Subobjects Functorial Construction¶
AUTHORS:
Nicolas M. Thiery (2010): initial revision
- class sage.categories.subobjects.SubobjectsCategory(category, *args)¶
Bases:
sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory
- classmethod default_super_categories(category)¶
Returns the default super categories of
category.Subobjects()
Mathematical meaning: if \(A\) is a subobject of \(B\) in the category \(C\), then \(A\) is also a subquotient of \(B\) in the category \(C\).
INPUT:
cls
– the classSubobjectsCategory
category
– a category \(Cat\)
OUTPUT: a (join) category
In practice, this returns
category.Subquotients()
, joined together with the result of the methodRegressiveCovariantConstructionCategory.default_super_categories()
(that is the join ofcategory
andcat.Subobjects()
for eachcat
in the super categories ofcategory
).EXAMPLES:
Consider
category=Groups()
, which hascat=Monoids()
as super category. Then, a subgroup of a group \(G\) is simultaneously a subquotient of \(G\), a group by itself, and a submonoid of \(G\):sage: Groups().Subobjects().super_categories() [Category of groups, Category of subquotients of monoids, Category of subobjects of sets]
Mind the last item above: there is indeed currently nothing implemented about submonoids.
This resulted from the following call:
sage: sage.categories.subobjects.SubobjectsCategory.default_super_categories(Groups()) Join of Category of groups and Category of subquotients of monoids and Category of subobjects of sets