Subobjects Functorial Construction - Category Framework

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# Subobjects Functorial Construction

AUTHORS:

> - Nicolas M. Thiery (2010): initial revision

class sage.categories.subobjects.SubobjectsCategory(category, *args)

  Bases: `RegressiveCovariantConstructionCategory`

  classmethod default_super_categories(category)

    Return 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 class `SubobjectsCategory`

    - `category` – a category \(Cat\)

    OUTPUT: a (join) category

    In practice, this returns `category.Subquotients()`, joined together with the result of the method `RegressiveCovariantConstructionCategory.default_super_categories()` (that is the join of `category` and `cat.Subobjects()` for each `cat` in the super categories of `category`).

    EXAMPLES:

    Consider `category=Groups()`, which has `cat=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

        sage: Groups().Subobjects().super_categories()
        [Category of groups, Category of subquotients of monoids, Category of subobjects of sets]

    Python

        >>> from sage.all import *
        >>> 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: 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

    [Python]

        >>> from sage.all import *
        >>> 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