Cat – Categories (package Combinat)

1. Introduction to the categories in MuPAD-Combinat

The MuPAD-Combinat package extends the category hierarchy of MuPAD. First, it defines some combinatorial categories that fix a common standard interface for combinatorial classes like compositions, permutations, trees, partitions, Young tableaux, etc (See Figure 1).

Graph of the combinatorial categories in MuPAD-Combinat

It also defines new algebraic categories for combinatorial (Hopf) algebras, operads, and so on (See Figure 1).

Graph of the algebraic categories in MuPAD-Combinat. For the sake of clarity, we have included a few preexisting categories of MuPAD; those appear with a gray background

Finally, it makes use of a few purely technical categories (Cat::FacadeDomain, Cat::UseOverloading).

Graph of the technical categories in MuPAD-Combinat

Cat::AlgebraWithBasis – the category of associative algebras with a distinguished basis

Cat::AlgebraWithSeveralBases – the category of algebras expanded on several bases

Cat::CombinatorialClass – the category of combinatorial classes

Cat::CombinatorialModule – the category of combinatorial modules

Cat::Crystal – Category for crystals

Cat::DecomposableClass – the category of decomposable combinatorial classes

Cat::FacadeDomain – the category of façade domains

Cat::FiniteDimensionalModuleWithBasis – The category of finite dimensional modules with basis

Cat::GradedAlgebraWithBasis – the category of graded algebras over a distinguished basis

Cat::GradedCombinatorialClass – the category of graded combinatorial classes

Cat::GradedHopfAlgebraWithBasis – the category of graded Hopf algebras over a distinguished basis

Cat::GradedModuleWithBasis – the category of graded modules with a distinguished basis

Cat::HopfAlgebraWithBasis – the category of Hopf algebras over a distinguished basis

Cat::HopfAlgebraWithSeveralBases – the category of the Hopf algebra expanded on several bases

Cat::Lattice – The category of Partially Ordered Set with sup and inf operation

Cat::ModuleWithBasis – the category of modules with a distinguished basis

Cat::ModuleWithSeveralBases – the category of modules with several bases

Cat::Object – The category of domains having objects as elements

Cat::Operad – the category of operads

Cat::OperadWithBasis – the category of operads over a distinguished basis

Cat::Poset – Category for Partially Ordered Sets

Cat::PermutationGroup – the category of permutation groups

Cat::SemiRing – the category of semi-rings

Cat::SemiRng – the category of semi-rings without unit

Cat::TowerOfPosets – Category for the tower of Posets

Cat::SkewCommutativeGroup – The category of skew commutative groups

Cat::TreesClass – The category of tree-like combinatorial classes that can be described by a grammar

Cat::UseOverloading – a helper category for domains using the conversion/overloading mechanism