Isomorphic Objects Functorial Construction#
AUTHORS:
Nicolas M. Thiery (2010): initial revision
- class sage.categories.isomorphic_objects.IsomorphicObjectsCategory(category, *args)#
Bases:
RegressiveCovariantConstructionCategory
- classmethod default_super_categories(category)#
Returns the default super categories of
category.IsomorphicObjects()
Mathematical meaning: if \(A\) is the image of \(B\) by an isomorphism in the category \(C\), then \(A\) is both a subobject of \(B\) and a quotient of \(B\) in the category \(C\).
INPUT:
cls
– the classIsomorphicObjectsCategory
category
– a category \(Cat\)
OUTPUT: a (join) category
In practice, this returns
category.Subobjects()
andcategory.Quotients()
, joined together with the result of the methodRegressiveCovariantConstructionCategory.default_super_categories()
(that is the join ofcategory
andcat.IsomorphicObjects()
for eachcat
in the super categories ofcategory
).EXAMPLES:
Consider
category=Groups()
, which hascat=Monoids()
as super category. Then, the image of a group \(G'\) by a group isomorphism is simultaneously a subgroup of \(G\), a subquotient of \(G\), a group by itself, and the image of \(G\) by a monoid isomorphism:sage: Groups().IsomorphicObjects().super_categories() [Category of groups, Category of subquotients of monoids, Category of quotients of semigroups, Category of isomorphic objects of sets]
Mind the last item above: there is indeed currently nothing implemented about isomorphic objects of monoids.
This resulted from the following call:
sage: sage.categories.isomorphic_objects.IsomorphicObjectsCategory.default_super_categories(Groups()) Join of Category of groups and Category of subquotients of monoids and Category of quotients of semigroups and Category of isomorphic objects of sets