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| 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.
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| == Symmetric compact closed category ==
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| 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>.
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| 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
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| :<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>
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| and
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| :<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>
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| where <math>\lambda,\rho</math> are the introduction of the unit on the left and right, respectively.
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| 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>:
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| :<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>
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| and
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| :<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>
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| == Definition ==
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| 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:
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| :<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>
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| :<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>
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| and
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| :<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>
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| :<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>
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| That is, in the general case, a compact closed category is both left and right-[[rigid category|rigid]], and [[biclosed monoidal category|biclosed]].
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| 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'''.
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| == Properties ==
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| 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]].
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| Compact closed categories are precisely the [[symmetric monoidal category|symmetric]] [[autonomous category|autonomous categories]]. They are also [[*-autonomous category|*-autonomous]].
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| 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
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| :<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>
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| which can be shown to be a proper trace. It helps to draw this diagrammatically:
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| <math>A\xrightarrow{\cong}A\otimes I\xrightarrow{\eta}A\otimes (C\otimes C^*)\xrightarrow{\cong}(A\otimes C)\otimes C^*
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| \xrightarrow{f}(B\otimes C)\otimes C^*\xrightarrow{\cong}B\otimes(C\otimes C^*)\xrightarrow{\epsilon}B\otimes I\xrightarrow{\cong}B.</math>
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| == Examples ==
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| 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>.
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| The category of finite-dimensional representations of any group is also compact closed.
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| The category '''Vect''', with ''all'' vector spaces as objects and linear maps as morphisms, is ''not'' compact closed.
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| ===Simplex category===
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| 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.
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| The left and right adjoints are the min and max operators; specifically, for a monotonic function ''f'' one has the right adjoint
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| :<math>f^r(n) = \sup \{m \in \mathbb{N} | f(m)\le n\}</math>
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| and the left adjoint
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| :<math>f^l(n) = \inf \{m \in \mathbb{N} | n \le f(m)\}</math>
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| The left and right units and counits are:
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| :<math>\mbox {id} \le f \circ f^l\qquad\mbox{(left unit)}</math>
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| :<math>\mbox {id} \le f^r \circ f\qquad\mbox{(right unit)}</math>
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| :<math>f^l \circ f \le \mbox {id}\qquad\mbox{(left counit)}</math>
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| :<math>f \circ f^r \le \mbox {id}\qquad\mbox{(right counit)}</math>
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| One of the yanking conditions is then
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| :<math>f = f \circ \mbox {id} \le f \circ (f^r \circ f)
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| = (f \circ f^r) \circ f \le \mbox {id} \circ f = f.</math>
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| 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>.
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| === Dagger compact category ===
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| A [[dagger symmetric monoidal category]] which is compact closed is a [[dagger compact category]].
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| ==Rigid category==
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| 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.
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| == References ==
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| {{cite journal
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| | last = Kelly
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| | first = G.M.
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| | authorlink = Max Kelly
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| | coauthors = Laplaza, M.L.
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| | title = Coherence for compact closed categories
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| | journal = Journal of Pure and Applied Algebra
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| | volume = 19
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| | pages = 193–213
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| | year = 1980
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| | doi = 10.1016/0022-4049(80)90101-2}}
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| [[Category:Monoidal categories]]
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| [[Category:Closed categories]]
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