I'm pursuing a DPhil in Mathematics at the University of Oxford. Broadly, I am interested in (higher) category theory, (higher) algebra, fusion categories, low dimensional topology, and the representation categories of vertex operator algebras. Currently, I am thinking about fusion 2-categories, how to generalize them, and how they can be used to produce 4D TQFTs.
Separable Fusion 2-Categories
We give various charaterizations of separability for fusion 2-categories. In particular, we prove that a fusion 2-category is separable if and only if its Drinfeld center is finite semisimple. We also develop Morita theory for fusion 2-categories.
Rigid and Separable Algebras in Compact Semisimple Monoidal 2-Categories
Given a monoidal 2-category that has right and left adjoints, we prove that the 2-categories of right module and of bimodules over a rigid algebra have right and left adjoints. Given a compact semisimple monoidal 2-category, we use this result to prove that the 2-categories of modules and of bimodules over a separable algebra are compact semisimple. Finally, we define the dimension of a connected rigid algebra in a compact semisimple 2-category, and prove that such an algebra is separable if and only if its dimension is non-zero.
The Relative Deligne Tensor Product over Pointed Braided Fusion Categories
We give a formula for the relative Deligne tensor product of two indecomposable finite semisimple module categories over a pointed braided fusion category over an algebraically closed field.
Compact Semisimple 2-Categories
Working over an arbitrary field, we define compact semisimple 2- categories, and show that every compact semisimple 2-category is equivalent to the 2-category of separable module 1-categories over a finite semisimple tensor 1-category. Then, we prove that, over an algebraically closed field or a real closed field, compact semisimple 2-categories are finite. Finally, we explain how a number of key results in the theory of finite semisimple 2-categories over an algebraically closed field of characteristic zero can be generalized to compact semisimple 2-categories.
Finite Semisimple Module 2-Categories
Given C a multifusion 2-category. We show that every finite semisimple C-module 2-category is canonically enriched over C. Using this enrichment, we prove that every finite semisimple C-module 2-category is equivalent to the 2-category of modules over an algebra in C.
The 2-Deligne Tensor Product
We prove that the 2-Deligne tensor product of two compact semisimple 2-categories exists. Further, under suitable hypotheses, we explain how to describe the Hom-categories, connected components, and simple objects of a 2-Deligne tensor product. Finally, we prove that the 2-Deligne tensor product of two compact semisimple tensor 2-categories is a compact semisimple tensor 2-category
Weak Fusion 2-Categories
We introduce a weakening of the notion of fusion 2-category given in arXiv:1812.11933. Then, we establish a number of properties of (multi)fusion 2-categories. Finally, we describe the fusion rule of the fusion 2-categories associated to certain pointed braided fusion categories.
Multifusion Categories and Finite Semisimple 2-Categories
The 3-categories of semisimple 2-categories and of multifusion categories are shown to be equivalent. A weakening of the notion of fusion 2-category is introduced. Multifusion 2-categories are defined, and the fusion rule of the fusion 2-categories associated to certain pointed braided fusion categories is described.
25.05.22. "Topological Orders" at the Junior Physics and Geometry Seminar.
06.05.22. "Once and Twice Categorified Algebra" at the Junior Algebra and Representation Theory Seminar.
16.07.21. "The Cobordism Hypothesis and Fusion (2-)Categories" at YTM21.
11.06.21. "Multifusion Categories vs Finite Semisimple 2-Categories" at CaCS2021.
10.06.21. "Higher Fusion Categories Described by Spaces" at the Junior Topology and Group Theory Seminar.
19.02.21. "Finite Semisimple 2-Categories and Fusion 2-Categories" at QSSS.
03.06.20. "An Introduction to Fusion Categories" at the Junior Topology and Group Theory Seminar.
Tutor for Homological Algebra, MT21.
Tutor for Category Theory, MT21.
TA for Introduction to Schemes, HT21.
TA for Homological Algebra, MT20.
TA for Category Theory, MT19.