Coverage for local/lib/python2.7/site-packages/sage/categories/isomorphic_objects.py : 17%

""" 

Isomorphic Objects Functorial Construction 

AUTHORS: 

- Nicolas M. Thiery (2010): initial revision 

""" 

#***************************************************************************** 

# Copyright (C) 2009 Nicolas M. Thiery <nthiery at users.sf.net> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.categories.category import Category 

from sage.categories.covariant_functorial_construction import RegressiveCovariantConstructionCategory 

class IsomorphicObjectsCategory(RegressiveCovariantConstructionCategory): 

_functor_category = "IsomorphicObjects" 

@classmethod 

def default_super_categories(cls, 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 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 

:meth:`RegressiveCovariantConstructionCategory.default_super_categories() <sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory.default_super_categories>` 

(that is the join of ``category`` and 

``cat.IsomorphicObjects()`` for each ``cat`` in 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] 

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 

""" 

return Category.join([category.Subobjects(), category.Quotients(), 

super(IsomorphicObjectsCategory, cls).default_super_categories(category)])