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| In [[mathematics]], especially in [[category theory]], a
| | == is to resist Luo Feng opponents attack. == |
| '''closed monoidal category''' is a context where we can take tensor products of objects and also form 'mapping objects'. A classic example is the [[category of sets]], '''Set''', where the tensor product of sets <math>A</math> and <math>B</math> is the usual [[cartesian product]] <math>A \times B</math>, and the mapping object <math>B^A</math> is the set of [[function (mathematics)|function]]s from <math>A</math> to <math>B</math>. Another example is the category '''FdVect''', consisting of [[finite-dimensional]] [[vector space]]s and [[linear map]]s. Here the tensor product is the usual [[tensor product]] of vector spaces, and the mapping object is the vector space of linear maps from one vector space to another.
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| The 'mapping object' referred to above is also called the '[[internal Hom]]'. The [[internal language]] of closed symmetric monoidal categories is the [[linear type system]].
| | Kill humans,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_27.htm サングラス オークリー 偏光].<br><br>a melee!<br><br>......<br><br>promise Arashiyama's death, indeed Luo Feng secretly calculating cause.<br><br>and Luo Feng Qiu addicted to killing a family of that name immortal, spiritual reading teacher,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_46.htm オークリー サングラス 登山], manipulation of the nine handle a sword! Luo Feng resist the manipulation of animal magic sword attack nine handle when the two desperately fighting the case, that dragon sword addicted family immortal God and beast claws hit,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_39.htm オークリー サングラス レンズ交換], knocked crooked sword out, burst Jian Qi natural out on crooked,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_52.htm オークリー 激安 サングラス], phantom beast God under the direction of a little attention at the time to stop, just crooked off Jian Qi Feng Luo calculated in accordance with a direction on good projection go through.<br><br>Phantom Beast God arrested a claw,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_22.htm オークリー サングラス 種類], claws resist when what angle ......<br><br>even then demanding the military system,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_19.htm オークリー サングラス 手入れ], there is no way that it is the Luo Feng killer, after all,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_17.htm 登山 サングラス オークリー], is to resist Luo Feng opponents attack.<br><br>so the ......<br><br>promise Arashiyama dead! He thought he was in the army is absolutely safe,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_44.htm 度付きサングラス オークリー], he did not know ...... provisions also have vulnerabilities, |
| | | 相关的主题文章： |
| ==Definition==
| | <ul> |
| A '''closed monoidal category''' is a [[monoidal category]] <math>\mathcal{C}</math> such that for every object <math>B</math> the [[functor]] given by right tensoring with <math>B</math>
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| :<math>A\mapsto A\otimes B</math>
| | <li>[http://bbs.sbwxlm.com/home.php?mod=space&uid=38813 http://bbs.sbwxlm.com/home.php?mod=space&uid=38813]</li> |
| has a [[right adjoint]], written
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| :<math>A\mapsto (B \Rightarrow A).</math>
| | <li>[http://www.yhgjs.com.cn/plus/feedback.php?aid=15 http://www.yhgjs.com.cn/plus/feedback.php?aid=15]</li> |
| This means that there exists a bijection, called '[[currying]]', between the [[Hom-set]]s
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| :<math>\text{Hom}_\mathcal{C}(A\otimes B, C)\cong\text{Hom}_\mathcal{C}(A,B\Rightarrow C)</math>
| | <li>[http://easycooker.com.cn/forum.php?mod=viewthread&tid=1264126 http://easycooker.com.cn/forum.php?mod=viewthread&tid=1264126]</li> |
| that is natural in both ''A'' and ''C''. In a different, but common notation, one would say that the functor
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| :<math>-\otimes B:\mathcal{C}\to\mathcal{C}</math>
| | </ul> |
| has a right adjoint
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| :<math>[B, -]:\mathcal{C}\to\mathcal{C}</math>
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| Equivalently, a closed monoidal category <math>\mathcal{C}</math> is a category equipped, for every two objects ''A'' and ''B'', with
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| * an object <math>A\Rightarrow B</math>,
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| * a morphism <math>\mathrm{eval}_{A,B} : (A\Rightarrow B) \otimes A \to B</math>,
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| satisfying the following universal property: for every morphism
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| :<math>f : X\otimes A\to B</math>
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| there exists a unique morphism
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| :<math>h : X \to A\Rightarrow B</math>
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| such that
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| :<math>f = \mathrm{eval}_{A,B}\circ(h \otimes \mathrm{id}_A).</math>
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| It can be shown that this construction defines a functor <math>\Rightarrow : C^{op} \otimes C \to C</math>. This functor is called the [[internal Hom functor]], and the object <math>A \Rightarrow B</math> is called the '''internal Hom''' of <math>A</math> and <math>B</math>. Many other notations are in common use for the internal Hom. When the tensor product on ''C'' is the cartesian product, the usual notation is <math>B^A</math> and this object is called the [[exponential object]].
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| ==Biclosed and symmetric categories==
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| Strictly speaking, we have defined a '''right closed''' monoidal category, since we required that ''right'' tensoring with any object <math>A</math> has a right adjoint. In a '''left closed''' monoidal category, we instead demand that the functor of left tensoring with any object <math>A</math>
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| :<math>B\mapsto A\otimes B</math>
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| have a right adjoint
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| :<math>B\mapsto(B\Leftarrow A)</math>
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| A '''biclosed''' monoidal category is a monoidal category that is both left and right closed.
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| A [[symmetric monoidal category]] is left closed if and only if it is right closed. Thus we may safely speak of a 'symmetric monoidal closed category' without specifying whether it is left or right closed. In fact, the same is true more generally for [[braided monoidal category|braided monoidal categories]]: since the braiding makes <math>A \otimes B</math> naturally isomorphic to <math>B \otimes A</math>, the distinction between tensoring on the left and tensoring on the right becomes immaterial, so every right closed braided monoidal category becomes left closed in a canonical way, and vice versa.
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| We have described closed monoidal categories as monoidal categories with an extra property. One can equivalently define a closed monoidal category to be a [[closed category]] with an extra property. Namely, we can demand the existence of a [[monoidal category|tensor product]] that is [[left adjoint]] to the [[internal Hom functor]].
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| In this approach, closed monoidal categories are also called '''monoidal closed categories'''.
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| ==Examples==
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| * The monoidal category '''Set''' of sets and functions, with cartesian product as the tensor product, is a closed monoidal category. Here, the internal hom <math>A \Rightarrow B</math> is the set of functions from <math>A</math> to <math>B</math>. In computer science, the bijection between tensoring and the internal hom is known as [[currying]], particularly in [[functional programming languages]]. Indeed, some languages, such as [[Haskell (programming language)|Haskell]] and [[Caml]], explicitly use an arrow notation to denote a function. This example is a [[Cartesian closed category]].
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| * More generally, every Cartesian closed category is a symmetric monoidal closed category, when the monoidal structure is the cartesian product structure. Here the internal hom <math>A \Rightarrow B</math> is usually written as the [[exponential object]] <math>B^A</math>.
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| * The monoidal category '''FdVect''' of finite-dimensional vector spaces and linear maps, with its usual tensor product, is a closed monoidal category. Here <math>A \Rightarrow B</math> is the vector space of linear maps from <math>A</math> to <math>B</math>. This example is a [[compact closed category]].
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| * More generally, every [[compact closed category]] is a symmetric monoidal closed category, in which the internal Hom functor <math>A\Rightarrow B</math> is given by <math>B\otimes A^*</math>.
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| ==References==
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| *[[Max Kelly|Kelly,G.M.]] [http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf "Basic Concepts of Enriched Category Theory"], London Mathematical Society Lecture Note Series No.64 (C.U.P., 1982)
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| *Paul-André Melliès, Categorical Semantics of Linear Logic, 2007
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| *{{nlab|id=closed+monoidal+category|title=Closed monoidal category}}
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| [[Category:Monoidal categories]]
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| [[Category:Closed categories]]
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is to resist Luo Feng opponents attack.
Kill humans,サングラス オークリー 偏光.
a melee!
......
promise Arashiyama's death, indeed Luo Feng secretly calculating cause.
and Luo Feng Qiu addicted to killing a family of that name immortal, spiritual reading teacher,オークリー サングラス 登山, manipulation of the nine handle a sword! Luo Feng resist the manipulation of animal magic sword attack nine handle when the two desperately fighting the case, that dragon sword addicted family immortal God and beast claws hit,オークリー サングラス レンズ交換, knocked crooked sword out, burst Jian Qi natural out on crooked,オークリー 激安 サングラス, phantom beast God under the direction of a little attention at the time to stop, just crooked off Jian Qi Feng Luo calculated in accordance with a direction on good projection go through.
Phantom Beast God arrested a claw,オークリー サングラス 種類, claws resist when what angle ......
even then demanding the military system,オークリー サングラス 手入れ, there is no way that it is the Luo Feng killer, after all,登山 サングラス オークリー, is to resist Luo Feng opponents attack.
so the ......
promise Arashiyama dead! He thought he was in the army is absolutely safe,度付きサングラス オークリー, he did not know ...... provisions also have vulnerabilities,
相关的主题文章：