories, which are often used as signature categories for institutions.
For the entire collection see [Zbl 1428.68025].Directed homotopy in non-positively curved spaceshttps://zbmath.org/1496.682252022-11-17T18:59:28.764376Z"Goubault, Éric"https://zbmath.org/authors/?q=ai:goubault.eric"Mimram, Samuel"https://zbmath.org/authors/?q=ai:mimram.samuelThe notion of non-positively curved precubical set, which can be thought of as an algebraic analogue of the well-known one for metric spaces, captures the geometric properties of the precubical sets associated with concurrent programs using only mutexes, which are the most widely used synchronization primitives. A precubical set is non-positively curved if it is geometric, satisfies the cube property and satisfies the unique \(n\)-cube property for \(n\geq 3\). Using this, as well as categorical rewriting techniques, the authors are then able to show that directed and non-directed homotopy coincide for directed paths in these precubical sets. Finally, they study the geometric realization of precubical sets in metric spaces, to show that the conditions on precubical sets actually coincide with those for metric spaces. Since the category of metric spaces is not cocomplete, they are led to work with generalized metric spaces and study some of their properties.
Reviewer: Philippe Gaucher (Paris)On the category of Petri net computationshttps://zbmath.org/1496.682452022-11-17T18:59:28.764376Z"Sassone, Vladimiro"https://zbmath.org/authors/?q=ai:sassone.vladimiroSummary: We introduce the notion of \textit{strongly concatenable process} as a refinement of concatenable processes
[\textit{P. Degano} et al., in: Proceedings of the 4th annual symposium on logic in computer science, LICS'89. Los Alamitos, CA: IEEE Computer Society. 175--185 (1989; Zbl 0722.68085)]
which can be expressed axiomatically via a functor \(\mathcal{Q}[\_]\) from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each net \(N\), the strongly concatenable processes of \(N\) are isomorphic to the arrows of \(\mathcal{Q}[N]\). In addition, we identify a \textit{coreflection} right adjoint to \(\mathcal{Q}[\_]\) and characterize its \textit{replete image}, thus yielding an axiomatization of the category of net computations.
For the entire collection see [Zbl 0835.68002].Longitudinal structure function \(F_L\) at low \(Q^2\) and low \(x\) with model for higher twist: an updatehttps://zbmath.org/1496.811002022-11-17T18:59:28.764376Z"Badełek, Barbara"https://zbmath.org/authors/?q=ai:badelek.barbara"Staśto, Anna M."https://zbmath.org/authors/?q=ai:stasto.anna-mSummary: A reanalysis of the model for the longitudinal structure function \(F_L(x, Q^2)\) at low \(x\) and low \(Q^2\) was undertaken, in view of the advent of the EIC. The model is based on the photon-gluon fusion mechanism suitably extrapolated to the region of low \(Q^2\). It includes the kinematic constraint \(F_L \sim Q^4\) as \(Q^2 \to 0\) and higher twist contribution which vanishes as \(Q^2 \to \infty \). Revised model was critically updated and compared to the presently available data.