Interface for Abelian Categories
Abelian categories are all over the place and very important. Thus we provide an Interface for (pre)additive and abelian categories. First recall the definitions:
A preadditive category is a category $\mathcal C$ such that all Hom-spaces are abelian groups and composition is bilinear. As a consequence all finite products are biproducts, also called direct sums.
Then $\mathcal C$ is called additive if it is preadditive and all finite products exist.
Additivity
To implement the preadditive structure you need the following methods
direct sum(X::YourObject...)::Tuple{YourObject, Vector{YourMorphism}, Yector{YourMorphism}returning the direct sum object, the inclusions and projections. You might only implement the binary operation while the package will compile a vector version. This might come with performance issues.+(f::YourMorphism, g::YourMorphism)::YourMorphismproviding the addition on morphisms.zero_morphism(X::YourObject, Y::YourObject)::YourMorphism
To complete additivity there has to be a zero object.
zero(C::YourCategory)::YourObject
Abelian Categories
A category is called abelian if
- it is additive,
- every morphism has a kernel and cokernel,
- every monomorphism and epimorphism is normal.
We need the following additional methods:
kernel(f::YourMorphism)::Tuple{YourObject, YourMorphism}providing the kernel tuple $(k,\phi)$ for $f \colon X \to Y$ where $\phi \colon k \hookrightarrow X$ is the embedding.cokernel(f::YourMorphism)::Tuple{YourObject, YourMorphism}providing the cokernel tuple $(c,\psi)$ for $f \colon X \to Y$ where $\psi \colon Y \twoheadrightarrow c$ is the projection.
Semisimple Categories
An abelian category is called semisimple if every object decomposes uniquely into a direct sum of simple objects. It might be useful to have the method
simples(C::YourCategory)::Vector{YourObject}
Categories with fibre functor
Whenever a category $\mathcal C$ has a fibre functor, i.e. an exact faithful functor $\mathcal C \to \mathrm{Vec}_k$, we can use matrix calculus to compute technical things we often need to implement certain constructions. Implement an existing fibre functor by providing the a method
matrix(f::MyMorphism)::MatElem