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Relevant bibliographies by topics / Symmetric monoidal categories

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Journal articles

Academic literature on the topic 'Symmetric monoidal categories'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 January 2023

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Journal articles on the topic "Symmetric monoidal categories"

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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