This ticket implements:

• Support for full subcategories defined by an axiom (Finite,
  Infinite, Facade, Commutative, Associative, Unital, Inverse,
  Distributive, NoZeroDivisors, Division, FiniteDimensional,
  Connected, WithBasis, Irreducible), and joins thereof:

    sage: Groups() & Sets().Finite()
    Category of finite groups

    sage: Algebras(QQ).Finite() & Monoids().Commutative()
    Category of finite commutative algebras over Rational Field

    sage: (Monoids() & CommutativeAdditiveGroups()).Distributive()
    Category of rings

    sage: Rings().Division() & Sets().Finite()
    Category of finite fields

• New categories:

  • AdditiveSemigroups, AdditiveMonoids, AdditiveGroups
  • DistributiveMagmasAndAdditiveMagmas
  • MagmaticAlgebras (will replace Algebras in Rename MagmaticAlgebras to Algebras and Algebras to AssociativeUnitalAlgebras #15043)
  • AssociativeAlgebras
  • UnitalAlgebras
  • Algebras of additive semigroups and monoids

• More mathematical rules:

  • A subquotient of a finite set is a finite set
  • The algebra of a finite set is finite dimensional
  • The algebra of a commutative magma is commutative
  • A finite division ring is a field
  • ...

• Documentation:

  • More documentation for IsomorphicObjects
  • Complete revamping of sage.categories.primer
  • Misc

• Use SubcategoryMethods to put the functorial constructions where
  they belong. E.g. DualObjects, TensorProducts, and Graded are now
  only defined for subcategories of Modules.

• More lazy imports, removed a bunch of unused imports, ...

This ticket is dedicated to the town of Megantic where I was so warmly welcomed and a good chunk of this ticket got implemented!

Depends on #11224
Depends on #8327
Depends on #10193
Depends on #12895
Depends on #14516
Depends on #14722
Depends on #13589
Depends on #14471
Depends on #15069
Depends on #15094
Depends on #11688
Depends on #13394
Depends on #15150
Depends on #15506
Depends on #15757
Depends on #15759
Depends on #16244
Depends on #16269

CC: @sagetrac-sage-combinat @simon-king-jena @saliola @anneschilling @vbraun @nbruin @zabrocki

Component: categories

Keywords: days54

Work Issues: To be merged simultaneously with #15801

Author: Nicolas M. Thiéry

Branch: c16f18b

Reviewer: Volker Braun, Nils Bruin, Peter Bruin, Frédéric Chapoton, Darij Grinberg, Florent Hivert, Simon King, Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/10963