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Difference between revisions of "Bicategory(2)"

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A category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b0160601.png" /> in which subcategories of epimorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b0160602.png" /> and of monomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b0160603.png" /> have been distinguished such that the following conditions are met:
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A category $\mathfrak K$ in which subcategories of epimorphisms $\mathfrak E$ and of monomorphisms $\mathfrak M$ have been distinguished such that the following conditions are met:
  
1) all morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b0160604.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b0160605.png" /> are decomposable into a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b0160606.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b0160607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b0160608.png" />;
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1) all morphisms $\alpha$ in $\mathfrak K$ are decomposable into a product $\alpha=\nu\mu$, where $\nu\in\mathfrak E$, $\mu\in\mathfrak M$;
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b0160609.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b01606010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b01606011.png" />, then there exists an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b01606012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b01606013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b01606014.png" />;
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2) if $\nu\mu=\rho\tau$, where $\nu,\rho\in\mathfrak E$, $\mu,\tau\in\mathfrak M$, then there exists an isomorphism $\theta$ such that $\rho=\nu\theta$, and $\tau=\theta^{-1}\mu$;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b01606015.png" /> coincides with the class of isomorphisms in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b01606016.png" />.
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3) $\mathfrak E\cap\mathfrak M$ coincides with the class of isomorphisms in the category $\mathfrak R$.
  
The epimorphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b01606017.png" /> (the monomorphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b01606018.png" />) are called the permissible epimorphisms (monomorphisms) of the bicategory.
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The epimorphisms in $\mathfrak E$ (the monomorphisms in $\mathfrak M$) are called the permissible epimorphisms (monomorphisms) of the bicategory.
  
 
The concept of a bicategory axiomatizes the possibility of a decomposition of an arbitrary mapping into a product of a surjective and an injective mapping. The category of sets, the category of sets with a marked point and the category of groups are bicategories with a unique bicategorical structure. In the category of all topological spaces and in the category of all associative rings there are proper classes of different bicategorical structures.
 
The concept of a bicategory axiomatizes the possibility of a decomposition of an arbitrary mapping into a product of a surjective and an injective mapping. The category of sets, the category of sets with a marked point and the category of groups are bicategories with a unique bicategorical structure. In the category of all topological spaces and in the category of all associative rings there are proper classes of different bicategorical structures.
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====Comments====
 
====Comments====
In the literature there has been much confusion about the terms bicategory and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016060/b01606019.png" />-category. Usually, bicategory is understood to mean  "generalized 2-category" , and a bicategory as defined above is called, e.g.,  "bicategory in the sense of Isbell" .
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In the literature there has been much confusion about the terms bicategory and $2$-category. Usually, bicategory is understood to mean  "generalized 2-category", and a bicategory as defined above is called, e.g.,  "bicategory in the sense of Isbell".
  
 
In this Encyclopaedia the term bicategory is always used as defined above.
 
In this Encyclopaedia the term bicategory is always used as defined above.

Latest revision as of 10:09, 23 August 2014

A category $\mathfrak K$ in which subcategories of epimorphisms $\mathfrak E$ and of monomorphisms $\mathfrak M$ have been distinguished such that the following conditions are met:

1) all morphisms $\alpha$ in $\mathfrak K$ are decomposable into a product $\alpha=\nu\mu$, where $\nu\in\mathfrak E$, $\mu\in\mathfrak M$;

2) if $\nu\mu=\rho\tau$, where $\nu,\rho\in\mathfrak E$, $\mu,\tau\in\mathfrak M$, then there exists an isomorphism $\theta$ such that $\rho=\nu\theta$, and $\tau=\theta^{-1}\mu$;

3) $\mathfrak E\cap\mathfrak M$ coincides with the class of isomorphisms in the category $\mathfrak R$.

The epimorphisms in $\mathfrak E$ (the monomorphisms in $\mathfrak M$) are called the permissible epimorphisms (monomorphisms) of the bicategory.

The concept of a bicategory axiomatizes the possibility of a decomposition of an arbitrary mapping into a product of a surjective and an injective mapping. The category of sets, the category of sets with a marked point and the category of groups are bicategories with a unique bicategorical structure. In the category of all topological spaces and in the category of all associative rings there are proper classes of different bicategorical structures.

References

[1] M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian)


Comments

In the literature there has been much confusion about the terms bicategory and $2$-category. Usually, bicategory is understood to mean "generalized 2-category", and a bicategory as defined above is called, e.g., "bicategory in the sense of Isbell".

In this Encyclopaedia the term bicategory is always used as defined above.

How to Cite This Entry:
Bicategory(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bicategory(2)&oldid=33106
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article