Mitchell Rowett's 2019 honours thesis in mathematics, on "Monoidal Ladder Categories", supervised by
Scott Morrison, is available as
thesis.pdf.
Outline
In this thesis, we construct a tensor product of module
categories over a linear rigid monoidal category, which we call
a ladder category. In the case of monoidal module categories
over a braided category, we exhibit a monoidal structure on the
ladder category.
We then give two major examples. For the first,
we show that given a fusion category $\mathcal{C}$ with a central functor
from $\mathsf{Rep}G$, the de-equivariantisation of $\mathcal{C}$ can be realised as the
idempotent completion of the ladder category of $\mathcal{C}$ with $\mathsf{Vec}$ over
$\mathsf{Rep}G$. We also give a proof that the definition of
de-equivariantisation by de-enrichment given in [MP19] is
equivalent to the standard definition of de-equivariantisation.
For the second example we give an explicit description of the
ladder category of the two unital $\mathsf{Ad} E_8$ fusion categories over
$\mathsf{Fib}$, which appears to be a fusion category not previously
studied. We also show that the ladder category construction is
equivalent to the Deligne tensor product in the case of fusion
categories.
