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A [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c1301201.png" /> is monoidal if it consists of the following data:
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A [[category]] $\mathcal{C}$ is monoidal if it consists of the following data:
  
1) a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c1301202.png" />;
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1) a category $\mathcal{C}$;
  
2) a [[Bifunctor|bifunctor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c1301203.png" />;
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2) a [[bifunctor]] $\otimes : \mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$;
  
3) an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c1301204.png" />; and
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3) an object $e\in\mathcal{C}$; and
  
4) three natural isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c1301205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c1301206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c1301207.png" /> such that
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4) three natural isomorphisms $\alpha,\lambda,\rho$ such that
  
A1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c1301208.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c1301209.png" /> is natural for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012010.png" /> and the diagram
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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
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$$
 +
\begin{array}{ccccc}
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a \otimes (b \otimes (c \otimes d)) & \stackrel{\alpha}{\rightarrow} & (a \otimes b) \otimes (c \otimes d) &  \stackrel{\alpha}{\rightarrow} & ((a \otimes b) \otimes c) \otimes d \\
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          \downarrow\mathrm{id}\otimes\alpha & & & & \uparrow \alpha\otimes\mathrm{id} \\
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a \otimes ((b \otimes c) \otimes d) &  & \stackrel{\alpha}{\rightarrow} &  & (a \otimes (b \otimes c)) \otimes d
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\end{array}
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$$
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commutes for all $a,b,c,d \in \mathcal{C}$;
  
<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/c13012011.png" /></td> </tr></table>
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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
 +
$$
 +
\begin{array}{ccc}
 +
a \otimes (e \otimes c) & \stackrel{\alpha}{\rightarrow} & (a \otimes e) \otimes c \\
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\downarrow\mathrm{id}\otimes\lambda & & \downarrow\rho\otimes\mathrm{id} \\
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a \otimes c & = & a \otimes c
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\end{array}
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$$
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commutes for all $a.c \in \mathcal{C}$;
  
commutes for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012012.png" />;
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A3) $\lambda_e = \rho_e : e \otimes e \rightarrow e$.
 
 
A2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012014.png" /> are natural and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012015.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012017.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012018.png" /> for all objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012019.png" /> and the diagram
 
 
 
<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/c13012020.png" /></td> </tr></table>
 
 
 
commutes for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012021.png" />;
 
 
 
A3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012022.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012023.png" />.
 
  
 
These axioms imply that all such diagrams commute.
 
These axioms imply that all such diagrams commute.
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Some examples of monoidal categories are:
 
Some examples of monoidal categories are:
  
E1) any category with finite products is monoidal if one takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012024.png" /> to be the (chosen) product of the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012026.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012027.png" /> the terminal object; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012030.png" /> are the unique isomorphisms that commute with the appropriate projections;
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130120/c13012031.png" />.
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E2) the usual  "tensor products"  give monoidal categories — whence the notation. Note that one cannot identify all isomorphic objects in $\mathcal{C}$.
  
 
==Closed categories.==
 
==Closed categories.==
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Barr,  C. Wells,  "Category theory for computing science" , CRM  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Barr,  C. Wells,  "Category theory for computing science" , CRM  (1990)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|part}}

Revision as of 19:46, 30 November 2017

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 is said to be symmetric if it comes with isomorphisms : natural on such that the following diagrams all commute:

, : :

A closed category is a symmetric monoidal category in which each functor has a specified right-adjoint .

Some examples of closed monoidal categories are:

E3) the category of relations, whose objects are sets and in which an arrow is a subset ; the object is the Cartesian product of the two sets, which is not the product in this category;

E4) the subsets of a monoid (a poset, hence a category); if , are two subsets of , then is while is .

References

[a1] M. Barr, C. Wells, "Category theory for computing science" , CRM (1990)
[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=42386
This article was adapted from an original article by Michel Eytan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article