W-shingling - Revision history

en>BD2412: minor fixes, mostly disambig links using AWB

2014-12-10T04:16:33Z

minor fixes, mostly disambig links using AWB

← Older revision		Revision as of 06:16, 10 December 2014
Line 1:		Line 1:
	In [[category theory]], compact closed categories are a general context for treating [[dual object]]s. The idea of a dual object generalizes the more familiar concept of the [[dual space\|dual]] of a finite-dimensional [[vector space]]. So, the motivating example of a compact closed category is ~~'''FdVect''', the category with finite dimensional [[vector spaces]] as objects and [[linear maps]] as morphisms.~~		Marvella is what you can call her but it's not [http://independentmumbaiescort.com/anamika/blogs/post/4358 over the counter std test] most female name out there. One of the extremely very best things in the globe for me is to do aerobics and now I'm trying to make cash with it. Hiring is her working day occupation now but she's always wanted her own company. For a while she's been in South Dakota.

	~~== Symmetric compact closed category ==~~

	A [[symmetric monoidal category]] <math>(\mathbf{C},\otimes,I)</math> is '''compact closed''' if every object <math>A \in C</math> has a [[dual object]]. If this holds, the dual object is unique up to canonical isomorphism, and it ~~is denoted <math>A^*</math>.~~

	~~In a bit more detail, an object <math>A^</math> is called the~~ '''[[dual object\|dual]]''' of '''A''' if it is equipped with two morphisms called the '''[[unit (category theory)\|unit]]''' <math>\eta_A:I\to A^\otimes A</math> and the '''counit''' <math>\varepsilon_A:A\otimes A^*\to I</math>, satisfying the equations
	~~:<math>\lambda_A\circ(\varepsilon_A\otimes A)\circ\alpha_{A,A^*,A}^{-1}\circ(A\otimes\eta_A)\circ\rho_A^{-1}=\mathrm{id}_A</math>~~
	~~and~~
	~~:<math>\rho_{A^}\circ(A^\otimes\varepsilon_A)\circ\alpha_{A^,A,A^}\circ(\eta_A\otimes A^)\circ\lambda_{A^}^{-1}=\mathrm{id}_{A^*},</math>~~
	~~where <math>\lambda,\rho</math> are the introduction of the unit on the left and right, respectively.~~

	For clarity, we rewrite the above compositions diagramatically. In order for <math>(\mathbf{C},\otimes,I)</math> to be compact closed, we need the following composites to be <math>\mathrm{id}_A</math>:
	~~:<math> A\xrightarrow{\cong} A\otimes I\xrightarrow{A\otimes\eta}A\otimes (A^\otimes A)\to (A\otimes A^)\otimes A\xrightarrow{\epsilon\otimes A} I\otimes A\xrightarrow{\cong} A</math>~~
	~~and~~
	~~:<math> A^\to I\otimes A^\xrightarrow{\eta}(A^\otimes A)\otimes A^\to A^\otimes (A\otimes A^)\xrightarrow{\epsilon} A^\otimes I\to A^</math>~~

	~~== Definition ==~~
	~~More generally, suppose <math>(\mathbf{C},\otimes,I)</math> is a monoidal category,~~ not ~~necessarily symmetric, such as in the case of a~~ [[pregroup grammar]]. The above notion of having a dual <math>A^*</math> for each object ''A'' is replaced by that of having both a left and a right adjoint, <math>A^l</math> and <math>A^r</math>, with a corresponding left unit <math>\eta^l_A:I\to A\otimes A^l</math>, right unit <math>\eta^r_A:~~I\to A^r\otimes A</math>, left counit <math>\varepsilon^l_A:A^l\otimes A\to I<~~/~~math>, and right counit <math>\varepsilon^r_A:A\otimes A^r\to I<~~/~~math>~~. ~~These must satisfy the four '''yanking conditions''', each of which are identities:~~

	~~:<math> A\to A\otimes I\xrightarrow{\eta^r}A\otimes (A^r\otimes A)\to (A\otimes A^r)\otimes A\xrightarrow{\epsilon^r} I\otimes A\to A<~~/~~math>~~

	~~:<math> A\to I\otimes A\xrightarrow{\eta^l}(A\otimes A^l)\otimes A\to A\otimes (A^l \otimes A)\xrightarrow{\epsilon^l} A\otimes I\to A<~~/~~math>~~
	~~and~~
	~~:<math> A^r\to I\otimes A^r\xrightarrow{\eta^r}(A^r\otimes A)\otimes A^r\to A^r\otimes (A\otimes A^r)\xrightarrow{\epsilon^r} A^r\otimes I\to A^r<~~/~~math>~~

	~~:<math> A^l\to A^l\otimes I\xrightarrow{\eta^l}A^l\otimes (A\otimes A^l)\to (A^l\otimes A)\otimes A^l \xrightarrow{\epsilon^l} I\otimes A^l\to A^l<~~/~~math>~~

	~~That is, in~~ the ~~general case, a compact closed category is both left and right-[[rigid category\|rigid]], and [[biclosed monoidal category\|biclosed]~~].

	~~Non-symmetric compact closed categories find applications in [[linguistics]], in the area~~ of ~~[[categorial grammar]]s and specifically in [[pregroup grammar]]s, where~~ the ~~distinct left and right adjoints are required to capture word-order~~ in ~~sentences. In this context, compact closed monoidal categories are called (Lambek) '''[[pregroup]]s'''.~~

	~~== Properties ==~~
	~~Compact closed categories are a special case of [[monoidal closed category\|monoidal closed categories]], which in turn are a special case of [[closed category\|closed categories]].~~

	~~Compact closed categories are precisely~~ the ~~[[symmetric monoidal category\|symmetric]] [[autonomous category\|autonomous categories]]. They are also [[-autonomous category\|-autonomous]].~~

	~~Every compact closed category '''C''' admits a [[traced monoidal category\|trace]]. Namely,~~ for ~~every morphism <math>f:A\otimes C\to B\otimes C</math>, one can define~~
	~~:<math>\mathrm{Tr_{A,B}^C}(f)=\rho_B\circ(id_B\otimes\varepsilon_C)\circ\alpha_{B,C,C^}\circ(f\otimes C^)\circ\alpha_{A,C,C^}^{-1}\circ(id_A\otimes\eta_{C^})\circ\rho_A^{-1}:A\to B</math>~~
	~~which can be shown to be a proper trace. It helps to draw this diagrammatically:~~
	<math>A\xrightarrow{\cong}A\otimes I\xrightarrow{\eta}A\otimes (C\otimes C^)\xrightarrow{\cong}(A\otimes C)\otimes C^
	~~\xrightarrow{f}(B\otimes C)\otimes C^\xrightarrow{\cong}B\otimes(C\otimes C^)\xrightarrow{\epsilon}B\otimes I\xrightarrow{\cong}B.</math>~~

	~~== Examples ==~~

	The canonical example is the category '''FdVect''' with finite dimensional [[vector spaces]] as objects and [[linear maps]] as morphisms. Here <math>A^*</math> is the usual dual of the vector space <math>A</math>.

	~~The category of finite-dimensional representations of any group is also compact closed.~~

	~~The category '''Vect''', with ''all'' vector spaces as objects and linear maps as morphisms, is ''not'' compact closed.~~

	~~===Simplex category===~~
	~~The [[simplex category]] provides an example of a (non-symmetric) compact closed category. The simplex category~~ is just the category of order-preserving ([[Monotone function\|monotone]]) maps of [[Finite ordinal number\|finite ordinals]] (viewed as totally ordered sets); its morphisms are order-preserving ([[monotone]]) maps of integers. We make it into a monoidal category by moving to ~~the [[Comma category\|arrow category]], so the objects are morphisms of the original category,~~ and ~~the morphisms are commuting squares. Then the tensor product of the arrow category is the original composition operator.~~
	~~The left and right adjoints are the min and max operators; specifically, for a monotonic function ''f~~'~~' one has the right adjoint~~

	~~:<math>f^r(n) = \sup \{m \in \mathbb{N} \| f(m)\le n\}</math>~~

	~~and the left adjoint~~
	~~:<math>f^l(n) = \inf \{~~m ~~\in \mathbb{N} \| n \le f(m)\}</math>~~

	~~The left and right units and counits are:~~
	~~:<math>\mbox {id} \le f \circ f^l\qquad\mbox{(left unit)}</math>~~
	~~:<math>\mbox {id} \le f^r \circ f\qquad\mbox{(right unit)}</math>~~
	~~:<math>f^l \circ f \le \mbox {id}\qquad\mbox{(left counit)}</math>~~
	~~:<math>f \circ f^r \le \mbox {id}\qquad\mbox{(right counit)}</math>~~

	~~One of the yanking conditions is then~~
	~~:<math>f = f \circ \mbox {id} \le f \circ (f^r \circ f)~~
	~~= (f \circ f^r) \circ f \le \mbox {id} \circ f = f.</math>~~
	~~The others follow similarly. The correspondence can be made clearer by writing the arrow <math>\~~to~~</math> instead of <math>\le</math>, and using <math>\otimes</math> for function composition <math>\circ</math>~~.

	~~=== Dagger compact category ===~~
	~~A [[dagger symmetric monoidal category]] which~~ is ~~compact closed is a [[dagger compact category]].~~

	~~==Rigid category==~~

	~~A [[monoidal category]] that is not symmetric,~~ but ~~otherwise obeys the duality axioms above, is known as a [[rigid category]]~~. ~~A monoidal category where every object has a left (resp. right) dual is also sometimes called~~ a '~~''left''' (resp~~. right) '''autonomous''' category. A monoidal category where every object has both a left and a right dual is sometimes called an [[autonomous category]]. An autonomous category that is also [[symmetric monoidal category\|symmetric]] is then a compact closed category.

	~~== References ==~~
	~~{{cite journal~~
	~~\| last = Kelly~~
	~~\| first = G.M.~~
	~~\| authorlink = Max Kelly~~
	~~\| coauthors = Laplaza, M.L.~~
	~~\| title = Coherence for compact closed categories~~
	~~\| journal = Journal of Pure and Applied Algebra~~
	~~\| volume = 19~~
	~~\| pages = 193–213~~
	~~\| year = 1980~~
	~~\| doi = 10.1016/0022-4049(80)90101-2}}~~

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

en>Wafitzge at 21:00, 1 July 2013

2013-07-01T21:00:26Z

← Older revision		Revision as of 23:00, 1 July 2013
Line 1:		Line 1:
	The ~~title~~ of the ~~author~~ is ~~Numbers~~. ~~Supervising~~ is ~~my occupation~~. ~~To do aerobics~~ is a ~~thing that~~ I'~~m totally addicted~~ to. For a ~~whilst she~~'s ~~been~~ in ~~South Dakota~~.<br><br>Here is ~~my blog post~~: ~~[http~~://~~premium~~.~~asslikethat~~.~~com~~/~~user~~/~~CSalerno premium~~.~~asslikethat~~.~~com~~]		In [[category theory]], compact closed categories are a general context for treating [[dual object]]s. The idea of a dual object generalizes the more familiar concept of the [[dual space\|dual]] of a finite-dimensional [[vector space]]. So, the motivating example of a compact closed category is '''FdVect''', the category with finite dimensional [[vector spaces]] as objects and [[linear maps]] as morphisms.

			== Symmetric compact closed category ==

			A [[symmetric monoidal category]] <math>(\mathbf{C},\otimes,I)</math> is '''compact closed''' if every object <math>A \in C</math> has a [[dual object]]. If this holds, the dual object is unique up to canonical isomorphism, and it is denoted <math>A^*</math>.

			In a bit more detail, an object <math>A^</math> is called the '''[[dual object\|dual]]''' of '''A''' if it is equipped with two morphisms called the '''[[unit (category theory)\|unit]]''' <math>\eta_A:I\to A^\otimes A</math> and the '''counit''' <math>\varepsilon_A:A\otimes A^*\to I</math>, satisfying the equations
			:<math>\lambda_A\circ(\varepsilon_A\otimes A)\circ\alpha_{A,A^*,A}^{-1}\circ(A\otimes\eta_A)\circ\rho_A^{-1}=\mathrm{id}_A</math>
			and
			:<math>\rho_{A^}\circ(A^\otimes\varepsilon_A)\circ\alpha_{A^,A,A^}\circ(\eta_A\otimes A^)\circ\lambda_{A^}^{-1}=\mathrm{id}_{A^*},</math>
			where <math>\lambda,\rho</math> are the introduction of the unit on the left and right, respectively.

			For clarity, we rewrite the above compositions diagramatically. In order for <math>(\mathbf{C},\otimes,I)</math> to be compact closed, we need the following composites to be <math>\mathrm{id}_A</math>:
			:<math> A\xrightarrow{\cong} A\otimes I\xrightarrow{A\otimes\eta}A\otimes (A^\otimes A)\to (A\otimes A^)\otimes A\xrightarrow{\epsilon\otimes A} I\otimes A\xrightarrow{\cong} A</math>
			and
			:<math> A^\to I\otimes A^\xrightarrow{\eta}(A^\otimes A)\otimes A^\to A^\otimes (A\otimes A^)\xrightarrow{\epsilon} A^\otimes I\to A^</math>

			== Definition ==
			More generally, suppose <math>(\mathbf{C},\otimes,I)</math> is a monoidal category, not necessarily symmetric, such as in the case of a [[pregroup grammar]]. The above notion of having a dual <math>A^*</math> for each object ''A'' is replaced by that of having both a left and a right adjoint, <math>A^l</math> and <math>A^r</math>, with a corresponding left unit <math>\eta^l_A:I\to A\otimes A^l</math>, right unit <math>\eta^r_A:I\to A^r\otimes A</math>, left counit <math>\varepsilon^l_A:A^l\otimes A\to I</math>, and right counit <math>\varepsilon^r_A:A\otimes A^r\to I</math>. These must satisfy the four '''yanking conditions''', each of which are identities:

			:<math> A\to A\otimes I\xrightarrow{\eta^r}A\otimes (A^r\otimes A)\to (A\otimes A^r)\otimes A\xrightarrow{\epsilon^r} I\otimes A\to A</math>

			:<math> A\to I\otimes A\xrightarrow{\eta^l}(A\otimes A^l)\otimes A\to A\otimes (A^l \otimes A)\xrightarrow{\epsilon^l} A\otimes I\to A</math>
			and
			:<math> A^r\to I\otimes A^r\xrightarrow{\eta^r}(A^r\otimes A)\otimes A^r\to A^r\otimes (A\otimes A^r)\xrightarrow{\epsilon^r} A^r\otimes I\to A^r</math>

			:<math> A^l\to A^l\otimes I\xrightarrow{\eta^l}A^l\otimes (A\otimes A^l)\to (A^l\otimes A)\otimes A^l \xrightarrow{\epsilon^l} I\otimes A^l\to A^l</math>

			That is, in the general case, a compact closed category is both left and right-[[rigid category\|rigid]], and [[biclosed monoidal category\|biclosed]].

			Non-symmetric compact closed categories find applications in [[linguistics]], in the area of [[categorial grammar]]s and specifically in [[pregroup grammar]]s, where the distinct left and right adjoints are required to capture word-order in sentences. In this context, compact closed monoidal categories are called (Lambek) '''[[pregroup]]s'''.

			== Properties ==
			Compact closed categories are a special case of [[monoidal closed category\|monoidal closed categories]], which in turn are a special case of [[closed category\|closed categories]].

			Compact closed categories are precisely the [[symmetric monoidal category\|symmetric]] [[autonomous category\|autonomous categories]]. They are also [[-autonomous category\|-autonomous]].

			Every compact closed category '''C''' admits a [[traced monoidal category\|trace]]. Namely, for every morphism <math>f:A\otimes C\to B\otimes C</math>, one can define
			:<math>\mathrm{Tr_{A,B}^C}(f)=\rho_B\circ(id_B\otimes\varepsilon_C)\circ\alpha_{B,C,C^}\circ(f\otimes C^)\circ\alpha_{A,C,C^}^{-1}\circ(id_A\otimes\eta_{C^})\circ\rho_A^{-1}:A\to B</math>
			which can be shown to be a proper trace. It helps to draw this diagrammatically:
			<math>A\xrightarrow{\cong}A\otimes I\xrightarrow{\eta}A\otimes (C\otimes C^)\xrightarrow{\cong}(A\otimes C)\otimes C^
			\xrightarrow{f}(B\otimes C)\otimes C^\xrightarrow{\cong}B\otimes(C\otimes C^)\xrightarrow{\epsilon}B\otimes I\xrightarrow{\cong}B.</math>

			== Examples ==

			The canonical example is the category '''FdVect''' with finite dimensional [[vector spaces]] as objects and [[linear maps]] as morphisms. Here <math>A^*</math> is the usual dual of the vector space <math>A</math>.

			The category of finite-dimensional representations of any group is also compact closed.

			The category '''Vect''', with ''all'' vector spaces as objects and linear maps as morphisms, is ''not'' compact closed.

			===Simplex category===
			The [[simplex category]] provides an example of a (non-symmetric) compact closed category. The simplex category is just the category of order-preserving ([[Monotone function\|monotone]]) maps of [[Finite ordinal number\|finite ordinals]] (viewed as totally ordered sets); its morphisms are order-preserving ([[monotone]]) maps of integers. We make it into a monoidal category by moving to the [[Comma category\|arrow category]], so the objects are morphisms of the original category, and the morphisms are commuting squares. Then the tensor product of the arrow category is the original composition operator.
			The left and right adjoints are the min and max operators; specifically, for a monotonic function ''f'' one has the right adjoint

			:<math>f^r(n) = \sup \{m \in \mathbb{N} \| f(m)\le n\}</math>

			and the left adjoint
			:<math>f^l(n) = \inf \{m \in \mathbb{N} \| n \le f(m)\}</math>

			The left and right units and counits are:
			:<math>\mbox {id} \le f \circ f^l\qquad\mbox{(left unit)}</math>
			:<math>\mbox {id} \le f^r \circ f\qquad\mbox{(right unit)}</math>
			:<math>f^l \circ f \le \mbox {id}\qquad\mbox{(left counit)}</math>
			:<math>f \circ f^r \le \mbox {id}\qquad\mbox{(right counit)}</math>

			One of the yanking conditions is then
			:<math>f = f \circ \mbox {id} \le f \circ (f^r \circ f)
			= (f \circ f^r) \circ f \le \mbox {id} \circ f = f.</math>
			The others follow similarly. The correspondence can be made clearer by writing the arrow <math>\to</math> instead of <math>\le</math>, and using <math>\otimes</math> for function composition <math>\circ</math>.

			=== Dagger compact category ===
			A [[dagger symmetric monoidal category]] which is compact closed is a [[dagger compact category]].

			==Rigid category==

			A [[monoidal category]] that is not symmetric, but otherwise obeys the duality axioms above, is known as a [[rigid category]]. A monoidal category where every object has a left (resp. right) dual is also sometimes called a '''left''' (resp. right) '''autonomous''' category. A monoidal category where every object has both a left and a right dual is sometimes called an [[autonomous category]]. An autonomous category that is also [[symmetric monoidal category\|symmetric]] is then a compact closed category.

			== References ==
			{{cite journal
			\| last = Kelly
			\| first = G.M.
			\| authorlink = Max Kelly
			\| coauthors = Laplaza, M.L.
			\| title = Coherence for compact closed categories
			\| journal = Journal of Pure and Applied Algebra
			\| volume = 19
			\| pages = 193–213
			\| year = 1980
			\| doi = 10.1016/0022-4049(80)90101-2}}

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

en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Clustering

2012-06-25T03:42:26Z

Repairing links to disambiguation pages - You can help! - Clustering

New page

The title of the author is Numbers. Supervising is my occupation. To do aerobics is a thing that I'm totally addicted to. For a whilst she's been in South Dakota.<br><br>Here is my blog post: [http://premium.asslikethat.com/user/CSalerno premium.asslikethat.com]