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Journal articles on the topic 'Symmetric monoidal categories'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 January 2023

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1

Nikolaus, Thomas, and Steffen Sagave. "Presentably symmetric monoidal ∞–categories are represented by symmetric monoidal model categories." Algebraic & Geometric Topology 17, no. 5 (2017): 3189–212. http://dx.doi.org/10.2140/agt.2017.17.3189.

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2

Banerjee, Abhishek. "Noetherian schemes over abelian symmetric monoidal categories." International Journal of Mathematics 28, no. 07 (2017): 1750051. http://dx.doi.org/10.1142/s0129167x17500513.

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In this paper, we develop basic results of algebraic geometry over abelian symmetric monoidal categories. Let [Formula: see text] be a commutative monoid object in an abelian symmetric monoidal category [Formula: see text] satisfying certain conditions and let [Formula: see text]. If the subobjects of [Formula: see text] satisfy a certain compactness property, we say that [Formula: see text] is Noetherian. We study the localization of [Formula: see text] with respect to any [Formula: see text] and define the quotient [Formula: see text] of [Formula: see text] with respect to any ideal [Formula: see text]. We use this to develop appropriate analogues of the basic notions from usual algebraic geometry (such as Noetherian schemes, irreducible, integral and reduced schemes, function field, the local ring at the generic point of a closed subscheme, etc.) for schemes over [Formula: see text]. Our notion of a scheme over a symmetric monoidal category [Formula: see text] is that of Toën and Vaquié.

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3

Breiner, Spencer, and John S. Nolan. "Symmetric Monoidal Categories with Attributes." Electronic Proceedings in Theoretical Computer Science 333 (February 8, 2021): 33–48. http://dx.doi.org/10.4204/eptcs.333.3.

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4

Ponto, Kate, and Michael Shulman. "Traces in symmetric monoidal categories." Expositiones Mathematicae 32, no. 3 (2014): 248–73. http://dx.doi.org/10.1016/j.exmath.2013.12.003.

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5

Morrison, Scott, and David Penneys. "Monoidal Categories Enriched in Braided Monoidal Categories." International Mathematics Research Notices 2019, no. 11 (2017): 3527–79. http://dx.doi.org/10.1093/imrn/rnx217.

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Abstract We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal{V}$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld centre of some monoidal category $\mathcal{T}$. Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features. Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather satisfy a braided interchange relation. Strikingly, our classification is slightly more general than what one might have anticipated in terms of strong monoidal functors $\mathcal{V}\to Z(\mathcal{T})$. We would like to understand this further; in a future article, we show that the functor is strong if and only if the enriched category is ‘complete’ in a certain sense. Nevertheless it remains to understand what non-complete enriched categories may look like. One should think of our construction as a generalization of de-equivariantization, which takes a strong monoidal functor ${\mathsf {Rep}}(G) \to Z(\mathcal{T})$ for some finite group $G$ and a monoidal category $\mathcal{T}$, and produces a new monoidal category $\mathcal{T} _{{/\hspace{-2px}/}G}$. In our setting, given any braided oplax monoidal functor $\mathcal{V} \to Z(\mathcal{T})$, for any braided $\mathcal{V}$, we produce $\mathcal{T} _{{/\hspace{-2px}/}\mathcal{V}}$: this is not usually an ‘honest’ monoidal category, but is instead $\mathcal{V}$-enriched. If $\mathcal{V}$ has a braided lax monoidal functor to ${\mathsf {Vec}}$, we can use this to reduce the enrichment to ${\mathsf {Vec}}$, and this recovers de-equivariantization as a special case. This is the published version of arXiv:1701.00567.

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6

ARDIZZONI, ALESSANDRO, and LAIACHI EL KAOUTIT. "INVERTIBLE BIMODULES, MIYASHITA ACTION IN MONOIDAL CATEGORIES AND AZUMAYA MONOIDS." Nagoya Mathematical Journal 225 (August 11, 2016): 1–63. http://dx.doi.org/10.1017/nmj.2016.25.

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In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita’s classical result and sheds light on other applications of geometric nature which cannot be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.

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7

DOšEN, KOSTA, and ZORAN PETRIĆ. "Isomorphic objects in symmetric monoidal closed categories." Mathematical Structures in Computer Science 7, no. 6 (1997): 639–62. http://dx.doi.org/10.1017/s0960129596002241.

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This paper presents a new and self-contained proof of a result characterizing objects isomorphic in the free symmetric monoidal closed category, i.e., objects isomorphic in every symmetric monoidal closed category. This characterization is given by a finitely axiomatizable and decidable equational calculus, which differs from the calculus that axiomatizes all arithmetical equalities in the language with 1, product and exponentiation by lacking 1c=1 and (a · b)c =ac · bc (the latter calculus characterizes objects isomorphic in the free cartesian closed category). Nevertheless, this calculus is complete for a certain arithmetical interpretation, and its arithmetical completeness plays an essential role in the proof given here of its completeness with respect to symmetric monoidal closed isomorphisms.

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8

Slodičak, Viliam. "Toposes are symmetric monoidal closed categories." Scientific Research of the Institute of Mathematics and Computer Science 11, no. 1 (2012): 107–16. http://dx.doi.org/10.17512/jamcm.2012.1.11.

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9

Rumynin, D. A. "Lie algebras in symmetric monoidal categories." Siberian Mathematical Journal 54, no. 5 (2013): 905–21. http://dx.doi.org/10.1134/s0037446613050145.

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10

Janelidze, George, and Ross Street. "Galois Theory in Symmetric Monoidal Categories." Journal of Algebra 220, no. 1 (1999): 174–87. http://dx.doi.org/10.1006/jabr.1999.7905.

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11

Cai, Chuanren, and Baoxin Jiang. "Symmetric centres of braided monoidal categories." Science in China Series A: Mathematics 43, no. 4 (2000): 384–90. http://dx.doi.org/10.1007/bf02897161.

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12

Guillou, Bertrand J., J. Peter May, Mona Merling, and Angélica M. Osorno. "SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION." Quarterly Journal of Mathematics 71, no. 1 (2019): 207–46. http://dx.doi.org/10.1093/qmathj/haz034.

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Abstract We give an operadic definition of a genuine symmetric monoidal $G$-category, and we prove that its classifying space is a genuine $E_\infty $$G$-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power and Lack to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal $G$-categories to genuine permutative $G$-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When $G$ is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal $G$-categories as input to an equivariant infinite loop space machine that gives genuine $\Omega $-$G$-spectra as output.

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13

Kelly, G. M., and F. Rossi. "Topological categories with many symmetric monoidal closed structures." Bulletin of the Australian Mathematical Society 31, no. 1 (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.

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14

Dosen, Kosta, and Zoran Petric. "Coherence of proof-net categories." Publications de l'Institut Math?matique (Belgrade) 78, no. 92 (2005): 1–33. http://dx.doi.org/10.2298/pim0578001d.

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The notion of proof-net category defined in this paper is closely related to graphs implicit in proof nets for the multiplicative fragment without constant propositions of linear logic. Analogous graphs occur in Kelly's and Mac Lane's coherence theorem for symmetric monoidal closed categories. A coherence theorem with respect to these graphs is proved for proof-net categories. Such a coherence theorem is also proved in the presence of arrows corresponding to the mix principle of linear logic. The notion of proof-net category catches the unit free fragment of the notion of star-autonomous category, a special kind of symmetric monoidal closed category.

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15

COCKETT, J. R. B., and J. S. LEMAY. "Integral categories and calculus categories." Mathematical Structures in Computer Science 29, no. 2 (2018): 243–308. http://dx.doi.org/10.1017/s0960129518000014.

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Differential categories are now an established abstract setting for differentiation. However, not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, sA: !A → !A ⊗ A, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorems is called a calculus category.Modifying an approach to antiderivatives by T. Ehrhard, we define having antiderivatives as the demand that a certain natural transformation, K: !A → !A, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category.When the coalgebra modality is monoidal, it is natural to demand an extra coherence between integration and the coalgebra modality. In the presence of this extra coherence, we show that a calculus category with a monoidal coalgebra modality has its integral transformation given by antiderivatives and, thus, that the integral structure is uniquely determined by the differential structure.The paper finishes by providing a suite of separating examples. Examples of differential categories, integral categories and calculus categories based on both monoidal and (mere) coalgebra modalities are presented. In addition, differential categories which are not integral categories are discussed and vice versa.

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16

GUIRAUD, YVES, and PHILIPPE MALBOS. "Coherence in monoidal track categories." Mathematical Structures in Computer Science 22, no. 6 (2012): 931–69. http://dx.doi.org/10.1017/s096012951100065x.

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We introduce homotopical methods based on rewriting on higher-dimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and use rewriting methods on polygraphs to solve it. The setting is extended to more general coherence problems, viewed as 3-dimensional word problems in a track category, including the case of braided monoidal categories.

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17

Pauwels, Bregje. "Quasi-Galois theory in symmetric monoidal categories." Algebra & Number Theory 11, no. 8 (2017): 1891–920. http://dx.doi.org/10.2140/ant.2017.11.1891.

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18

Péroux, Maximilien, and Brooke Shipley. "Coalgebras in symmetric monoidal categories of spectra." Homology, Homotopy and Applications 21, no. 1 (2019): 1–18. http://dx.doi.org/10.4310/hha.2019.v21.n1.a1.

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19

Banerjee, Abhishek. "Affine group schemes over symmetric monoidal categories." Pacific Journal of Mathematics 255, no. 1 (2012): 25–40. http://dx.doi.org/10.2140/pjm.2012.255.25.

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20

Hyland, Martin, and John Power. "Symmetric Monoidal Sketches and Categories of Wirings." Electronic Notes in Theoretical Computer Science 100 (October 2004): 31–46. http://dx.doi.org/10.1016/j.entcs.2004.09.004.

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21

Nicas, Andrew. "Trace and Duality in Symmetric Monoidal Categories." K-Theory 35, no. 3-4 (2005): 273–339. http://dx.doi.org/10.1007/s10977-005-3466-y.

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22

Hassanzadeh, Mohammad, Masoud Khalkhali, and Ilya Shapiro. "Monoidal Categories, 2-Traces, and Cyclic Cohomology." Canadian Mathematical Bulletin 62, no. 02 (2019): 293–312. http://dx.doi.org/10.4153/cmb-2018-016-4.

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AbstractIn this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category ( $\mathscr{C},\otimes$ ) endowed with a symmetric 2-trace, i.e., an $F\in \text{Fun}(\mathscr{C},\text{Vec})$ satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in $F$ ”. Furthermore, we observe that if $\mathscr{M}$ is a $\mathscr{C}$ -bimodule category and $(F,M)$ is a stable central pair, i.e., $F\in \text{Fun}(\mathscr{M},\text{Vec})$ and $M\in \mathscr{M}$ satisfy certain conditions, then $\mathscr{C}$ acquires a symmetric 2-trace. The dual notions of symmetric 2-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories.

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23

Banerjee, Abhishek. "On the Lie transformation algebra of monoids in symmetric monoidal categories." Rendiconti del Seminario Matematico della Università di Padova 131 (2014): 151–57. http://dx.doi.org/10.4171/rsmup/131-8.

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24

Elias, Ben, and Mikhail Khovanov. "Diagrammatics for Soergel Categories." International Journal of Mathematics and Mathematical Sciences 2010 (2010): 1–58. http://dx.doi.org/10.1155/2010/978635.

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The monoidal category of Soergel bimodules can be thought of as a categorification of the Hecke algebra of a finite Weyl group. We present this category, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations.

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25

Pourkia, Arash. "Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories." Mathematics and Statistics 3, no. 6 (2015): 157–63. http://dx.doi.org/10.13189/ms.2015.030604.

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26

Banerjee, Abhishek. "Schemes over symmetric monoidal categories and torsion theories." Journal of Pure and Applied Algebra 220, no. 9 (2016): 3017–47. http://dx.doi.org/10.1016/j.jpaa.2016.02.001.

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27

Böhm, Gabriella. "The Gray Monoidal Product of Double Categories." Applied Categorical Structures 28, no. 3 (2019): 477–515. http://dx.doi.org/10.1007/s10485-019-09587-5.

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AbstractThe category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $${\mathbb {A}}$$A, the corresponding internal hom functor "Equation missing" sends a double category $${\mathbb {B}}$$B to the double category whose 0-cells are the double functors $${\mathbb {A}} \rightarrow {\mathbb {B}}$$A→B, whose horizontal and vertical 1-cells are the horizontal and vertical pseudo transformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.

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28

Kimura, Kenichiro, Shun-ichi Kimura, and Nobuyoshi Takahashi. "Motivic zeta functions in additive monoidal categories." Journal of K-theory 9, no. 3 (2011): 459–73. http://dx.doi.org/10.1017/is011011006jkt174.

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AbstractLet C be a pseudo-abelian symmetric monoidal category, and X a Schur-finite object of C. We study the problem of rationality of the motivic zeta function ζx(t) of X. Since the coefficient ring is not a field, there are several variants of rationality — uniform, global, determinantal and pointwise rationality. We show that ζx(t) is determinantally rational, and we give an example of C and X for which the motivic zeta function is not uniformly rational.

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29

Vogel, Hans-Jürgen. "On generalized Hom-functors of certain symmetric monoidal categories." Discussiones Mathematicae - General Algebra and Applications 22, no. 1 (2002): 47. http://dx.doi.org/10.7151/dmgaa.1047.

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30

Hosseini, E., and A. Zaghian. "Purity and flatness in symmetric monoidal closed exact categories." Journal of Algebra and Its Applications 19, no. 01 (2019): 2050004. http://dx.doi.org/10.1142/s0219498820500048.

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Let [Formula: see text] be a symmetric monoidal closed exact category. This category is a natural framework to define the notions of purity and flatness. When [Formula: see text] is endowed with an injective cogenerator with respect to the exact structure, we show that an object [Formula: see text] in [Formula: see text] is flat if and only if any conflation ending in [Formula: see text] is pure. Furthermore, we prove a generalization of the Lambek Theorem (J. Lambek, A module is flat if and only if its character module is injective, Canad. Math. Bull. 7 (1964) 237–243) in [Formula: see text]. In the case [Formula: see text] is a quasi-abelian category, we prove that [Formula: see text] has enough pure injective objects.

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31

Schmitt, Vincent. "Completions of Non-Symmetric Metric Spaces Via Enriched Categories." gmj 16, no. 1 (2009): 157–82. http://dx.doi.org/10.1515/gmj.2009.157.

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Abstract It is known from [Lawvere, Repr. Theory Appl. Categ. 1: 1–37 2002] that nonsymmetric metric spaces correspond to enrichments over the monoidal closed category [0, ∞]. We use enriched category theory and in particular a generic notion of flatness to describe various completions for these spaces. We characterise the weights of colimits commuting in the base category [0, ∞] with the conical terminal object and cotensors. Those can be interpreted in metric terms as very general filters, which we call filters of type 1. This correspondence extends the one between minimal Cauchy filters and weights which are adjoint as modules. Translating elements of enriched category theory into the metric context, one obtains a notion of convergence for filters of type 1 with a related completeness notion for spaces, for which there exists a universal completion. Another smaller class of flat presheaves is also considered both in the context of both metric spaces and preorders. (The latter being enrichments over the monoidal closed category 2.) The corresponding completion for preorders is the so-called dcpo completion.

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32

ISAAC, P. S., W. P. JOYCE, and J. LINKS. "AN ALGEBRAIC APPROACH TO SYMMETRIC PRE-MONOIDAL STATISTICS." Journal of Algebra and Its Applications 06, no. 01 (2007): 49–69. http://dx.doi.org/10.1142/s0219498807002065.

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Recently, generalized Bose–Fermi statistics was studied in a category theoretic framework and to accommodate this endeavor the notion of a pre-monoidal category was developed. Here we describe an algebraic approach for the construction of such categories. We introduce a procedure called twining which breaks the quasi-bialgebra structure of the universal enveloping algebras of semi-simple Lie algebras and renders the category of finite-dimensional modules pre-monoidal. The category is also symmetric, meaning that each object of the category provides representations of the symmetric groups, which allows for a generalized boson-fermion statistic to be defined. Exclusion and confinement principles for systems of indistinguishable particles are formulated as an invariance with respect to the actions of the symmetric group. We apply the procedure to suggest that the symmetries which can be associated to color, spin and flavor degrees of freedom lead to confinement of states.

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33

Dosen, Kosta, and Zoran Petric. "Relevant categories and partial functions." Publications de l'Institut Math?matique (Belgrade), no. 96 (2007): 17–23. http://dx.doi.org/10.2298/pim0796017d.

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A relevant category is a symmetric monoidal closed category with a diagonal natural transformation that satisfies some coherence conditions. Every cartesian closed category is a relevant category in this sense. The denomination relevant comes from the connection with relevant logic. It is shown that the category of sets with partial functions, which is isomorphic to the category of pointed sets, is a category that is relevant, but not cartesian closed.

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34

Zhang, Tao, Yue Gu, and Shuanhong Wang. "Hopf Quasimodules and Yetter-Drinfeld Modules over Hopf Quasigroups." Algebra Colloquium 28, no. 02 (2021): 213–42. http://dx.doi.org/10.1142/s1005386721000183.

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We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup [Formula: see text] in a symmetric monoidal category [Formula: see text]. If [Formula: see text] possesses an adjoint quasiaction, we show that symmetric Yetter-Drinfeld categories are trivial, and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over [Formula: see text] and the category of four-angle Hopf modules over [Formula: see text] under some suitable conditions.

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35

Dell’Ambrogio, Ivo. "The unitary symmetric monoidal model category of small C*-categories." Homology, Homotopy and Applications 14, no. 2 (2012): 101–27. http://dx.doi.org/10.4310/hha.2012.v14.n2.a7.

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36

Dǒsen, Kosta, and Zoran Petrić. "Associativity as commutativity." Journal of Symbolic Logic 71, no. 1 (2006): 217–26. http://dx.doi.org/10.2178/jsl/1140641170.

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AbstractIt is shown that coherence conditions for monoidal categories concerning associativity are analogous to coherence conditions for symmetric strictly monoidal categories, where associativity arrows are identities. Mac Lane's pentagonal coherence condition for associativity is decomposed into conditions concerning commutativity, among which we have a condition analogous to naturality and a degenerate case of Mac Lane's hexagonal condition for commutativity. This decomposition is analogous to the derivation of the Yang-Baxter equation from Mac Lane's hexagon and the naturality of commutativity. The pentagon is reduced to an inductive definition of a kind of commutativity.

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37

Borceux, Francis, and Carmen Quinteriro. "Enriched accessible categories." Bulletin of the Australian Mathematical Society 54, no. 3 (1996): 489–501. http://dx.doi.org/10.1017/s0004972700021900.

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We consider category theory enriched in a locally finitely presentable symmetric monoidal closed category ν. We define the ν-filtered colimits as those colimits weighted by a ν-flat presheaf and consider the corresponding notion of ν-accessible category. We prove that ν-accessible categories coincide with the categories of ν-flat presheaves and also with the categories of ν-points of the categories of ν-presheaves. Moreover, the ν-locally finitely presentable categories are exactly the ν-cocomplete finitely accessible ones. To prove this last result, we show that the Cauchy completion of a small ν-category Cis equivalent to the category of ν-finitely presentable ν-flat presheaves on C.

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38

Groth, Moritz, Kate Ponto, and Michael Shulman. "The additivity of traces in monoidal derivators." Journal of K-theory 14, no. 3 (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.

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39

Cisinski, Denis-Charles, and Gonçalo Tabuada. "Symmetric monoidal structure on non-commutative motives." Journal of K-Theory 9, no. 2 (2011): 201–68. http://dx.doi.org/10.1017/is011011005jkt169.

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AbstractIn this article we further the study of non-commutative motives, initiated in [12, 43]. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Motlocdg of dg categories. As an application, we obtain : (1) a computation of the spectra of morphisms in Motlocdg in terms of non-connective algebraic K-theory; (2) a fully-faithful embedding of Kontsevich's category KMMk of non-commutative mixed motives into the base category Motlocdg(e) of the localizing motivator; (3) a simple construction of the Chern character maps from non-connective algebraic K-theory to negative and periodic cyclic homology; (4) a precise connection between Toën's secondary K-theory and the Grothendieck ring of KMMk; (5) a description of the Euler characteristic in KMMk in terms of Hochschild homology.

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40

Patterson, Evan, David I. Spivak, and Dmitry Vagner. "Wiring diagrams as normal forms for computing in symmetric monoidal categories." Electronic Proceedings in Theoretical Computer Science 333 (February 8, 2021): 49–64. http://dx.doi.org/10.4204/eptcs.333.4.

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41

Meir, Ehud. "Descent, fields of invariants, and generic forms via symmetric monoidal categories." Journal of Pure and Applied Algebra 220, no. 6 (2016): 2077–111. http://dx.doi.org/10.1016/j.jpaa.2015.10.016.

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42

Blumberg, Andrew J., and Michael A. Hill. "G-symmetric monoidal categories of modules over equivariant commutative ring spectra." Tunisian Journal of Mathematics 2, no. 2 (2020): 237–86. http://dx.doi.org/10.2140/tunis.2020.2.237.

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43

Moeller, Joe. "Noncommutative network models." Mathematical Structures in Computer Science 30, no. 1 (2019): 14–32. http://dx.doi.org/10.1017/s0960129519000161.

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AbstractNetwork models, which abstractly are given by lax symmetric monoidal functors, are used to construct operads for modeling and designing complex networks. Many common types of networks can be modeled with simple graphs with edges weighted by a monoid. A feature of the ordinary construction of network models is that it imposes commutativity relations between all edge components. Because of this, it cannot be used to model networks with bounded degree. In this paper, we construct the free network model on a given monoid, which can model networks with bounded degree. To do this, we generalize Green’s graph products of groups to pointed categories which are finitely complete and cocomplete.

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44

Elmendorf, A. D. "Function spectra." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 1 (1990): 31–34. http://dx.doi.org/10.1017/s0305004100068924.

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Boardman's stable category (see [5]) is a closed category ([4], VII·7), and in the best of all possible worlds, the category of spectra underlying the stable category would be closed as well; this would make life considerably easier for those doing calculations in stable homotopy theory. Unfortunately none of the categories of spectra introduced to date are closed; only S, the category introduced in [2], is even symmetric monoidal. The problem with making S closed is that it comes equipped with an augmentation to I, the category of universes and linear isometries (called Un in [2]), which preserves the symmetric monoidal structure. Since I is not closed, this makes it difficult to see how S might be closed.

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TILLMANN, ULRIKE. "Discrete models for the category of Riemann surfaces." Mathematical Proceedings of the Cambridge Philosophical Society 121, no. 1 (1997): 39–49. http://dx.doi.org/10.1017/s0305004196001065.

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We study algebraic models of the Segal category of Riemann surfaces [11]. These are 2-categories based on the 1+1 dimensional cobordism category enriched by diffeomorphisms of surfaces (or isotopy classes of such). In particular, we show that one of these models is a symmetric strict monoidal strict 2-category. The main technical tools are quotient constructions on 2-categories.

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46

Iwanari, Isamu. "Tannakization in derived algebraic geometry." Journal of K-theory 14, no. 3 (2014): 642–700. http://dx.doi.org/10.1017/is014008019jkt278.

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AbstractIn this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present basic properties of derived affine group schemes in the Appendix.

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Baez, John C., and Jade Master. "Open Petri nets." Mathematical Structures in Computer Science 30, no. 3 (2020): 314–41. http://dx.doi.org/10.1017/s0960129520000043.

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AbstractThe reachability semantics for Petri nets can be studied using open Petri nets. For us, an “open” Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category Open(Petri), which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category ${\mathbb O}$ pen(Petri). We describe two forms of semantics for open Petri nets using symmetric monoidal double functors out of ${\mathbb O}$ pen(Petri). The first, an operational semantics, gives for each open Petri net a category whose morphisms are the processes that this net can carry out. This is done in a compositional way, so that these categories can be computed on smaller subnets and then glued together. The second, a reachability semantics, simply says which markings of the outputs can be reached from a given marking of the inputs.

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Bulacu, Daniel, and Blas Torrecillas. "On Doi-Hopf modules and Yetter-Drinfeld modules in symmetric monoidal categories." Bulletin of the Belgian Mathematical Society - Simon Stevin 21, no. 1 (2014): 89–115. http://dx.doi.org/10.36045/bbms/1394544297.

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Brochier, Adrien. "Virtual tangles and fiber functors." Journal of Knot Theory and Its Ramifications 28, no. 07 (2019): 1950044. http://dx.doi.org/10.1142/s0218216519500445.

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We define a category [Formula: see text] of tangles diagrams drawn on surfaces with boundaries. On the one hand, we show that there is a natural functor from the category of virtual tangles to [Formula: see text] which induces an equivalence of categories. On the other hand, we show that [Formula: see text] is universal among ribbon categories equipped with a strong monoidal functor to a symmetric monoidal category. This is a generalization of the Shum–Reshetikhin–Turaev theorem characterizing the category of ordinary tangles as the free ribbon category. This gives a straightforward proof that all quantum invariants of links extend to framed oriented virtual links. This also provides a clear explanation of the relation between virtual tangles and Etingof–Kazhdan formalism suggested by Bar-Natan. We prove a similar statement for virtual braids, and discuss the relation between our category and knotted trivalent graphs.

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50

Pavlov, Dmitri, and Jakob Scholbach. "SYMMETRIC OPERADS IN ABSTRACT SYMMETRIC SPECTRA." Journal of the Institute of Mathematics of Jussieu 18, no. 4 (2018): 707–58. http://dx.doi.org/10.1017/s1474748017000202.

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This paper sets up the foundations for derived algebraic geometry, Goerss–Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën–Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith’s stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and $\text{E}_{\infty }$-ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of $\text{E}_{\infty }$-rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent. This paper is also available at arXiv:1410.5699v2.

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