View source for Bicategory(2)

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:

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" />;

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" />;

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" />.

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.

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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Sh. Tsalenko,   E.G. Shul'geifer,   "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR></table>



====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" .

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