Isomorphic Objects Functorial Construction¶
AUTHORS:
Nicolas M. Thiery (2010): initial revision
- class sage.categories.isomorphic_objects.IsomorphicObjectsCategory(category, *args)[source]¶
- Bases: - RegressiveCovariantConstructionCategory- classmethod default_super_categories(category)[source]¶
- Return 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 class- IsomorphicObjectsCategory
- category– a category \(Cat\)
 - OUTPUT: a (join) category - In practice, this returns - category.Subobjects()and- category.Quotients(), joined together with the result of the method- RegressiveCovariantConstructionCategory.default_super_categories()(that is the join of- categoryand- cat.IsomorphicObjects()for each- catin the super categories of- category).- EXAMPLES: - Consider - category=Groups(), which has- cat=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] - >>> from sage.all import * >>> 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 - >>> from sage.all import * >>> 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