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Difference between revisions of "Dual category"

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''to a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d0340901.png" />''
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''to a category $\mathcal{C}$''
  
The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d0340902.png" /> with the same objects as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d0340903.png" /> and with morphism sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d0340904.png" /> ( "reversal of arrows" ). Composition of two morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d0340905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d0340906.png" /> in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d0340907.png" /> is defined as composition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d0340908.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d0340909.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d03409010.png" />. The concepts and the statements in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d03409011.png" /> are replaced by dual concepts and statements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d03409012.png" />. Thus, the concept of an epimorphism is dual to the concept of a monomorphism; the concept of a projective object is dual to that of an injective object, that of the direct product to that of the direct sum, etc. A contravariant functor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d03409013.png" /> becomes covariant on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d03409014.png" />.
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The category $\mathcal{C}^o$ with the same objects as $\mathcal{C}$ and with morphism sets $\mathrm{Hom}_{\mathcal{C}^o}(A,B) = \mathrm{Hom}_{\mathcal{C}}(B,A)$ ( "reversal of arrows" ). Composition of two morphisms $u$ and $v$ in the category $\mathcal{C}^o$ is defined as composition of $v$ with $u$ in $\mathcal{C}$. The concepts and the statements in the category $\mathcal{C}$ are replaced by dual concepts and statements in $\mathcal{C}^o$. Thus, the concept of an [[epimorphism]] is dual to the concept of a [[monomorphism]]; the concept of a projective object is dual to that of an injective object, that of the direct product to that of the direct sum, etc. A contravariant [[functor]] on $\mathcal{C}$ becomes covariant on $\mathcal{C}^o$.
  
A dual category may sometimes have a direct realization; thus, the category of discrete Abelian groups is equivalent to the category dual to the category of compact Abelian groups ([[Pontryagin duality|Pontryagin duality]]), while the category of affine schemes is equivalent to the category dual to the category of commutative rings with a unit element.
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A dual category may sometimes have a direct realization; thus, the category of discrete Abelian groups is equivalent to the category dual to the category of compact Abelian groups ([[Pontryagin duality]]), while the category of affine schemes is equivalent to the category dual to the category of commutative rings with a unit element.
  
  
  
 
====Comments====
 
====Comments====
The dual category to a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d03409015.png" /> is also called the opposite category, and one also uses the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034090/d03409016.png" /> (see [[Category|Category]]).
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The dual category to a category $\mathcal{C}$ is also called the '''opposite category''', and one also uses the notation $\mathcal{C}^{\mathrm{op}}$ (see [[Category]]).
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Revision as of 21:40, 3 March 2018

to a category $\mathcal{C}$

The category $\mathcal{C}^o$ with the same objects as $\mathcal{C}$ and with morphism sets $\mathrm{Hom}_{\mathcal{C}^o}(A,B) = \mathrm{Hom}_{\mathcal{C}}(B,A)$ ( "reversal of arrows" ). Composition of two morphisms $u$ and $v$ in the category $\mathcal{C}^o$ is defined as composition of $v$ with $u$ in $\mathcal{C}$. The concepts and the statements in the category $\mathcal{C}$ are replaced by dual concepts and statements in $\mathcal{C}^o$. Thus, the concept of an epimorphism is dual to the concept of a monomorphism; the concept of a projective object is dual to that of an injective object, that of the direct product to that of the direct sum, etc. A contravariant functor on $\mathcal{C}$ becomes covariant on $\mathcal{C}^o$.

A dual category may sometimes have a direct realization; thus, the category of discrete Abelian groups is equivalent to the category dual to the category of compact Abelian groups (Pontryagin duality), while the category of affine schemes is equivalent to the category dual to the category of commutative rings with a unit element.


Comments

The dual category to a category $\mathcal{C}$ is also called the opposite category, and one also uses the notation $\mathcal{C}^{\mathrm{op}}$ (see Category).

How to Cite This Entry:
Dual category. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Dual_category&oldid=18831
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article