en>Rjwilmsi: /* Biology */Journal cites, added 1 PMID, added 1 PMC, added 1 issue number using AWB (10482)

2014-11-02T19:10:13Z

Biology: Journal cites, added 1 PMID, added 1 PMC, added 1 issue number using AWB (10482)

en>Jonesey95: Fixing deprecated parameter using AutoEd

2014-01-28T22:32:19Z

Fixing deprecated parameter using AutoEd

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	~~Nice~~ to ~~meet you~~, ~~I am Marvella Shryock~~. ~~Managing individuals~~ is ~~his profession~~. ~~What I love performing~~ is to ~~gather badges but I~~'~~ve been using~~ on ~~new issues recently~~. ~~Years ago he moved~~ to ~~North Dakota~~ and ~~his family enjoys it~~.<br><br>~~my site;~~ [http://www.~~jphuahua~~.~~com~~/~~xueyouhui~~/~~football~~/~~?action~~-~~viewnews-itemid-238780 std testing at home~~]		In [[mathematics]], especially in [[category theory]], a
			'''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.

			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]].

			==Definition==
			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>
			:<math>A\mapsto A\otimes B</math>
			has a [[right adjoint]], written
			:<math>A\mapsto (B \Rightarrow A).</math>
			This means that there exists a bijection, called '[[currying]]', between the [[Hom-set]]s
			:<math>\text{Hom}_\mathcal{C}(A\otimes B, C)\cong\text{Hom}_\mathcal{C}(A,B\Rightarrow C)</math>
			that is natural in both ''A'' and ''C''. In a different, but common notation, one would say that the functor
			:<math>-\otimes B:\mathcal{C}\to\mathcal{C}</math>
			has a right adjoint
			:<math>[B, -]:\mathcal{C}\to\mathcal{C}</math>

			Equivalently, a closed monoidal category <math>\mathcal{C}</math> is a category equipped, for every two objects ''A'' and ''B'', with
			* an object <math>A\Rightarrow B</math>,
			* a morphism <math>\mathrm{eval}_{A,B} : (A\Rightarrow B) \otimes A \to B</math>,
			satisfying the following universal property: for every morphism
			:<math>f : X\otimes A\to B</math>
			there exists a unique morphism
			:<math>h : X \to A\Rightarrow B</math>
			such that
			:<math>f = \mathrm{eval}_{A,B}\circ(h \otimes \mathrm{id}_A).</math>

			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]].

			==Biclosed and symmetric categories==
			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>
			:<math>B\mapsto A\otimes B</math>
			have a right adjoint
			:<math>B\mapsto(B\Leftarrow A)</math>

			A '''biclosed''' monoidal category is a monoidal category that is both left and right closed.

			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.

			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]].
			In this approach, closed monoidal categories are also called '''monoidal closed categories'''.

			==Examples==

			* 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]].

			* 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>.

			* 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]].

			* 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>.

			==References==
			*[[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)
			*Paul-André Melliès, Categorical Semantics of Linear Logic, 2007
			*{{nlab\|id=closed+monoidal+category\|title=Closed monoidal category}}

			[[Category:Monoidal categories]]
			[[Category:Closed categories]]

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-09T17:53:38Z

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Nice to meet you, I am Marvella Shryock. Managing individuals is his profession. What I love performing is to gather badges but I've been using on new issues recently. Years ago he moved to North Dakota and his family enjoys it.<br><br>my site; [http://www.jphuahua.com/xueyouhui/football/?action-viewnews-itemid-238780 std testing at home]

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en>Jonesey95: Fixing deprecated parameter using AutoEd

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