Generic Methods
Many constructions for additive, abelian, linear or monoidal categories are entangled. Thus we provide a vast list of methods to compute objects or morphisms using other methods.
Keep in mind that the performance will usually be much better if the following methods are overwritten form custom types.
AbstractAlgebra.image — Method
image(f::Morphism)Return the image $Im(f)$ of $f:X → Y$ together with a monomorphism $Im(f) ↪ Y$.
Hecke.central_primitive_idempotents — Method
central_primitive_idempotents(H::AbstractHomSpace)Compute the central primitive idempotents of an endomorphism space $H$.
Hecke.decompose — Method
decompose(X::Object)Decompose an object $X$ in an abelian category.
Hecke.decompose — Method
decompose(X::Object, S::Vector{Object})Decompose an object $X$ in a semisimple category into simple objects of $S$.
Hecke.is_simple — Method
is_simple(X::Object)Check whether $X$ is a simple object.
Nemo.eigenvalues — Method
eigenvalues(f::Morphism)Compute the eigenvalues of $f$. Return a dictonary with entries λ => ker(f - λid).
TensorCategories.endomorphism_ring — Function
endomorphism_ring(X::Object)
Return the endomorphism ring of $X$ as a matrix algebra.
TensorCategories.express_in_basis — Function
express_in_basis(f::Morphism)Return a vector of coefficients expressing $f: X → Y$ in the basis o f $\mathrm{Hom}(X,Y)$.
TensorCategories.express_in_basis — Method
express_in_basis(f::Morphism, B::Vector{Morphism})Return a vector of coefficients expressing $f$in the basis $B$.
TensorCategories.horizontal_direct_sum — Method
function horizontal_direct_sum(f::Morphism, g::Morphism)Return the sum of $f:X → Z$, $g:Y → Z$ as ``f+g:X⊕Y → Z.
TensorCategories.is_epimorphism — Method
is_epimorphism(f::Morphism)Check wether $f$is epi.
TensorCategories.is_monomorphism — Method
is_monomoprhism(f::Morphism)Check whether $f$ mono.
TensorCategories.left_inverse — Method
left_inverse(f::Morphism)Compute a morphism $g$ such that $g ∘ f = id$. Errors if $f$is not mono.
TensorCategories.right_inverse — Method
right_inverse(f::Morphism)Compute a morphism $g$ such that $f ∘ g = id$. Errors if $f$ is not epi.
TensorCategories.vertical_direct_sum — Method
function vertical_direct_sum(f::Morphism, g::Morphism)Return the sum of $f:X → Y$, $g:X → Z$ as ``f+g: X → Y⊕Z.
TensorCategories.zero_morphism — Method
zero_morphism(X::Object, Y::Object)Compute the zero morphism between $X$and $Y$.
TensorCategories.zero_morphism — Method
zero_morphism(X::Object)Compute the zero morphism on $X$.
TensorCategories.:× — Method
×(X::Object...)Return the product Object and an array containing the projection morphisms.
TensorCategories.∐ — Method
∐(X::Object...)Return the coproduct Object and an array containing the injection morphisms.