Purpose

Introduce a framework for testing whether or not something is a morphism in a category. See the discussion on sage-algebra. Here is a summary of the discussion.

Methods for categories

Categories C should have a method C.has_morphism(f) answering whether f is a morphism in C. By symmetry, we want a method C.has_object(X), answering whether X is an object in C.

Note that we want X in C to be true if and only if X is an object of C (so, it is synonymous to C.has_object(X)). This currently is not always the case:

sage: P.<x,y> = QQ[]
sage: f = P.hom(reversed(P.gens()))
sage: f in Rings().hom_category()
True

but of course f is not an object of the hom-category (it is only contained in an object of the hom-category).

Class/Set of objects and morphisms

It would be nice to have container classes for the objects and for the morphisms of a category. Then, f in C.morphisms() would be a very natural notation for C.has_morphism(f), and X in C.objects() would be another way of saying X in C.

Of course, since f in C.morphisms() and f in C.objects() are nice notations, they should be as fast as possible -- otherwise, people wouldn't use them.

Further discussion should be put in comments to this ticket.

Depends on #9138
Depends on #11115
Depends on #11780

CC: @nilesjohnson @jpflori

Component: categories

Keywords: objects morphisms containment sd34

Work Issues: Cope with non-unique number fields

Author: Simon King

Issue created by migration from https://trac.sagemath.org/ticket/10667