Zeitschriftenartikel: „Monoidal structures“

1

Riehl, Emily. „Monoidal algebraic model structures“. Journal of Pure and Applied Algebra 217, Nr. 6 (Juni 2013): 1069–104. http://dx.doi.org/10.1016/j.jpaa.2012.09.029.

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2

Logar, Alessandro, und Fabio Rossi. „Monoidal closed structures on categories with constant maps“. Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 38, Nr. 2 (April 1985): 175–85. http://dx.doi.org/10.1017/s144678870002303x.

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AbstractThe purpose of this paper is to study the so-called canonical monoidal closed structures on concrete categories with constant maps. First of all we give an example of a category of this kind where there exists a non canonical monoidal closed structure. Later, we give a technique to construct a class of suitable full subcategories of the category of T0-spaces, such that all monoidal closed structures on them are canonical. Finally we show that “almost all” useful categories of topological compact spaces admit no monoidal closed structures whatsoever.

3

Schneider, Hans-Jürgen, und Blas Torrecillas. „Monoidal structures for N-complexes“. Journal of Pure and Applied Algebra 223, Nr. 12 (Dezember 2019): 5083–90. http://dx.doi.org/10.1016/j.jpaa.2019.03.011.

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4

Mesablishvili, Bachuki. „Entwining structures in monoidal categories“. Journal of Algebra 319, Nr. 6 (März 2008): 2496–517. http://dx.doi.org/10.1016/j.jalgebra.2007.08.030.

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5

García-Calcines, José Manuel, Luis Javier Hernández-Paricio und María Teresa Rivas-Rodríguez. „Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and Joins“. Mathematics 10, Nr. 4 (14.02.2022): 590. http://dx.doi.org/10.3390/math10040590.

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In this work, we analyze the combinatorial properties of the category of augmented semi-simplicial sets. We consider various monoidal structures induced by the co-product, the product, and the join operator in this category. In addition, we also consider monoidal structures on augmented sequences of integers induced by the sum and product of integers and by the join of augmented sequences. The cardinal functor that associates to each finite set X its cardinal |X| induces the sequential cardinal that associates to each augmented semi-simplicial finite set X an augmented sequence |X|n of non-negative integers. We prove that the sequential cardinal functor is monoidal for the corresponding monoidal structures. This allows us to easily calculate the number of simplices of cones and suspensions of an augmented semi-simplicial set as well as other augmented semi-simplicial sets which are built by joins. In this way, the monoidal structures of the augmented sequences of numbers may be thought of as an algebraization of the augmented semi-simplicial sets that allows us to do a simpler study of the combinatorics of the augmented semi-simplicial finite sets.

6

Kelly, G. M., und F. Rossi. „Topological categories with many symmetric monoidal closed structures“. Bulletin of the Australian Mathematical Society 31, Nr. 1 (Februar 1985): 41–59. http://dx.doi.org/10.1017/s0004972700002264.

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It would seem from results of Foltz, Lair, and Kelly that symmetric monoidal closed structures, and even monoidal biclosed ones, are quite rare on one-sorted algebraic or essentially-algebraic categories. They showed many such categories to admit no such structures at all, and others to admit only one or two; no such category is known to admit an infinite set of such structures.Among concrete categories, topological ones are in some sense at the other extreme from essentially-algebraic ones; and one is led to ask whether a topological category may admit many such structures. On the category of topological spaces itself, only one such structure - in fact symmetric - is known; although Greve has shown it to admit a proper class of monoidal closed structures. One of our main results is a proof that none of these structures described by Greve, except the classical one, is biclosed.Our other main result is that, nevertheless, there exist topological categories (of quasi-topological spaces) which admit a proper class of symmetric monoidal closed structures. Even if we insist (like most authors) that topological categories must be wellpowered, we can still exhibit ones with more such structures than any small cardinal.

7

Bulacu, D., S. Caenepeel und B. Torrecillas. „The braided monoidal structures on the category of vector spaces graded by the Klein group“. Proceedings of the Edinburgh Mathematical Society 54, Nr. 3 (14.06.2011): 613–41. http://dx.doi.org/10.1017/s0013091509001746.

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AbstractLet k be a field, let k* = k \ {0} and let C2 be a cyclic group of order 2. We compute all of the braided monoidal structures on the category of k-vector spaces graded by the Klein group C2 × C2. For the monoidal structures we compute the explicit form of the 3-cocycles on C2 × C2 with coefficients in k*, while, for the braided monoidal structures, we compute the explicit form of the abelian 3-cocycles on C2 × C2 with coefficients in k*. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras with underlying vector space k[C2 × C2].

8

Dorta, Joseph, Samantha Jarvis und Nelson Niu. „Monoidal Structures on Generalized Polynomial Categories“. Electronic Proceedings in Theoretical Computer Science 397 (14.12.2023): 84–97. http://dx.doi.org/10.4204/eptcs.397.6.

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9

Groth, Moritz, Kate Ponto und Michael Shulman. „The additivity of traces in monoidal derivators“. Journal of K-theory 14, Nr. 3 (14.07.2014): 422–94. http://dx.doi.org/10.1017/is014005011jkt262.

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AbstractMotivated by traces of matrices and Euler characteristics of topological spaces, we expect abstract traces in a symmetric monoidal category to be “additive”. When the category is “stable” in some sense, additivity along cofiber sequences is a question about the interaction of stability and the monoidal structure.May proved such an additivity theorem when the stable structure is a triangulation, based on new axioms for monoidal triangulated categories. in this paper we use stable derivators instead, which are a different model for “stable homotopy theories”. We define and study monoidal structures on derivators, providing a context to describe the interplay between stability and monoidal structure using only ordinary category theory and universal properties. We can then perform May's proof of the additivity of traces in a closed monoidal stable derivator without needing extra axioms, as all the needed compatibility is automatic.

10

ESTRADA, SERGIO, JAMES GILLESPIE und SINEM ODABAŞI. „Pure exact structures and the pure derived category of a scheme“. Mathematical Proceedings of the Cambridge Philosophical Society 163, Nr. 2 (23.11.2016): 251–64. http://dx.doi.org/10.1017/s0305004116000980.

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AbstractLet$\mathcal{C}$be closed symmetric monoidal Grothendieck category. We define the pure derived category with respect to the monoidal structure via a relative injective model category structure on the categoryC($\mathcal{C}$) of unbounded chain complexes in$\mathcal{C}$. We use λ-Purity techniques to get this. As application we define the stalkwise pure derived category of the category of quasi–coherent sheaves on a quasi-separated scheme. We also give a different approach by using the category of flat quasi–coherent sheaves.

11

Fiedorowicz, Zbigniew, Steven Gubkin und Rainer M. Vogt. „Associahedra and weak monoidal structures on categories“. Algebraic & Geometric Topology 12, Nr. 1 (20.03.2012): 469–92. http://dx.doi.org/10.2140/agt.2012.12.469.

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12

Chen, Yuanyuan, und Liangyun Zhang. „Hom-Yang-Baxter Equations and Frobenius Monoidal Hom-Algebras“. Advances in Mathematical Physics 2018 (01.08.2018): 1–10. http://dx.doi.org/10.1155/2018/2912578.

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It is shown that quasi-Frobenius Hom-Lie algebras are connected with a class of solutions of the classical Hom-Yang-Baxter equations. Moreover, a similar relation is discussed on Frobenius (symmetric) monoidal Hom-algebras and solutions of quantum Hom-Yang-Baxter equations. Monoidal Hom-Hopf algebras with Frobenius structures are studied at last.

13

ZHANG, XIAOHUI, und LIHONG DONG. „BRAIDED MIXED DATUMS AND THEIR APPLICATIONS ON HOM-QUANTUM GROUPS“. Glasgow Mathematical Journal 60, Nr. 1 (04.09.2017): 231–51. http://dx.doi.org/10.1017/s0017089517000088.

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AbstractIn this paper, we mainly provide a categorical view on the braided structures appearing in the Hom-quantum groups. Let $\mathcal{C}$ be a monoidal category on which F is a bimonad, G is a bicomonad, and ϕ is a distributive law, we discuss the necessary and sufficient conditions for $\mathcal{C}^G_F(\varphi)$, the category of mixed bimodules to be monoidal and braided. As applications, we discuss the Hom-type (co)quasitriangular structures, the Hom–Yetter–Drinfeld modules, and the Hom–Long dimodules.

14

Lacabanne, Abel, Daniel Tubbenhauer und Pedro Vaz. „Asymptotics in finite monoidal categories“. Proceedings of the American Mathematical Society, Series B 10, Nr. 34 (08.11.2023): 398–412. http://dx.doi.org/10.1090/bproc/198.

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15

BUCKLEY, MITCHELL, RICHARD GARNER, STEPHEN LACK und ROSS STREET. „The Catalan simplicial set“. Mathematical Proceedings of the Cambridge Philosophical Society 158, Nr. 2 (03.12.2014): 211–22. http://dx.doi.org/10.1017/s0305004114000498.

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AbstractThe Catalan numbers are well known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory.

16

BARNES, DAVID. „A monoidal algebraic model for rational SO(2)-spectra“. Mathematical Proceedings of the Cambridge Philosophical Society 161, Nr. 1 (11.04.2016): 167–92. http://dx.doi.org/10.1017/s0305004116000219.

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AbstractThe category of rational SO(2)–equivariant spectra admits an algebraic model. That is, there is an abelian category ${\mathcal A}$(SO(2)) whose derived category is equivalent to the homotopy category of rational SO(2)–equivariant spectra. An important question is: does this algebraic model capture the smash product of spectra?The category ${\mathcal A}$(SO(2)) is known as Greenlees' standard model, it is an abelian category that has no projective objects and is constructed from modules over a non–Noetherian ring. As a consequence, the standard techniques for constructing a monoidal model structure cannot be applied. In this paper a monoidal model structure on ${\mathcal A}$(SO(2)) is constructed and the derived tensor product on the homotopy category is shown to be compatible with the smash product of spectra. The method used is related to techniques developed by the author in earlier joint work with Roitzheim. That work constructed a monoidal model structure on Franke's exotic model for the K(p)–local stable homotopy category.A monoidal Quillen equivalence to a simpler monoidal model category R•-mod that has explicit generating sets is also given. Having monoidal model structures on ${\mathcal A}$(SO(2)) and R•-mod removes a serious obstruction to constructing a series of monoidal Quillen equivalences between the algebraic model and rational SO(2)–equivariant spectra.

17

Zhang, Xiaohui, und Hui Wu. „The cosemisimplicity and cobraided structures of monoidal comonads“. Frontiers of Mathematics in China 17, Nr. 3 (18.05.2022): 485–99. http://dx.doi.org/10.1007/s11464-022-1019-9.

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18

Hu, Nick, und Jamie Vicary. „Traced Monoidal Categories as Algebraic Structures in Prof“. Electronic Proceedings in Theoretical Computer Science 351 (29.12.2021): 84–97. http://dx.doi.org/10.4204/eptcs.351.6.

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19

Fiore, M., N. Gambino, M. Hyland und G. Winskel. „Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures“. Selecta Mathematica 24, Nr. 3 (20.11.2017): 2791–830. http://dx.doi.org/10.1007/s00029-017-0361-3.

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20

Hovey, Mark. „Additive closed symmetric monoidal structures on R-modules“. Journal of Pure and Applied Algebra 215, Nr. 5 (Mai 2011): 789–805. http://dx.doi.org/10.1016/j.jpaa.2010.06.024.

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21

Manin, Yuri Ivanovich, und Bruno Vallette. „Monoidal Structures on the Categories of Quadratic Data“. Documenta Mathematica 25 (2020): 1727–86. http://dx.doi.org/10.4171/dm/784.

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22

KOCK, JOACHIM. „Elementary remarks on units in monoidal categories“. Mathematical Proceedings of the Cambridge Philosophical Society 144, Nr. 1 (Januar 2008): 53–76. http://dx.doi.org/10.1017/s0305004107000679.

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AbstractWe explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellable idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of left-right constraints, and is motivated by higher category theory. To start, we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and observe that it is contractible (if non-empty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent non-algebraic notion of fair monoidal category, where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered. The paper is self-contained. All arguments are elementary, some of them of a certain beauty.

23

Wang, Shuan-hong, und Hai-xing Zhu. „On Braided Lie Structures of Algebras in the Categories of Weak Hopf Bimodules“. Algebra Colloquium 17, Nr. 04 (Dezember 2010): 685–98. http://dx.doi.org/10.1142/s1005386710000659.

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Let H be a weak Hopf algebra. In this paper, it is proved that the monoidal category [Formula: see text] of weak Hopf bimodules studied in Wang [19] is equivalent to the monoidal category [Formula: see text] of weak Yetter–Drinfel'd modules introduced in Böhm [2]. When H has a bijective antipode, a braiding in the category [Formula: see text] is constructed by the braiding on [Formula: see text], generalizing the main result in Schauenburg [14]. Finally, the braided Lie structures of an algebra A in the category [Formula: see text] are investigated, by showing that if A is a sum of two braided commutative subalgebras, then the braided commutator ideal of A is nilpotent.

24

Joana CIRICI und Geoffroy HOREL. „Mixed Hodge structures and formality of symmetric monoidal functors“. Annales scientifiques de l'École normale supérieure 53, Nr. 4 (2020): 1071–104. http://dx.doi.org/10.24033/asens.2440.

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25

Zhang, Xiaohui, Wei Wang und Xiaofan Zhao. „Smash coproducts of monoidal comonads and Hom-entwining structures“. Rocky Mountain Journal of Mathematics 49, Nr. 6 (Oktober 2019): 2063–105. http://dx.doi.org/10.1216/rmj-2019-49-6-2063.

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26

Lowen, R., und M. Sioen. „On the multitude of monoidal closed structures on UAP“. Topology and its Applications 137, Nr. 1-3 (Februar 2004): 215–23. http://dx.doi.org/10.1016/s0166-8641(03)00211-6.

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27

Iğde, Elif, und Koray Yılmaz. „Tensor Products and Crossed Differential Graded Lie Algebras in the Category of Crossed Complexes“. Symmetry 15, Nr. 9 (25.08.2023): 1646. http://dx.doi.org/10.3390/sym15091646.

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The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has resulted in many areas, including algebraic topology, representation theory, and quantum physics. In this paper, we present analogous definitions for Lie algebras within the framework of whiskered structures, bimorphisms, crossed complexes, crossed differential graded algebras, and tensor products. These definitions, given for groupoids in existing literature, have been adapted to establish a direct correspondence between these algebraic structures and Lie algebras. We show that a 2-truncation of the crossed differential graded Lie algebra, obtained from our adapted definitions, gives rise to a braided crossed module of Lie algebras. We also construct a functor to simplicial Lie algebras, enabling a systematic mapping between different Lie algebraic categories, which supports the validity of our adapted definitions and establishes their compatibility with established categories.

28

Zhang, Guo-Qiang. „Some monoidal closed categories of stable domains and event structures“. Mathematical Structures in Computer Science 3, Nr. 2 (Juni 1993): 259–76. http://dx.doi.org/10.1017/s0960129500000207.

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This paper introduces the following new constructions on stable domains and event structures: the tensor product; the linear function space; and the exponential. These give rise to a monoidal closed category of dI-domains and to stable event structures, which can be used to interpret intuitionistic linear logic. Finally, the usefulness of the category of stable event structures for modeling concurrency and its relation to other models are discussed.

29

Batanin, Mikhail A. „Homotopy coherent category theory and A∞-structures in monoidal categories“. Journal of Pure and Applied Algebra 123, Nr. 1-3 (Januar 1998): 67–103. http://dx.doi.org/10.1016/s0022-4049(96)00084-9.

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30

Hoehnke, Hans-Jürgen. „Skew-monoidal semigroup structures on simple rings with several objects“. Semigroup Forum 34, Nr. 1 (Dezember 1986): 69–87. http://dx.doi.org/10.1007/bf02573153.

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31

Bulacu, D., und S. Caenepeel. „Monoidal Ring and Coring Structures Obtained from Wreaths and Cowreaths“. Algebras and Representation Theory 17, Nr. 4 (27.06.2013): 1035–82. http://dx.doi.org/10.1007/s10468-013-9431-1.

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32

IONESCU, LUCIAN M. „A NOTE ON DUALITY, FROBENIUS ALGEBRAS AND TQFTS“. Journal of Knot Theory and Its Ramifications 13, Nr. 08 (Dezember 2004): 999–1006. http://dx.doi.org/10.1142/s0218216504003676.

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Topological quantum field theories (TQFTs) represent the structure present in cobordism categories. As an example, we review the correspondence between Frobenius algebras and (1+1)TQFTs. It is a corollary of the self-duality of the cobordism category, which is a rigid monoidal category generated by a Frobenius object (the circle). A self-dual definition of a Frobenius object without the use of a prefered dual is considered. The issue of duality as part of the definition of a TQFT is addressed. Note that duality is preserved by monoidal functors. Hermitian structures are modeled as a conjugation compatible with duality. It is the structure cobordism categories posses. A definition of generalized cobordism categories is proposed.

33

Chen, Quan-Guo, und Wen-Jing Cheng. „Twisted partial actions of monoidal Hom-Hopf algebras“. Filomat 36, Nr. 9 (2022): 2991–3011. http://dx.doi.org/10.2298/fil2209991c.

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In this work, the notion of a twisted partial Hom-Hopf action is introduced, and the conditions on partial cocycles are established in order to construct partial Hom-crossed products. Also, the equivalence of partial Hom-crossed products is discussed. Finally, we shall describe the coquasitriangular structures on partial Hom-crossed products

34

Srivastava, Arun K., und S. P. Tiwari. „On Categories of Fuzzy Petri Nets“. Advances in Fuzzy Systems 2011 (2011): 1–5. http://dx.doi.org/10.1155/2011/812040.

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We introduce the concepts of fuzzy Petri nets and marked fuzzy Petri nets along with their appropriate morphisms, which leads to two categories of such Petri nets. Some aspects of the internal structures of these categories are then explored, for example, their reflectiveness/coreflectiveness and symmetrical monoidal closed structure.

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Huang, Hua-Lin, Gongxiang Liu und Yu Ye. „The Braided Monoidal Structures on a Class of Linear Gr-Categories“. Algebras and Representation Theory 17, Nr. 4 (25.08.2013): 1249–65. http://dx.doi.org/10.1007/s10468-013-9445-8.

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36

BAUES, HANS–JOACHIM, MAMUKA JIBLADZE und TEIMURAZ PIRASHVILI. „Third Mac Lane cohomology“. Mathematical Proceedings of the Cambridge Philosophical Society 144, Nr. 2 (März 2008): 337–67. http://dx.doi.org/10.1017/s030500410700076x.

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AbstractMacLane cohomology is an algebraic version of the topological Hochschild cohomology. Based on the computation of the third author (see Appendix) we obtain an interpretation of the third Mac Lane cohomology of rings using certain kind of crossed extensions of rings in the quadratic world. Actually we obtain two such interpretations corresponding to the two monoidal structures on the category of square groups.

37

Marty, Florian. „Smoothness in Relative Geometry“. Journal of K-theory 12, Nr. 3 (15.11.2013): 461–91. http://dx.doi.org/10.1017/is013010008jkt242.

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AbstractIn [TVa], Bertrand Toën and Michel Vaquié defined a scheme theory for a closed monoidal category ( ⊗1). In this article, we define a notion of smoothness in this relative (and not necessarily additive) context which generalizes the notion of smoothness in the category of rings. This generalisation consists in replacing homological finiteness conditions by homotopical ones, using the Dold-Kan correspondence. To do this, we provide the category s of simplicial objects in a monoidal category and all the categories sA-mod, sA-alg (A ∈ sComm()) with compatible model structures using the work of Rezk [R]. We then give a general notion of smoothness in sComm(). We prove that this notion is a generalisation of the notion of smooth morphism in the category of rings and is stable under composition and homotopy pushouts. Finally we provide some examples of smooth morphisms, in particular in ℕ-alg and Comm(Set).

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Kock, Joachim, und Bertrand Toën. „Simplicial localization of monoidal structures, and a non-linear version of Deligne's conjecture“. Compositio Mathematica 141, Nr. 01 (01.12.2004): 253–61. http://dx.doi.org/10.1112/s0010437x04001009.

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39

Jiao, Zhengming, und Gongyu Huang. „Solutions of the Hom-Yang–Baxter Equation from Monoidal Hom-(Co)Algebra Structures“. Mathematical Notes 104, Nr. 1-2 (Juli 2018): 121–34. http://dx.doi.org/10.1134/s0001434618070131.

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40

Bonchi, Filippo, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski und Fabio Zanasi. „String Diagram Rewrite Theory I: Rewriting with Frobenius Structure“. Journal of the ACM 69, Nr. 2 (30.04.2022): 1–58. http://dx.doi.org/10.1145/3502719.

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String diagrams are a powerful and intuitive graphical syntax, originating in theoretical physics and later formalised in the context of symmetric monoidal categories. In recent years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology. In several of these proposals, transformations of systems are modelled as rewrite rules of diagrams. These developments require a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the two-dimensional nature of string diagrams poses quite a few additional challenges. This work systematises and expands a series of recent conference papers, laying down such a foundation. As a first step, we focus on the case of rewrite systems for string diagrammatic theories that feature a Frobenius algebra. This common structure provides a more permissive notion of composition than the usual one available in monoidal categories, and has found many applications in areas such as concurrency, quantum theory, and electrical circuits. Notably, this structure provides an exact correspondence between the syntactic notion of string diagrams modulo Frobenius structure and the combinatorial structure of hypergraphs. Our work introduces a combinatorial interpretation of string diagram rewriting modulo Frobenius structures in terms of double-pushout hypergraph rewriting. We prove this interpretation to be sound and complete and we also show that the approach can be generalised to rewriting modulo multiple Frobenius structures. As a proof of concept, we show how to derive from these results a termination strategy for Interacting Bialgebras, an important rewrite theory in the study of quantum circuits and signal flow graphs.

41

Cheng, Eugenia. „Distributive laws for Lawvere theories“. Compositionality 2 (25.05.2020): 1. http://dx.doi.org/10.32408/compositionality-2-1.

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Distributive laws give a way of combining two algebraic structures expressed as monads; in this paper we propose a theory of distributive laws for combining algebraic structures expressed as Lawvere theories. We propose four approaches, involving profunctors, monoidal profunctors, an extension of the free finite-product category 2-monad from Cat to Prof, and factorisation systems respectively. We exhibit comparison functors between CAT and each of these new frameworks to show that the distributive laws between the Lawvere theories correspond in a suitable way to distributive laws between their associated finitary monads. The different but equivalent formulations then provide, between them, a framework conducive to generalisation, but also an explicit description of the composite theories arising from distributive laws.

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Böhm, Gabriella. „Comodules over weak multiplier bialgebras“. International Journal of Mathematics 25, Nr. 05 (Mai 2014): 1450037. http://dx.doi.org/10.1142/s0129167x14500372.

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This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

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Fresse, Benoit. „Props in model categories and homotopy invariance of structures“. gmj 17, Nr. 1 (März 2010): 79–160. http://dx.doi.org/10.1515/gmj.2010.007.

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Abstract We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object 𝑋 homotopy equivalent to an algebra 𝐴 over a cofibrant prop P inherits a P-algebra structure so that 𝑋 defines a model of 𝐴 in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of 𝐴∞-algebras.

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BUCKLEY, MITCHELL. „THREE STUDIES IN HIGHER CATEGORY THEORY: FIBRATIONS, SKEW-MONOIDAL STRUCTURES AND EXCISION OF EXTREMALS“. Bulletin of the Australian Mathematical Society 94, Nr. 2 (21.07.2016): 337–38. http://dx.doi.org/10.1017/s0004972716000356.

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SCHWEDE, STEFAN. „Stable homotopical algebra and Γ-spaces“. Mathematical Proceedings of the Cambridge Philosophical Society 126, Nr. 2 (März 1999): 329–56. http://dx.doi.org/10.1017/s0305004198003272.

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In this paper we advertise the category of Γ-spaces as a convenient framework for doing ‘algebra’ over ‘rings’ in stable homotopy theory. Γ-spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (−1)-connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for Γ-spaces. The study of ‘rings, modules and algebras’ based on Γ-spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their Eilenberg–MacLane spectra counterparts.

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Gálvez-Carrillo, Imma, Joachim Kock und Andrew Tonks. „Decomposition Spaces and Restriction Species“. International Mathematics Research Notices 2020, Nr. 21 (12.09.2018): 7558–616. http://dx.doi.org/10.1093/imrn/rny089.

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Abstract We show that Schmitt’s restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital $2$-Segal spaces), and that their associated coalgebras are an instance of the general construction of incidence coalgebras of decomposition spaces. We introduce directed restriction species that subsume Schmitt’s restriction species and also induce decomposition spaces. Whereas ordinary restriction species are presheaves on the category of finite sets and injections, directed restriction species are presheaves on the category of finite posets and convex maps. We also introduce the notion of monoidal (directed) restriction species, which induce monoidal decomposition spaces and hence bialgebras, most often Hopf algebras. Examples of this notion include rooted forests, directed graphs, posets, double posets, and many related structures. A prominent instance of a resulting incidence bialgebra is the Butcher–Connes–Kreimer Hopf algebra of rooted trees. Both ordinary and directed restriction species are shown to be examples of a construction of decomposition spaces from certain cocartesian fibrations over the category of finite ordinals that are also cartesian over convex maps. The proofs rely on some beautiful simplicial combinatorics, where the notion of convexity plays a key role. The methods developed are of independent interest as techniques for constructing decomposition spaces.

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BARNES, DAVID, und CONSTANZE ROITZHEIM. „STABLE LEFT AND RIGHT BOUSFIELD LOCALISATIONS“. Glasgow Mathematical Journal 56, Nr. 1 (25.02.2013): 13–42. http://dx.doi.org/10.1017/s0017089512000882.

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AbstractWe study left and right Bousfield localisations of stable model categories which preserve stability. This follows the lead of the two key examples: localisations of spectra with respect to a homology theory and A-torsion modules over a ring R with A a perfect R-algebra. We exploit stability to see that the resulting model structures are technically far better behaved than the general case. We can give explicit sets of generating cofibrations, show that these localisations preserve properness and give a complete characterisation of when they preserve monoidal structures. We apply these results to obtain convenient assumptions under which a stable model category is spectral. We then use Morita theory to gain an insight into the nature of right localisation and its homotopy category. We finish with a correspondence between left and right localisation.

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Abbadini, Marco. „On the Axiomatisability of the Dual of Compact Ordered Spaces“. Bulletin of Symbolic Logic 27, Nr. 4 (Dezember 2021): 526. http://dx.doi.org/10.1017/bsl.2021.54.

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AbstractWe prove that the category of Nachbin’s compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we observe that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to (i) any finitely accessible category, (ii) any first-order definable class of structures, and (iii) any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by Mundici: our result—whose proof is independent of Mundici’s theorem—asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras.Abstract taken directly from the thesis.E-mail: marco.abbadini.uni@gmail.comURL: https://air.unimi.it/retrieve/handle/2434/812809/1698986/phd_unimi_R11882.pdf

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Elmanto, Elden, und Rune Haugseng. „On distributivity in higher algebra I: the universal property of bispans“. Compositio Mathematica 159, Nr. 11 (18.09.2023): 2326–415. http://dx.doi.org/10.1112/s0010437x23007388.

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Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ( $\infty$ -)categories of spans (or correspondences). In this paper, we study the more complicated setup where we have two pushforwards (an ‘additive’ and a ‘multiplicative’ one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(\infty,2)$ -categories of bispans, characterized by a universal property: they corepresent functors out of $\infty$ -categories of spans where the pullbacks have left adjoints and certain canonical 2-morphisms (encoding base change and distributivity) are invertible. This gives a universal way to obtain functors from bispans, which amounts to upgrading ‘monoid-like’ structures to ‘ring-like’ ones. For example, symmetric monoidal $\infty$ -categories can be described as product-preserving functors from spans of finite sets, and if the tensor product is compatible with finite coproducts our universal property gives the canonical semiring structure using the coproduct and tensor product. More interestingly, we encode the additive and multiplicative transfers on equivariant spectra as a functor from bispans in finite $G$ -sets, extend the norms for finite étale maps in motivic spectra to a functor from certain bispans in schemes, and make $\mathrm {Perf}(X)$ for $X$ a spectral Deligne–Mumford stack a functor of bispans using a multiplicative pushforward for finite étale maps in addition to the usual pullback and pushforward maps. Combining this with the polynomial functoriality of $K$ -theory constructed by Barwick, Glasman, Mathew, and Nikolaus, we obtain norms on algebraic $K$ -theory spectra.

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Linzi, Alessandro. „Polygroup objects in regular categories“. AIMS Mathematics 9, Nr. 5 (2024): 11247–77. http://dx.doi.org/10.3934/math.2024552.

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<abstract><p>We express the fundamental properties of commutative polygroups (also known as canonical hypergroups) in category-theoretic terms, over the category $ \mathbf{Set} $ formed by sets and functions. For this, we employ regularity as well as the monoidal structure induced on the category $ {\mathbf{Rel}} $ of sets and relations by cartesian products. We highlight how our approach can be generalised to any regular category. In addition, we consider the theory of partial multirings and find fully faithful functors between certain slice or coslice categories of the category of partial multirings and other categories formed by well-known mathematical structures and their morphisms.</p></abstract>

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