Isomorphic Objects Functorial Construction - Category Framework

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# 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)

    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 `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

        sage: Groups().IsomorphicObjects().super_categories()
        [Category of groups,
         Category of subquotients of monoids,
         Category of quotients of semigroups,
         Category of isomorphic objects of sets]

    Python

        >>> 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: 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

    [Python]

        >>> 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