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Symmetric monoidal category

About: Symmetric monoidal category is a research topic. Over the lifetime, 981 publications have been published within this topic receiving 22044 citations. The topic is also known as: symmetric tensor category.


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Book ChapterDOI
TL;DR: In this article, a reference guide to various notions of monoidal categories and their associated string diagrams is presented, which is useful not only to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning.
Abstract: This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning We have opted for a somewhat informal treatment of topological notions, and have omitted most proofs Nevertheless, the exposition is sufficiently detailed to make it clear what is presently known, and to serve as a starting place for more in-depth study Where possible, we provide pointers to more rigorous treatments in the literature Where we include results that have only been proved in special cases, we indicate this in the form of caveats

732 citations

Posted Content
16 Jan 1998
TL;DR: In this article, the authors define and study the model category of symmetric spectra, based on simplicial sets and topological spaces, and prove that the category is closed symmetric monoidal.
Abstract: The long hunt for a symmetric monoidal category of spectra finally ended in success with the simultaneous discovery of the third author's discovery of symmetric spectra and the Elmendorf-Kriz-Mandell-May category of S-modules. In this paper we define and study the model category of symmetric spectra, based on simplicial sets and topological spaces. We prove that the category of symmetric spectra is closed symmetric monoidal and that the symmetric monoidal structure is compatible with the model structure. We prove that the model category of symmetric spectra is Quillen equivalent to Bousfield and Friedlander's category of spectra. We show that the monoidal axiom holds, so that we get model categories of ring spectra and modules over a given ring spectrum.

625 citations

Journal ArticleDOI
TL;DR: In this paper, the authors provide a general method for constructing model category structures for categories of ring, algebra, and module spectra, and provide the necessary input for obtaining model categories of symmetric ring spectra and functors with smash product.
Abstract: In recent years the theory of structured ring spectra (formerly known as A$_{\infty}$- and E$_{\infty}$-ring spectra) has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra. In this paper we provide a general method for constructing model category structures for categories of ring, algebra, and module spectra. This provides the necessary input for obtaining model categories of symmetric ring spectra, functors with smash product, $\Gamma$-rings, and diagram ring spectra. Algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra. 1991 Mathematics Subject Classification: primary 55U35; secondary 18D10.

572 citations

Journal ArticleDOI
TL;DR: In this article, a theory of module categories over monoidal categories was developed, and it was shown that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi).
Abstract: We develop a theory of module categories over monoidal categories (this is a straightforward categorization of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and we classify module categories over the fusion category of sl(2) at a positive integer level where we meet once again the ADE classification pattern.

537 citations

Book
27 Apr 2008
TL;DR: In this article, the authors define a general notion of geometric stack over a base symmetric monoidal model category C, and prove that this notion satisfies the expected properties of the algebraic n-stacks of Simpson.
Abstract: This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category, such as derivations, etale and smooth maps, flat and projective modules, etc. We then use the theory of stacks over model categories introduced in \cite{hagI} in order to define a general notion of geometric stack over a base symmetric monoidal model category C, and prove that this notion satisfies the expected properties. The rest of the paper consists in specializing C to several different contexts. First of all, when C=k-Mod is the category of modules over a ring k, with the trivial model structure, we show that our notion gives back the algebraic n-stacks of C. Simpson. Then we set C=sk-Mod, the model category of simplicial k-modules, and obtain this way a notion of geometric derived stacks which are the main geometric objects of Derived Algebraic Geometry. We give several examples of derived version of classical moduli stacks, as for example the derived stack of local systems on a space, of algebra structures over an operad, of flat bundles on a projective complex manifold, etc. Finally, we present the cases where C=(k) is the model category of unbounded complexes of modules over a char 0 ring k, and C=Sp^{\Sigma} the model category of symmetric spectra. In these two contexts, called respectively Complicial and Brave New Algebraic Geometry, we give some examples of geometric stacks such as the stack of associative dg-algebras, the stack of dg-categories, and a geometric stack constructed using topological modular forms.

501 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202324
202240
202115
202028
201930
201832