Objects, morphisms, and composition

The three basic kinds of values are categories, objects, and morphisms, represented by subtypes of Category, Object, and Morphism.

Parents and endpoints

For an object $X$, parent(X) returns its category. For a morphism $f$, domain(f) and codomain(f) return its source and target, and parent(f) is inferred from domain(f). A morphism constructor must therefore ensure that its codomain belongs to the same category; the generic parent(f) method does not perform that check. A linear category also supplies base_ring(C).

The parent category is part of the represented data. Two objects that print in the same way need not be compatible if they belong to different category values or use different coefficient fields. When a construction changes the parent, use the model-specific functor or scalar-extension operation supplied for that construction.

Composition

For $f:X\to Y$ and $g:Y\to Z$, compose(f,g) returns $g\circ f$. The infix form g ∘ f has the usual mathematical order. The call id(X) returns $\operatorname{id}_X$; every category implementation must satisfy the identity and associativity laws for these represented morphisms.

For an endomorphism $f$, composition_power(f,n) accepts a nonnegative Julia Int $n$ and returns $f^n$, with exponent zero equal to the identity. The operation inv(f) returns a represented inverse when the model can compute one and otherwise throws an error. Its available keywords are model-dependent; the generic Hom-space solver has check=false by default and rechecks both inverse identities when called with check=true.

Categories need not come with matrices. When a concrete model supplies matrix(f), its coordinate and composition conventions are explained under Matrix coordinates.

Equality and isomorphism

X == Y and f == g mean equality in the chosen representation, including the parent or endpoints when the model implements those checks. Object equality is not isomorphism. Where isomorphism testing is supported, is_isomorphic(X,Y) returns (true,f) with $f:X\to Y$ an isomorphism, or (false,nothing). Assign the two entries separately; the tuple itself is not a Boolean.

For example, in the supplied category of finite-dimensional vector spaces:

using TensorCategories, Oscar
V = vector_spaces(QQ)
X = VectorSpaceObject(V, 2)
ok, h = is_isomorphic(X, X)
@assert ok && inv(h) ∘ h == id(X)
ok

true

Products and coproducts

In additive models, the generic product(X,Y) returns the product object and its projections, while the generic coproduct(X,Y) returns the coproduct and its injections. The finite-set model returns only the object unless its optional third argument is true. In additive categories use direct sums when both families of maps are needed.

Continue with Hom spaces and linear algebra.