3,745 research outputs found
Finite Semisimple Module 2-Categories
Let be a multifusion 2-category. We show that every finite
semisimple -module 2-category is canonically enriched over
. Using this enrichment, we prove that every finite semisimple
-module 2-category is equivalent to the 2-category of modules
over an algebra in .Comment: Many figure
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.Comment: Added reference
The Morita Theory of Fusion 2-Categories
We develop the Morita theory of fusion 2-categories. In order to do so, we
begin by proving that the relative tensor product of modules over a separable
algebra in a fusion 2-category exists. We use this result to construct the
Morita 3-category of separable algebras in a fusion 2-category. Then, we go on
to explain how module 2-categories form a 3-category. After that, we define
separable module 2-categories over a fusion 2-category, and prove that the
Morita 3-category of separable algebras is equivalent to the 3-category of
separable module 2-categories. As a consequence, we show that the dual tensor
2-category with respect to a separable module 2-category, that is the
associated 2-category of module 2-endofunctors, is a multifusion 2-category.
Finally, we give three equivalent characterizations of Morita equivalence
between fusion 2-categories.Comment: Many figures. Added new section on the Morita theory of fusion
2-categorie
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 -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
On the Drinfeld Centers of Fusion 2-Categories
We prove that the Drinfeld center of a fusion 2-category is invariant under
Morita equivalence and under taking the 2-Deligne tensor product with an
invertible fusion 2-category. We go on to show that the concept of Morita
equivalence between connected fusion 2-categories recovers exactly the notion
of Witt equivalence between braided fusion 1-categories. Then, we introduce the
notion of separable fusion 2-category. Conjecturally, separability ensures that
a fusion 2-category is 4-dualizable. We define the dimension of a fusion
2-category, which is a scalar whose non-vanishing is equivalent to
separability. In addition, we prove that a fusion 2-category is separable if
and only if its Drinfeld center is finite semisimple. We then establish the
separability of every strongly fusion 2-category, that is fusion 2-category
whose braided fusion 1-category of endomorphisms of the monoidal unit is
or . We proceed to show that every fusion
2-category is Morita equivalent to the 2-Deligne tensor product of a strongly
fusion 2-category and an invertible fusion 2-category. Finally, we prove that
every fusion 2-category is separable
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.Comment: Minor corrections. previously part of arXiv:2012.15774v
Gauging Noninvertible Defects: A 2-Categorical Perspective
We generalize the notion of an anomaly for a symmetry to a noninvertible
symmetry enacted by surface operators using the framework of condensation in
2-categories. Given a multifusion 2-category, potentially with some additional
levels of monoidality, we prove theorems about the structure of the 2-category
obtained by condensing a suitable algebra object. We give examples where the
resulting category displays grouplike fusion rules and through a cohomology
computation, find the obstruction to condensing further to the vacuum theory.Comment: 26 pages, v2 a new theorem about symmetric fusion 2-categories is
adde
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