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==Closed categories.==
 
==Closed categories.==
A monoidal category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012032.png" /> is said to be symmetric if it comes with isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012033.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012034.png" /> natural on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012035.png" /> such that the following diagrams all commute:
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A monoidal category $\mathcal{C}$ is said to be ''symmetri''c if it comes with isomorphisms $\gamma_{a.b} : a \otimes b \cong b \otimes a$, natural on $a,b \in \mathcal{C}$ such that the following diagrams all commute:
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$$
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\gamma_{a,b} \circ \gamma_{b,a} = \mathrm{id}\,;
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$$
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$$
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\rho_b = \lambda_b \circ \gamma_{b,e} : b\otimes e \cong b\,;
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$$
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$$
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\begin{array}{ccccc}
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a \otimes (b \otimes c) & \stackrel{\alpha}{\rightarrow} & (a \otimes b) \otimes c & \stackrel{\gamma}{\rightarrow} & c \otimes (a \otimes b) \\
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\downarrow_{\mathrm{id}\otimes\gamma} & & & & \downarrow_\alpha \\
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a \otimes (c \otimes b) & \stackrel{\alpha}{\rightarrow} & (a \otimes c) \otimes b & \stackrel{\gamma\otimes\mathrm{id}}{\rightarrow} & (c \otimes a) \otimes b
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\end{array}
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$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012037.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012038.png" />:
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A '''closed category''' $\mathcal{V}$ is a symmetric monoidal category in which each functor ${-}\otimes b: \mathcal{V} \rightarrow \mathcal{V}$ has a specified [[Adjoint functor|right-adjoint]] $({-})^b : \mathcal{V} \rightarrow \mathcal{V}$.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012039.png" /></td> </tr></table>
 
 
 
A closed category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012040.png" /> is a symmetric monoidal category in which each functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012041.png" /> has a specified right-adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012042.png" />.
 
  
 
Some examples of closed monoidal categories are:
 
Some examples of closed monoidal categories are:
  
E3) the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012043.png" /> of relations, whose objects are sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012044.png" /> and in which an arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012045.png" /> is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012046.png" />; the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012047.png" /> is the Cartesian product of the two sets, which is not the product in this category;
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E3) the category $\mathsf{Rel}$ of relations, whose objects are sets $a,b,c,\ldots$ and in which an arrow $\sigma:a\rightarrow b$ is a subset $\sigma \subseteq a \otimes b$, the object $a \otimes b$ being the [[Cartesian product]] of the two sets (which is not the product in this category);
  
E4) the subsets of a monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012048.png" /> (a poset, hence a category); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012050.png" /> are two subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012051.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012052.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012053.png" /> while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012054.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012055.png" />.
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E4) the subsets of a monoid $M$ (partially ordered by inclusion, hence a category); if $A$, $B$ are two subsets of $M$, then $A \otimes B$ is $\{ab : a \in A,\,b \in B\}$ while $C^A$ is $\{b \in M : ab\in C\ \text{for all}\ a \in A\}$.
  
 
====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Barr,   C. Wells,   "Category theory for computing science" , CRM (1990)</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Barr,C. Wells, "Category theory for computing science", CRM (1990) {{ZBL|0714.18001}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> S. MacLane,   "Categories for the working mathematician" , Springer (1971)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top"> S. MacLane, "Categories for the working mathematician", Springer (1971)</TD></TR>
 
</table>
 
</table>
  
{{TEX|part}}
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{{TEX|done}}

Latest revision as of 14:59, 6 April 2023

A category $\mathcal{C}$ is monoidal if it consists of the following data:

1) a category $\mathcal{C}$;

2) a bifunctor $\otimes : \mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$;

3) an object $e\in\mathcal{C}$; and

4) three natural isomorphisms $\alpha,\lambda,\rho$ such that

A1) $\alpha_{a,b,c} : a \otimes (b \otimes c) \cong (a \otimes b) \otimes c$ is natural for all $a,b,c \in \mathcal{C}$ and the diagram $$ \begin{array}{ccccc} a \otimes (b \otimes (c \otimes d)) & \stackrel{\alpha}{\rightarrow} & (a \otimes b) \otimes (c \otimes d) & \stackrel{\alpha}{\rightarrow} & ((a \otimes b) \otimes c) \otimes d \\ \downarrow\mathrm{id}\otimes\alpha & & & & \uparrow \alpha\otimes\mathrm{id} \\ a \otimes ((b \otimes c) \otimes d) & & \stackrel{\alpha}{\rightarrow} & & (a \otimes (b \otimes c)) \otimes d \end{array} $$ commutes for all $a,b,c,d \in \mathcal{C}$;

A2) $\lambda$ and $\rho$ are natural and $\lambda : e \otimes a \cong a$, $\rho : a \otimes e \cong a$ for all objects $a \in \mathcal{C}$ and the diagram $$ \begin{array}{ccc} a \otimes (e \otimes c) & \stackrel{\alpha}{\rightarrow} & (a \otimes e) \otimes c \\ \downarrow\mathrm{id}\otimes\lambda & & \downarrow\rho\otimes\mathrm{id} \\ a \otimes c & = & a \otimes c \end{array} $$ commutes for all $a.c \in \mathcal{C}$;

A3) $\lambda_e = \rho_e : e \otimes e \rightarrow e$.

These axioms imply that all such diagrams commute.

Some examples of monoidal categories are:

E1) any category with finite products is monoidal if one takes $a\otimes b$ to be the (chosen) product of the objects $a$ and $b$, with $e$ the terminal object; $\alpha,\lambda,\rho$ are the unique isomorphisms that commute with the appropriate projections;

E2) the usual "tensor products" give monoidal categories — whence the notation. Note that one cannot identify all isomorphic objects in $\mathcal{C}$.

Closed categories.

A monoidal category $\mathcal{C}$ is said to be symmetric if it comes with isomorphisms $\gamma_{a.b} : a \otimes b \cong b \otimes a$, natural on $a,b \in \mathcal{C}$ such that the following diagrams all commute: $$ \gamma_{a,b} \circ \gamma_{b,a} = \mathrm{id}\,; $$ $$ \rho_b = \lambda_b \circ \gamma_{b,e} : b\otimes e \cong b\,; $$ $$ \begin{array}{ccccc} a \otimes (b \otimes c) & \stackrel{\alpha}{\rightarrow} & (a \otimes b) \otimes c & \stackrel{\gamma}{\rightarrow} & c \otimes (a \otimes b) \\ \downarrow_{\mathrm{id}\otimes\gamma} & & & & \downarrow_\alpha \\ a \otimes (c \otimes b) & \stackrel{\alpha}{\rightarrow} & (a \otimes c) \otimes b & \stackrel{\gamma\otimes\mathrm{id}}{\rightarrow} & (c \otimes a) \otimes b \end{array} $$

A closed category $\mathcal{V}$ is a symmetric monoidal category in which each functor ${-}\otimes b: \mathcal{V} \rightarrow \mathcal{V}$ has a specified right-adjoint $({-})^b : \mathcal{V} \rightarrow \mathcal{V}$.

Some examples of closed monoidal categories are:

E3) the category $\mathsf{Rel}$ of relations, whose objects are sets $a,b,c,\ldots$ and in which an arrow $\sigma:a\rightarrow b$ is a subset $\sigma \subseteq a \otimes b$, the object $a \otimes b$ being the Cartesian product of the two sets (which is not the product in this category);

E4) the subsets of a monoid $M$ (partially ordered by inclusion, hence a category); if $A$, $B$ are two subsets of $M$, then $A \otimes B$ is $\{ab : a \in A,\,b \in B\}$ while $C^A$ is $\{b \in M : ab\in C\ \text{for all}\ a \in A\}$.

References

[a1] M. Barr,C. Wells, "Category theory for computing science", CRM (1990) Zbl 0714.18001
[a2] S. MacLane, "Categories for the working mathematician", Springer (1971)
How to Cite This Entry:
Closed monoidal category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closed_monoidal_category&oldid=42575
This article was adapted from an original article by Michel Eytan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article