# Full subcategory

From Encyclopedia of Mathematics

A subcategory of a category such that for any objects and from one has the equality

Thus, a full subcategory is completely defined by the class of its objects. Conversely, any subclass of the class of objects of a category uniquely defines a full subcategory, for which it serves as the class of objects. This subcategory contains only those morphisms for which the sources and targets belong to that subclass. In particular, the full subcategory corresponding to a single object consists of the set .

Many important classes of subcategories are full subcategories (reflective and co-reflective subcategories, varieties, etc.).

#### Comments

#### References

[a1] | B. Mitchell, "Theory of categories" , Acad. Press (1965) |

**How to Cite This Entry:**

Full subcategory.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Full_subcategory&oldid=18127

This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article