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| In [[mathematics]], a '''monoidal category''' (or '''tensor category''') is a [[category (mathematics)|category]] '''C''' equipped with a [[bifunctor]]
| | Are you always having problems with your PC? Are you constantly searching for techniques to grow PC performance? Next this might be the article you're seeking. Here we will discuss a few of the most asked concerns when it comes to having we PC serve you well; how can I make my computer faster for free? How to create my computer run quicker?<br><br>Before actually ordering the software it's best to check on the companies that make the software. If you will find details on the kind of reputation every firm has, maybe the risk of malicious programs is reduced. Software from reputed firms have aided me, and several other users, to make my PC run quicker.. If the product description does not look good to you, does not include details about the software, refuses to include the scan functions, you need to go for another 1 which ensures you're paying for what we need.<br><br>Naturally, the upcoming logical step is to receive these false entries cleaned out. Fortunately, this really is not a difficult task. It is the second thing you should do when you observed the computer has lost speed. The first will be to make certain there are no viruses or severe spyware present.<br><br>Paid registry cleaners on the different hand, I have found, are frequently cheap. They provide normal, free changes or at least cheap updates. This follows considering the software maker requirements to ensure their product is best inside staying ahead of its competitors.<br><br>There are many [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities] s found on the market now. How do you know that one to choose? Well, when you purchased the car we did certain research on it, didn't you? We didn't simply go out and buy the initially red convertible we saw. The same thing functions with registry products. On any look engine, sort inside "registry cleaner reviews" plus they may receive posted for we to read about.<br><br>The software finds these problems and takes care of them in the shortest time possible. Your difficult drive might furthermore result issues sometimes, specifically if you have one that is almost maxed. When a begin your machine, the are numerous booting processes involved and having an almost full storage space does not enable a bit. You will always have a slow PC as there are thus many items in the difficult disk being processed simultaneously. The best way to solve this issue is to upgrade. This allows the PC some time to breath and functioning quicker instantly.<br><br>Across the best of the scan results display page you see the tabs... Registry, Junk Files, Privacy, Bad Active X, Performance, etc. Every of these tabs might show we the results of which area. The Junk Files are mostly temporary files such as web data, pictures, web pages... And they are merely taking up storage.<br><br>A registry cleaner is a program which cleans the registry. The Windows registry constantly gets flooded with junk information, information which has not been removed from uninstalled programs, erroneous file association and alternative computer-misplaced entries. These neat small program software tools are quite prevalent nowadays and you can find many wise ones found on the Internet. The good ones give you choice to keep, clean, update, backup, and scan the System Registry. When it finds supposedly unwanted elements inside it, the registry cleaner lists them plus recommends the user to delete or repair these orphaned entries and corrupt keys. |
| :⊗ : '''C''' × '''C''' → '''C'''
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| which is [[associative]], [[up to]] a [[natural isomorphism]], and an object ''I'' which is both a [[left identity|left]] and [[right identity]] for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain [[coherence condition]]s which ensure that all the relevant diagrams commute.
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| In a monoidal category, analogs of usual [[monoid]]s from [[abstract algebra]] can be defined using the same commutative diagrams. In fact, usual monoids are exactly the [[monoid object]]s in the monoidal category of sets with Cartesian product.
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| The ordinary [[tensor product]] makes [[vector space]]s, [[abelian group]]s, [[module (mathematics)|''R''-modules]], or [[algebra (ring theory)|''R''-algebras]] into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples.
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| In [[category theory]], monoidal categories can be used to define the concept of a [[monoid object]] and an associated action on the objects of the category. They are also used in the definition of an [[enriched category]].
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| Monoidal categories have numerous applications outside of category theory proper. They are used to define models for the multiplicative fragment of intuitionistic [[linear logic]]. They also form the mathematical foundation for the [[topological order]] in condensed matter. [[Braided monoidal categories]] have applications in [[quantum field theory]] and [[string theory]].
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| ==Formal definition==
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| A '''monoidal category''' is a category <math>\mathbf C</math> equipped with
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| *a [[bifunctor]] <math>\otimes \colon \mathbf C\times\mathbf C\to\mathbf C</math> called the ''[[tensor product]]'' or ''monoidal product'',
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| *an object <math>I</math> called the ''unit object'' or ''identity object'',
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| *three [[natural isomorphism]]s subject to certain [[coherence condition]]s expressing the fact that the tensor operation
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| **is associative: there is a natural isomorphism <math>\alpha</math>, called ''associator'', with components <math>\alpha_{A,B,C} \colon (A\otimes B)\otimes C \cong A\otimes(B\otimes C)</math>,
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| **has <math>I</math> as left and right identity: there are two natural isomorphisms <math>\lambda</math> and <math>\rho</math>, respectively called ''left'' and ''right unitor'', with components <math>\lambda_A \colon I\otimes A\cong A</math> and <math>\rho_A \colon A\otimes I\cong A</math>.
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| :
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| The coherence conditions for these natural transformations are:
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| * for all <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> in <math>\mathbf C</math>, the pentagon [[diagram (category theory)|diagram]]
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| :[[Image:monoidal-category-pentagon.png]]
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| [[Commutative diagram|commutes]];
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| * for all <math>A</math> and <math>B</math> in <math>\mathbf C</math>, the triangle [[diagram (category theory)|diagram]]
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| :[[Image:monoidal-category-triangle.png]]
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| [[Commutative diagram|commutes]];
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| It follows from these three conditions that ''a large class'' of such diagrams (i.e. diagrams whose morphisms are built using <math>\alpha</math>, <math>\lambda</math>, <math>\rho</math>, identities and tensor product) commute: this is [[Saunders Mac Lane|Mac Lane's]] "[[coherence theorem]]". It is sometimes inaccurately stated that ''all'' such diagrams commute.
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| A '''strict monoidal category''' is one for which the natural isomorphisms α, λ and ρ are identities. Every monoidal category is monoidally [[equivalence of categories|equivalent]] to a strict monoidal category.
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| ==Examples==
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| *Any category with finite [[product (category theory)|product]]s is monoidal with the product as the monoidal product and the [[terminal object]] as the unit. Such a category is sometimes called a '''cartesian monoidal category'''. For example:
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| **'''Set''', the category of sets with the Cartesian product, one-element sets serving as the unit.
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| *Any category with finite [[coproduct]]s is monoidal with the coproduct as the monoidal product and the [[initial object]] as the unit.
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| *'''''R''-Mod''', the [[category of modules]] over a [[commutative ring]] ''R'', is a monoidal category with the [[tensor product of modules]] ⊗<sub>''R''</sub> serving as the monoidal product and the ring ''R'' (thought of as a module over itself) serving as the unit. As special cases one has:
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| **'''''K''-Vect''', the [[category of vector spaces]] over a [[field (mathematics)|field]] ''K'', with the one-dimensional vector space ''K'' serving as the unit.
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| **'''Ab''', the [[category of abelian groups]], with the group of [[integer]]s '''Z''' serving as the unit.
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| *For any commutative ring ''R'', the category of [[R-algebra|''R''-algebras]] is monoidal with the [[tensor product of algebras]] as the product and ''R'' as the unit.
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| *The [[category of pointed spaces]] is monoidal with the [[smash product]] serving as the product and the pointed [[0-sphere]] (a two-point discrete space) serving as the unit.
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| *The category of all [[endofunctor]]s on a category '''C''' is a strict monoidal category with the composition of functors as the product and the identity functor as the unit.
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| *Just like for any category '''E''', the [[Subcategory#Embeddings|full subcategory]] spanned by any given object is a monoid, it is the case that for any [[2-category]] '''E''', and any object '''C'''∈Ob('''E'''), the full 2-subcategory of '''E''' spanned by {'''C'''} is a monoidal category. In the case '''E'''='''Cat''', we get the [[endofunctor]]s example above.
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| * [[Semilattice|Bounded-above meet semilattices]] are strict [[symmetric monoidal category|symmetric monoidal categories]]: the product is meet and the identity is the top element.
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| == Free strict monoidal category ==
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| For every category '''C''', the [[free category|free]] strict monoidal category Σ('''C''') can be constructed as follows:
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| * its objects are lists (finite sequences) ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> of objects of '''C''';
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| * there are arrows between two objects ''A''<sub>1</sub>, ..., ''A''<sub>''m''</sub> and ''B''<sub>1</sub>, ..., ''B''<sub>''n''</sub> only if ''m'' = ''n'', and then the arrows are lists (finite sequences) of arrows ''f''<sub>1</sub>: ''A''<sub>1</sub> → ''B''<sub>1</sub>, ..., ''f''<sub>''n''</sub>: ''A''<sub>''n''</sub> → ''B''<sub>''n''</sub> of '''C''';
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| * the tensor product of two objects ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> and ''B''<sub>1</sub>, ..., ''B''<sub>''m''</sub> is the concatenation ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub>, ''B''<sub>1</sub>, ..., ''B''<sub>''m''</sub> of the two lists, and, similarly, the tensor product of two morphisms is given by the concatenation of lists.
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| This operation Σ mapping category '''C''' to Σ('''C''') can be extended to a strict 2-monad on '''Cat'''.
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| == See also ==
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| * Many monoidal categories have additional structure such as [[braided monoidal category|braiding]], [[symmetric monoidal category|symmetry]] or [[closed monoidal category|closure]]: the references describe this in detail.
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| * [[Monoidal functor]]s are the functors between monoidal categories which preserve the tensor product and [[monoidal natural transformation]]s are the natural transformations, between those functors, which are "compatible" with the tensor product.
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| * There is a general notion of [[monoid object]] in a monoidal category, which generalizes the ordinary notion of [[monoid]]. In particular, a strict monoidal category can be seen as a monoid object in the category of categories '''Cat''' (equipped with the monoidal structure induced by the cartesian product).
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| * A monoidal category can also be seen as the category '''B'''(□, □) of a [[bicategory]] '''B''' with only one object, denoted □.
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| * [[Rigid category|Rigid categories]] are monoidal categories in which duals with nice properties exist.
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| * [[Autonomous category|Autonomous categories]] (or [[compact closed category|compact closed categories]]) are monoidal categories in which inverses exist; they abstract the idea of '''FdVect''', finite-dimensional vector spaces.
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| * [[Dagger symmetric monoidal category|Dagger symmetric monoidal categories]], equipped with an extra dagger functor, abstracting the idea of '''FdHilb''', finite-dimensional Hilbert spaces. These include the [[dagger compact category|dagger compact categories]].
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| * A category '''C''' [[Enriched category|enriched]] in a monoidal category '''M''' replaces the notion of a set of morphisms between pairs of objects in '''C''' with the notion of an '''M'''-object of morphisms between every two objects in '''C'''.
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| * [[Tannakian category|Tannakian categories]] are monoidal categories enriched over a field which are very similar to representation categories of linear algebraic groups.
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| * [[Spherical category]]
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| == References ==
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| * [[André Joyal|Joyal, André]]; [[Ross Street|Street, Ross]] (1993). "Braided Tensor Categories". ''Advances in Mathematics'' ''102'', 20–78.
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| * [[Max Kelly|Kelly, G. Max]] (1964). "On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc." ''Journal of Algebra'' ''1'', 397–402
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| * {{cite book | last = Kelly | first = G. Max | url = http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf | title = Basic Concepts of Enriched Category Theory | series = London Mathematical Society Lecture Note Series No. 64 | publisher = Cambridge University Press | year = 1982}}
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| * [[Saunders Mac Lane|Mac Lane, Saunders]] (1963). "Natural Associativity and Commutativity". ''Rice University Studies'' ''49'', 28–46.
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| * Mac Lane, Saunders (1998), ''[[Categories for the Working Mathematician]]'' (2nd ed.). New York: Springer-Verlag.
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| *{{nlab|id=monoidal+category|title=Monoidal category}}
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| {{Portal|Category theory}}
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| [[Category:Monoidal categories| ]]
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Are you always having problems with your PC? Are you constantly searching for techniques to grow PC performance? Next this might be the article you're seeking. Here we will discuss a few of the most asked concerns when it comes to having we PC serve you well; how can I make my computer faster for free? How to create my computer run quicker?
Before actually ordering the software it's best to check on the companies that make the software. If you will find details on the kind of reputation every firm has, maybe the risk of malicious programs is reduced. Software from reputed firms have aided me, and several other users, to make my PC run quicker.. If the product description does not look good to you, does not include details about the software, refuses to include the scan functions, you need to go for another 1 which ensures you're paying for what we need.
Naturally, the upcoming logical step is to receive these false entries cleaned out. Fortunately, this really is not a difficult task. It is the second thing you should do when you observed the computer has lost speed. The first will be to make certain there are no viruses or severe spyware present.
Paid registry cleaners on the different hand, I have found, are frequently cheap. They provide normal, free changes or at least cheap updates. This follows considering the software maker requirements to ensure their product is best inside staying ahead of its competitors.
There are many tuneup utilities s found on the market now. How do you know that one to choose? Well, when you purchased the car we did certain research on it, didn't you? We didn't simply go out and buy the initially red convertible we saw. The same thing functions with registry products. On any look engine, sort inside "registry cleaner reviews" plus they may receive posted for we to read about.
The software finds these problems and takes care of them in the shortest time possible. Your difficult drive might furthermore result issues sometimes, specifically if you have one that is almost maxed. When a begin your machine, the are numerous booting processes involved and having an almost full storage space does not enable a bit. You will always have a slow PC as there are thus many items in the difficult disk being processed simultaneously. The best way to solve this issue is to upgrade. This allows the PC some time to breath and functioning quicker instantly.
Across the best of the scan results display page you see the tabs... Registry, Junk Files, Privacy, Bad Active X, Performance, etc. Every of these tabs might show we the results of which area. The Junk Files are mostly temporary files such as web data, pictures, web pages... And they are merely taking up storage.
A registry cleaner is a program which cleans the registry. The Windows registry constantly gets flooded with junk information, information which has not been removed from uninstalled programs, erroneous file association and alternative computer-misplaced entries. These neat small program software tools are quite prevalent nowadays and you can find many wise ones found on the Internet. The good ones give you choice to keep, clean, update, backup, and scan the System Registry. When it finds supposedly unwanted elements inside it, the registry cleaner lists them plus recommends the user to delete or repair these orphaned entries and corrupt keys.