## You are here

Homewellpowered category

## Primary tabs

# wellpowered category

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 $e:E\to A$ 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 $f:M_{i}\to M$ 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.

# References

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

## Mathematics Subject Classification

18A05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections