Objects¶
- class sage.categories.objects.Objects[source]¶
- Bases: - Category_singleton- The category of all objects the basic category - EXAMPLES: - sage: Objects() Category of objects sage: Objects().super_categories() [] - >>> from sage.all import * >>> Objects() Category of objects >>> Objects().super_categories() [] - class SubcategoryMethods[source]¶
- Bases: - object- Endsets()[source]¶
- Return the category of endsets between objects of this category. - EXAMPLES: - sage: Sets().Endsets() Category of endsets of sets sage: Rings().Endsets() Category of endsets of unital magmas and additive unital additive magmas - >>> from sage.all import * >>> Sets().Endsets() Category of endsets of sets >>> Rings().Endsets() Category of endsets of unital magmas and additive unital additive magmas - See also 
 - Homsets()[source]¶
- Return the category of homsets between objects of this category. - EXAMPLES: - sage: Sets().Homsets() Category of homsets of sets sage: Rings().Homsets() Category of homsets of unital magmas and additive unital additive magmas - >>> from sage.all import * >>> Sets().Homsets() Category of homsets of sets >>> Rings().Homsets() Category of homsets of unital magmas and additive unital additive magmas - Note - Background - Information, code, documentation, and tests about the category of homsets of a category - Csshould go in the nested class- Cs.Homsets. They will then be made available to homsets of any subcategory of- Cs.- Assume, for example, that homsets of - Csare- Csthemselves. This information can be implemented in the method- Cs.Homsets.extra_super_categoriesto make- Cs.Homsets()a subcategory of- Cs().- Methods about the homsets themselves should go in the nested class - Cs.Homsets.ParentMethods.- Methods about the morphisms can go in the nested class - Cs.Homsets.ElementMethods. However it’s generally preferable to put them in the nested class- Cs.MorphimMethods; indeed they will then apply to morphisms of all subcategories of- Cs, and not only full subcategories.- See also - FunctorialConstruction- Todo - Design a mechanism to specify that an axiom is compatible with taking subsets. Examples: - Finite,- Associative,- Commutative(when meaningful), but not- Infinitenor- Unital.
- Design a mechanism to specify that, when \(B\) is a subcategory of \(A\), a \(B\)-homset is a subset of the corresponding \(A\) homset. And use it to recover all the relevant axioms from homsets in super categories. 
- For instances of redundant code due to this missing feature, see: - AdditiveMonoids.Homsets.extra_super_categories()
- HomsetsCategory.extra_super_categories()(slightly different nature)
- plus plenty of spots where this is not implemented.