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Topic: Symmetric monoidal category

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	Monoidal category - Wikipedia, the free encyclopedia
		In mathematics, a monoidal category (or tensor category) is a bicategory with one object.
		Monoidal functors are the functors between monoidal categories which preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product.
		There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid.
	en.wikipedia.org /wiki/Monoidal_category (627 words)

	PRO (category theory) - Wikipedia, the free encyclopedia
		In category theory, a PRO is a strict monoidal category whose objects are the natural integers and whose tensor product is given on objects by the addition on integers.
		An algebra of a PRO P in a category C is a strict monoidal functor from P to C.
		More precisely, what we mean here by "the algebras of Δ in C are the monoid objects in C" for example is that the category of algebras of P in C is equivalent to the category of monoids in C.
	en.wikipedia.org /wiki/PRO_(category_theory) (222 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.
		Suppose C is a cofibrantly generated symmetric monoidal model cat- egory such that the unit S is cofibrant and the domains of the generating cofib* *rations can be taken to be cofibrant.
	hopf.math.purdue.edu /Hovey/mon-mod.txt (8088 words)

	[No title]
		A reformulation of the definition of a symmetric spectrum is gi* ven in Section 2.2 where we recall the definition of monoids* and modules in a symme* tric monoidal* category.
		C, and the category of symmetric sequences of objects in C is the functor category C.
		The category of symmetric spectra with the class of stable equiv- alences, the class of stable cofibrations, and the class of stable fibrations i* s a model category*.
	hopf.math.purdue.edu /Hovey-Shipley-Smith/symm.txt (15448 words)

	Symmetric spectra
		The stable homotopy category of spectra, much studied by algebraic topologists, is a closed symmetric monoidal category (or a category with a tensor product).
		In this paper, we present such a category of spectra: the category of symmetric spectra.
		The category of symmetric spectra has a combinatorial construction which allows application to different settings, e.g.
	www.math.uic.edu /~bshipley/symmetric.html (97 words)

	Bosker Blog » Blog Archive » Monoidal centres
		The centre of a monoid M is defined to be the submonoid consisting of those elements that commute with everything in M.
		The idea is to construct a monoidal category that has two braidings, one of which is a symmetry and the other of which is not.
		Let B be the free symmetric strict monoidal category on one generator: so an object is just a natural number, and a morphism m → n is a bijection from the m-element set to the n-element-set.
	bosker.wordpress.com /2006/06/06/monoidal-centres (826 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)

	Wikipedia: Monoidal category
		A strict monoidal category is a category with a product operation × on objects that has properties analogous to those of the tensor product (it is not assumed to be a categorical product).
		The concept of braided monoidal category has been much studied from the 1980s onwards; it occurs in string theory applications, and is a more 'relaxed' theory defined by fewer such coherence conditions.
		Given a field (or commutative ring) K, the category K-Vect is a symmetric monoidal category with product ⊗ and identity K.
	www.factbook.org /wikipedia/en/m/mo/monoidal_category.html (202 words)

	[No title]
		Let the monoid be G. When G is an abelian group, the M and j seem to be determined by elements N_a,b depending on two objects of A. There is more meat in a V-functor.
		If V is the commutative monoid, then a V-enriched category is a set A plus two functions [-,-,-]: A x A x A ---> V [-]: A ---> V satisfying [a,c,d] + [a,b,c] = [a,b,d] + [b,c,d] [a,a,b] + [a] = 0 = [a,b,b] + [b] for all a, b, c, d.
		Subject: categories: Re: one-object closed categories Concerning categories enriched in monoidal categories with a single object: another example is given by cocycles.
	www.mta.ca /~cat-dist/catlist/1999/comm-monoid (1465 words)

	[No title]
		Symmetric spectra form a monoidal model category, unlike ordinary spectra, but we are unable to prove that the monoid axiom holds in general.
		Symmetric spectra and ordinary spectra are not always Quillen equivalent; we need the cyclic permutation map on K tensor K tensor K to be homotopic to the identity.
		Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993.
	www.lehigh.edu /~dmd1/h616 (960 words)

	When projective does not imply flat, and other homological anomalies (Site not responding. Last check: 2007-10-29)
		If $\cal M$ is both an abelian category and a symmetric monoidal closed category, then it is natural to ask whether projective objects in $\cal M$ are flat, and whether the tensor product of two projective objects is projective.
		However, the category $\cal M_G$ of Mackey functors for a compact Lie group $G$ is a category of this type in which projective objects need not be so well-behaved.
		Similar misbehavior occurs in two categories of global Mackey functors which are widely used in the study of classifying spaces of finite groups.
	www.emis.de /journals/TAC/volumes/1999/n9/5-09abs.html (394 words)

	Symmetric spectra, by Mark Hovey, Brooke Shipley, and Jeff Smith (Site not responding. Last check: 2007-10-29)
		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.
		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.
	www.math.uiuc.edu /K-theory/0251 (147 words)

	[No title]
		Orthogonal spectra are intermediate between symmetric spectra and S-modules: they are defined in the same diagrammatic fashion as symmetric spectra, but, as with S-modules, their stable weak equivalences are just the maps that induce isomorphisms on homotopy groups.
		We prove that the categories of orthogonal spectra and S-modules are Quillen equivalent and that this equivalence induces Quillen equivalences between the respective categories of ring spectra, of modules over a ring spectrum, and of commutative ring spectra.
		In the case of symmetric spectra, a stable weak equivalence f: X >--> Y is a map such that f^:[Y,E] >--> [X,E] is an isomorphism for all symmetric* Omega-spectra E, where the brackets refer to the levelwise homotopy category.
	www.lehigh.edu /~dmd1/h17 (2038 words)

	Paddy McCrudden - Representations of quantum categories
		with extra structure which ensures that the category of representations is a braided monoidal category with a twist.
		is the category of vector spaces, a quantum monoid is a quasi-triangular quasi-bialgebra with a twist and a quantum group is a quantum monoid with an antipode.
		and the category of representations of M is a braided monoidal object with a twist (in a certain symmetric monoidal bicategory) then M is a quantum monoid.
	www.cms.math.ca /CMS/Events/summer98/s98-abs/node116.html (145 words)

	Symmetric ring spectra and topological Hochschild homology, by Brooke Shipley (Site not responding. Last check: 2007-10-29)
		Symmetric spectra were introduced by Jeff Smith as a symmetric monoidal category of spectra.
		In this paper, a detection functor is defined which detects stable equivalences of symmetric spectra.
		One of the advantages of a symmetric monoidal category of spectra is that one can define topological Hochschild homology on ring spectra simply by mimicking the Hochschild complex from algebra.
	www.math.uiuc.edu /K-theory/0252 (134 words)

	Open problems on model categories
		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)

	[No title]
		The standard definition of the category of symmetric spectra in Cop in the case when C is the category* of sets is usually phrased in terms of the smash product of b* ased sim- plicial sets, which is a special case of the smash product in C introduced in * *Section 5.
		The formulation of the category of symmetric spectra that follows is therefore * a simple generalization of the category* of symmetric spectra of [9].
		Ring Categories, Bipermutative Categories, and the Operads * and E * This section is devoted to the proofs of Theorems 3.4 and 3.8.
	www.math.purdue.edu /research/atopology/Elmendorf-Mandell/RMA2.txt (17071 words)

	Lauda & Pfeiffer on Open-Closed Topological Strings \| The String Coffee Table
		In particular, one makes a (dg) category whose objects are labelled by the nonnegative integers (as usual), but whose morphisms are the space of chain complexes on the moduli space of Riemann surfaces with the relevant set of boundaries.
		A closed Toplogical CFT (TCFT) is then a differential graded symmetric monoidal functor from this category to the categories of chain complexes (ie, dg-Vect, I think).
		There’s a category, whose objects are the integers, and whose morphisms are the singular chains on the moduli spaces of Riemann surfaces with
	golem.ph.utexas.edu /string/archives/000680.html (1796 words)

	Citations: Axiomatizing the algebra of net computations and processes - Degano, Meseguer, Montanari (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		In this approach, a net N is analogous to a signature Sigma, and the symmetric monoidal category P(N) associated to N is analogous to the cartesian category L(Sigma) of terms and substitutions freely generated by Sigma.
		The class of symmetric spaces with source and target 0, containing no generators, is in one to one correspondence with the class of (sort preserving) bijective functions between the components of and those of 0.
		....T (of a strictly symmetric (strict) monoidal category 2 (the arrows of T (N) represent the commutative processes of the net N) and (2) at the logical level, to an adjunction induced by a suitable morphism between theories in PMEqtl.
	citeseer.ist.psu.edu /context/97694/0 (3188 words)

	First Order Linear Logic in Symmetric Monoidal Closed Categories
		That is, the terms of the logic are given by a linear tyoe theory LTT corresponding to the algebraic idea of a symmetric monoidal closed category.
		The study of logic in such categories is motivated by two examples which are derived as linear analogues of presheaf topoi and Heyting valued sets respectively.
		A monoidal factorisation system then gives rise to a structure preserving fibration between symmetric monoidal closed categories, which we term a linear doctrine.
	www.lfcs.inf.ed.ac.uk /reports/92/ECS-LFCS-92-194 (378 words)

	[TYPES] Announcement of paper on differential categories (Site not responding. Last check: 2007-10-29)
		Differential categories by R.F. Blute, J.R.B. Cockett and R.A.G. Seely This paper is available at http://www.math.mcgill.ca/rags/difftl/difftl.ps.gz Abstract Following work of Ehrhard and Regnier, we introduce the notion of a differential category: an additive symmetric monoidal category with a comonad (a "coalgebra modality") and a differential combinator, satisfying a number of coherence conditions.
		In such a category, one should imagine the morphisms in the base category as being linear maps and the morphisms in the coKleisli category as being smooth (infinitely differentiable).
		Although such categories do not necessarily arise from models of linear logic, one should think of this as replacing the usual dichotomy of linear vs. stable maps established for coherence spaces.
	lists.seas.upenn.edu /pipermail/types-list/2005/000844.html (335 words)

	Linear Logic and Typed Lambda-Calculus Workshop (Site not responding. Last check: 2007-10-29)
		To support these new definitions, we prove that every symmetric premonoidal category embeds fully into a closed symmetric premonoidal category, and we characterize closed symmetric premonoidal categories in terms of strong monads on the base cartesian category.
		We also show that this structure is category theoretically natural, being given by the category of algebras for a monad on a mild variant of Cat.
		Finally, we show that to give a symmetric premonoidal category is equivalent to giving a fibration with specified structure; and that extra structure on a premonoidal closed category, such as that used to model continuations, transfers elegantly to corresponding structure on the fibration.
	iml.univ-mrs.fr /~ehrhard/Abstracts/Power.html (323 words)

	Forschungsseminar
		The goal is to construct a symmetric monoidal model category of such algebras and to verify the conditions for a HA-context.
		The category of simplicial algebras should be related to the categories of dg-algebras and E_\infty-algebras.
		In this talk a symmetric monoidal model category of spectra (symmetric spectra) should be constructed and the properties of a HA-context should be verified.
	www.uni-math.gwdg.de /bunke/Forschungsseminar.html (506 words)

	Casual Category Theory - Spring 2000
		Now we will see the definition of symmetric monoidal category and monoidal closed category V, and the properties of the correspondent V-categories.
		A V-enriched category, in few words, is a category having hom-objects in V, where V is a monoidal category.
		Proofs of some well-known properties of presheave categories, namely: presheave categories are free colimit completions or any presheave functor is isomorphic to a colimit of representables.
	www.brics.dk /~varacca/CCT/cct-spring00.html (504 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		The structural framework we use in our results is the concept of multicategory, which is familiar to category theorists and computer scientists, but perhaps less so to topologists.
		The joke is that while permutative, and more generally symmetric monoidal categories merely form the objects of a multicategory, multicategories form the objects of a symmetric monoidal category.
		We conjecture that this symmetric monoidal category, which is also closed and bicomplete, is a model for the connective stable homotopy category.
	www.math.wayne.edu /research/seminars/topabstr/W04/03.09.04.html (145 words)

	AMCA: Vassiliev Theory as Deformation Theory by David N. Yetter (Site not responding. Last check: 2007-10-29)
		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)

	MyPHPblog (Site not responding. Last check: 2007-10-29)
		It explains what a category is and all that sort of stuff, and includes some of his ideas about the correct way to think about these things (which can often be different from the actual definitions).
		They consider monoidal categories in which there is an inversion functor, and monoidal categories in which all objects just happen to be invertible up to isomorphism (but without a specified inverse) and show an equivalence between the two resulting notions (more precisely, a 2-equivalence between the resulting 2-categories).
		I think I can extend this to theories involving symmetries without too much difficulty, to give something similar to the result that a symmetric monoidal category is equivalent to a symmetric strict monoidal category.
	assyrian.org.uk /blog/archive.php?blogid=1&eid=29 (491 words)

	Axiomatizing Petri Net Concatenable Processes - SASSONE (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		In this paper we give a fully equational description of the category of concatenable processes of N, thus yielding an axiomatic theory of the noninterleaving behaviour of Petri nets.
		identifies such structures as symmetric monoidal categories where objects are states, i.e.
		Recall that a symmetric strict monoidal category (see [12] for a thorough elementary introduction) is a category C together with a...
	citeseer.ist.psu.edu /sassone95axiomatizing.html (560 words)

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