Bicategory(2)

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.