Monoidal Ladder Categories Mitchell Rowett

Mitchell Rowett's 2019 honours thesis in mathematics, on "Monoidal Ladder Categories", supervised by Scott Morrison, is available as thesis.pdf.


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.

last modified: 2019-10-25