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# Topic: Monoidal category

###### In the News (Tue 21 May 13)

 Monoidal category - Wikipedia, the free encyclopedia In mathematics, a monoidal category (or tensor category) is a 2-category with one object (a 2-monoid). Monoidal categories are used to define models for the multiplicative fragment of intuitionistic linear logic. A monoidal category may be regarded as a bicategory with one object. en.wikipedia.org /wiki/Monoidal_category   (560 words)

 Monoidal category: Encyclopedia topic   (Site not responding. Last check: 2007-11-03) In mathematics (mathematics: A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement), a monoidal category (or tensor category) is a category (category: A general concept that marks divisions or coordinations in a conceptual scheme) equipped with a binary 'tensor' functor and a unit object. A monoidal category may be regarded as a bicategory (bicategory: a bicategory is a concept in category theory used to extend the notion of sameness (i.e.... Monoidal categories are used to define models for linear logic (linear logic: in mathematical logic, linear logic is a type of substructural logic that denies... www.absoluteastronomy.com /reference/monoidal_category   (574 words)

 Enriched category In category theory and its applications to mathematics, an enriched category is a category whose hom-sets are replaced by objects from some other category, in a well-behaved manner. If M is the category 2 with Ob(2) = {0,1}, a single nonidentity morphism (from 0 to 1), and ordinary multiplication of numbers as the monoidal operation, then C can be interpreted as a preordered set. If there is a monoidal functor[?] from a monoidal category M to a monoidal category N, then any category enriched over M can be reinterpreted as a category enriched over N. www.ebroadcast.com.au /lookup/encyclopedia/en/Enriched_category.html   (580 words)

 [No title] A monoid in this symmetric monoidal category is a differential g* *raded algebra (DGA). Hence, M is cofibrant i* *n the underlying category C. Proof of lemma 5.2 The main ingredient is a filtration of a certain kind of pus* *hout in the monoid category. P is a map of monoids and (iii)P has the universal property of the pushout in the category of monoids. www.math.purdue.edu /research/atopology/Schwede-Shipley/last.txt   (8006 words)

 Preadditive category   (Site not responding. Last check: 2007-11-03) A preadditive category is a category that is enriched over the monoidal category of abelian groups. An additive category is a preadditive category with all biproducts. A pre-Abelian category is an additive category with all and cokernels. www.freeglossary.com /Module_category   (1324 words)

 [No title] In particular, there is a homotopy category of monoids of topological symmetric spectra, and this homotopy category is equivalent to the homotopy category of monoids of simplicial symmetric spectra. Y is an isomorphism in the homotopy category. X of monoids is a countable composition of maps Pi -fi!Pi+1, where fi is the pushout in C of a map X^(i+1)^ gi. hopf.math.purdue.edu /Hovey/mon-mod.txt   (8088 words)

 [No title] A monoidal category is a bicategory with one object. A monoid in monoidal categories is a braided monoidal category. A braided monoidal category is a tri-category with one object and one map. www.mta.ca /~cat-dist/catlist/1999/braided   (1985 words)

 Homotopical algebra and higher categories The category of fair \$2\$-categories is shown to be equivalent to the category of bicatgeories with strict composition law. The unit compatibility condition for a (strong) monoidal functor is shown to be precisely the condition for the functor to lift to the categories of units. The notion of Saavedra unit leads naturally to the equivalent non-algebraic notion of fair monoidal category (treated elsewhere), where the contractible multitude of units is considered as a whole instead of choosing one unit. mat.uab.es /~kock/cat.html   (775 words)

 The Dimensional Ladder Categories from Spaces The fundamental groupoid of a topological space The fundamental group of a pointed space categories from chain complexes: a 2-term chain complex is a category in AbGp. A commutative monoid is a strict monoidal category with one object. Quantum Groups algebras, coalgebras, bialgebras (in a general monoidal category) the category of representations of an algebra the monoidal category of representations of a bialgebra the braided monoidal category of representations of a quasitriangular bialgebra the symmetric monoidal category of representations of a triangular bialgebra the monoidal (resp. math.ucr.edu /home/baez/hda/dimensional_ladder.html   (2262 words)

 [No title] In the case wh* *ere A is a strict symmetric monoidal category, this lax functor is in fact a pseudofu* *nctor, that associated to A in [Th2] Appendix. In the case where A is a strict symmetric monoidal category the construction* * coincides with that given in [Th2], except that instead of considering a pseudofunctor on* * op as an op-lax functor and applying Street's first construction for op-lax functors, on* *e considers Theory and Applications of Categories, Vol. Null_=A of* * 5.2.3 is a lax morphism of lax symmetric monoidal categories. hopf.math.purdue.edu /Thomason/thomason_SymMon_equals_Spectra.txt   (7868 words)

 Closed monoidal category - Wikipedia, the free encyclopedia In mathematics, a monoidal closed category C is a closed category with an associative tensor product which is left adjoint to the internal Hom functor, that is a monoidal category equipped with a functor Equivalently, a monoidal closed category C is a category equipped, for every two objects A and B, with In particular, every cartesian closed category is a monoidal closed category. en.wikipedia.org /wiki/Closed_monoidal_category   (125 words)

 [No title] Categories of relations are deﬁned in the context of symmetric monoidal categories. Symmetries for monoidal categories is introduced as a further set of constraints on monoidal categories. The equation from the previous proposition is clearly equivalent to the Yang-Baxter equation in a symmetric monoidal category. ljm.ksu.ru /vol17/jal.xml   (1647 words)

 Open problems on model categories   (Site not responding. Last check: 2007-11-03) This clearly ought to be the homotopy category of a model structure on the category of chain complexes of comodules, but we have been unable to build such a model structure. Given a symmetric monoidal model category C, Schwede and Shipley have given conditions under which the category of monoids in C is again a model category (with underlying fibrations and weak equivalences). This would remove the loose end in my book on model categories, where I am unable to show that the homotopy category of a monoidal model category is a central algebra over the homotopy category of simplicial sets. claude.math.wesleyan.edu /~mhovey/problems/model.html   (1291 words)

 AMCA: Monoidal category theory applied to quasi-Hopf algebras by Peter Schauenburg   (Site not responding. Last check: 2007-11-03) A quasi-Hopf algebra is by definition an algebra H whose modules (on either or both sides) form a monoidal category. We discuss how in many instances a consistent use of monoidal category and actegory theory, particularly of ring and module theory within such categories, leads to more conceptual proofs. The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. at.yorku.ca /c/a/j/f/36.htm   (145 words)

 Citations: Iterated monoidal categories - Balteanu, Fiedorowicz, Schwanzl, Vogt (ResearchIndex) introduced a theory of n fold monoidal categories that mimics and models categorically the tautology that an (n 1) fold loop space is a loop space in the category of n fold loop spaces. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k fold monoidal categories and their higher dimensional counterparts. Noting the correspondence between loop spaces and monoidal categories, we iteratively defined the notion of n fold.... citeseer.ist.psu.edu /context/482996/0   (1682 words)

 [No title] Has anyone proved that if you take an "algebra" (actually monoid) object in a monoidal biclosed category that has equalizers and coequalizers, then the category of two-sided modules for that algebra is again a monoidal biclosed category. In response to Michael Barr's question: Theorem: If V is a closed braided monoidal category which is complete and cocomplete then the bicategory V-Mod of V-categories, V-modules (sometimes called V-bimodules, V-distributors or V-profunctors), and V-module morphisms is a monoidal bicategory (meaning the hom of a tricategory with one object). I didn't include the older fact well known to enriched category theorists: even without any symmetry or braiding, V-Mod is a bicategory in which all right extensions and right liftings exist (I don't like the word "biclosed"; I use "left closed", "right closed" and "closed" for both when dealing with a monoidal category). www.mta.ca /~cat-dist/catlist/1999/bimod-biclosed   (1236 words)

 Citebase - Frobenius algebras and ambidextrous adjunctions   (Site not responding. Last check: 2007-11-03) Authors: Lauda, Aaron D. In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. In particular, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. We then categorify this theorem by replacing the monoidal category M with a semistrict monoidal 2-category M, and replacing the 2-category D into which it embeds by a semistrict 3-category. www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0502550   (609 words)

 Journal of the American Mathematical Society Abstract: The stable homotopy category, much studied by algebraic topologists, is a closed symmetric monoidal category. For many years, however, there has been no well-behaved closed symmetric monoidal category of spectra whose homotopy category is the stable homotopy category. In this paper, we present such a category of spectra; the category of symmetric spectra. www.ams.org /jams/2000-13-01/S0894-0347-99-00320-3/home.html   (580 words)

 CTRC Seminar Abstract: Yang-Baxter systematically   (Site not responding. Last check: 2007-11-03) In a braided monoidal category, the Yang-Baxter hexagon commutes for each triple of objects. Using the 2-functor st: MCat -> MCat I constructed a braided monoidal category from a monoidal category with a Yang-Baxter system, with naturality of the braiding in each variable being proven via the two subdivisions of the Yang-Baxter hexagon respectively. This construction gives the object part of a biequivalence between the 2-category of braided monoidal categories whose only arrows are composites of R's and the 2-category of Yang-Baxter systems where Y consists of all objects of the monoidal category. www.math.mcgill.ca /rags/seminar/ab19990309ctsa.html   (199 words)

 week137 A braided monoidal category is simple algebraic gadget that captures a bit of the essence of 3-dimensionality in its rawest form. The center of a braided monoidal category is obviously a symmetric monoidal category. For example, the center of a monoidal category is a braided monoidal category. math.ucr.edu /home/baez/week137.html   (1569 words)

 ATCAT 1997-1998 In the second, an adjunction is established for monoidal bicategories, and then applied to the bicategory of V-categories and bimodules. Thus, for example, the category set of sets together with familiar relations is an equipment but so too is set together with partial functions and set together with spans. Another motivating example is cat, the (mere) category of categories, together with profunctors which specializes somewhat to ord, the category of ordered sets, together with ordered ideals. www.mscs.dal.ca /~pare/Sem97-98.html   (808 words)

 AMCA: Vassiliev Theory as Deformation Theory by David N. Yetter   (Site not responding. Last check: 2007-11-03) We describe the construction of cochain complexes associated to monoidal categories, monoidal functors, and braided monoidal categories, and theorems relating the cohomology of the category (functor) to infinitesimal deformations of its structure maps. When a symmetric monoidal category with duals is deformed to give rise to a ribbon category, the k Ëxtrinsic deformations" in which a braided monoidal category is deformed a subcategory of a larger category are shown to provide a setting for the consideration of universal Vassiliev invariants over general coefficient rings. at.yorku.ca /c/a/e/a/30.htm   (189 words)

 Free Monoidal Category Coherence property for a certain category means that we know that diagrams of a certain class commute. First theorem of this class was the proved by S. Mac Lane for monoidal categories (see [Mac Lane 71, pp.161-165]) His proof was based on normalisation of object expressions, but involved many other ingredients and was far from elementary. Extracting a proof of coherence for monoidal categories from a formal proof of normalization for monoids www.cs.chalmers.se /pub/users/ilya/FTP/FTP/FMC   (330 words)

 Citations: On closed categories of functors - Day (ResearchIndex) The basic idea is to use the monoidal structure to define the semantics of the multiplicative, or substructural, connectives (I, in BI s notation) in the standard way,.... Given a small (symmetric) monoidal category (C; I) there is a (symmetric) monoidal structure on the category [C ; Set] defined as follows: The unit I of the monoidal structure is C[ I ] Given functors E and F, the formula for the tensor product is written using co ends: E F)X =.... In this paper we have chosen to take the monoid rather than ternary relation semantics as primitive, partly because the latter is difficult to justify in intuitive terms. citeseer.ist.psu.edu /context/97128/0   (1662 words)

 Citebase - Combinatorial n-fold monoidal categories and n-fold operads   (Site not responding. Last check: 2007-11-03) After a review of the role of operads in loop space theory and higher categories we go over definitions of iterated monoidal categories and introduce a large family of simple examples. Then we generalize the definition of operad by defining n-fold operads and their algebras in an iterated monoidal category. Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy. www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0411561   (560 words)

 [No title]   (Site not responding. Last check: 2007-11-03) Enrichment as Categorical Delooping I: Enrichment Over Iterated Monoidal Categories The 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception is the case in which V is symmetric, which leads to V-Cat being symmetric as well. The main result is that for V a k-fold monoidal category, V-Cat becomes a (k-1)-fold monoidal 2-category in a canonical way. Higher Dimensional Enrichment Lyubashenko has described enriched 2-categories as categories enriched over V-Cat, the 2-category of categories enriched over a symmetric monoidal V. I have generalized this to the k-fold monoidal V. The symmetric case can easily be recovered. www.lehigh.edu /dmd1/public/www-data/sf620.txt   (291 words)

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