Objects¶
- class sage.categories.objects.Objects[source]¶
Bases:
Category_singletonThe 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 classCs.Homsets. They will then be made available to homsets of any subcategory ofCs.Assume, for example, that homsets of
CsareCsthemselves. This information can be implemented in the methodCs.Homsets.extra_super_categoriesto makeCs.Homsets()a subcategory ofCs().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 classCs.MorphimMethods; indeed they will then apply to morphisms of all subcategories ofCs, and not only full subcategories.See also
FunctorialConstructionTodo
Design a mechanism to specify that an axiom is compatible with taking subsets. Examples:
Finite,Associative,Commutative(when meaningful), but notInfinitenorUnital.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.