Wellpoweredness is a kind of smallness condition on a category.

Let $M$ be a class of monomorphisms. A category is said to be $M$ -wellpowered if for any object any class of parwise non-isomorphism $M$ -subobjects is a set. (By a $M$ -subobject of an object $A$ we understand a pair $(E,e)$ , where $\Map eEA$ is a morphism belonging to $M$ .) In other words, if we consider isomorphic objects as the same object, the class of all $M$ -subobjects is a set.

More precisely, for any $A$ there exists a set of $M$ -subobjects $(M_i,m_i)$ , $i\in I$ such that for any extremal subobject $(M,m)$ of the object $A$ there exists $i\in I$ and an isomorphism $\Map f{M_i}M$ such that $m_i=m\circ f$ .

If $M$ is the class of all regular monomorphisms, extremal monomorphisms, monomorphisms, we speak about regular wellpowered, extremally wellpowered, wellpowered category.

Dual notions: regular cowellpowered, extremally cowellpowered, cowellpowered category.

Bibliography

1

J. Adámek, H. Herrlich, and G. Strecker.
Abstract and Concrete Categories.
Wiley, New York, 1990.