Inconsistent dual flags in spaces

[borisdevos]

There seems to be some inconsistency in tracking which spaces are dual and which aren't.

using TensorKit
P = Vect[IsingAnyon](:σ => 1)
t = ones(ComplexF64, P ← P)
space(t) # Vect[IsingAnyon](:σ => 1) ← Vect[IsingAnyon](:σ => 1)
space(t, 1) # Vect[IsingAnyon](…) of dim 1.4142135623730951: \n :σ => 1
space(t, 2) # Vect[IsingAnyon](…)' of dim 1.4142135623730951: \n :σ => 1
isdual(space(t, 1)), isdual(space(t, 2)) # (false, true)
isdual(codomain(t, 1)) # false
isdual(domain(t, 1)) # false

What really confused me at first was that the flags in the domain always seemed to reverse, but it wouldn't show explicitly.

Just for clarity I show the following as well:

t2 = ones(ComplexF64, P' ← P)
space(t2) # Vect[IsingAnyon](:σ => 1)' ← Vect[IsingAnyon](:σ => 1)
space(t2, 1) # Vect[IsingAnyon](…)' of dim 1.4142135623730951: \n :σ => 1, space(t2, 2) is the same
isdual(space(t2, 1)), isdual(space(t2, 2)) # (true, true)
isdual(codomain(t2, 1)) # true
isdual(domain(t2, 1)) # false

The same behavior persists for ComplexSpace as well.

[lkdvos]

This is actually working as intended, space(t, i) returns all spaces as if they would have been in the codomain, so that means space(t, i) = i <= numout(t) ? codomain(t)[i] : dual(domain(t)[i - numout(t)]).

I agree this is not necessarily the most intuitive, but this is a historic convention that makes it such that permuted tensors get the same space(t, i) outputs, and changing this would be quite painful.

Documented here: https://quantumkithub.github.io/TensorKit.jl/latest/man/tensors/#ss_tensor_properties

[Jutho]

Indeed, this was to ease the codomain / domain distinction for people that come from a more traditional (i.e. non-categorical) tensor background. Think of it as isdual(space(t, i)) encodes whether the tensor index i is outgoing (false) or incoming (true) with respect to to the tensor t.