Modular Categories

Abstract

We limit ourselves to recalling some basic definitions. A category is called an Ab-category if it is enriched over (the symmetric tensor category Ab of) abelian groups. I.e., all hom-sets come with abelian group structures and the composition • of morphisms is a homomorphism w.r.t. both arguments. An Ab-category is called additive if it has a zero object and every pair of objects has a direct sum. If k is a field (or more generally, a commutative unital ring) then a category C is called k-linear if all hom-sets are k-vector spaces (or k-modules) and • is bilinear. An object X is called simple if every monic morphism Y ֒→ X is an isomorphism and absolutely simple if End X ∼ = kid X . If k is an algebraically closed field, as we will mostly assume, the two notions coincide. A k-linear category is semisimple if every object is a finite direct sum of simple objects. A semisimple category is called finite if the number of isomorphism classes of simple objects is finite. (There is a notion of finiteness for non-semisimple categories, cf.[34], but we will not need it.) A positive * -operation on a C-linear category C is a contravariant endofunctor of C that acts like the identity on objects, is involutive ( * * = id) and anti-linear.(Dropping the C-linearity, one arrives at the notion of a dagger-category.) A * -operation is called positive if s * • s = 0 implies s = 0. A category equipped with a (positive) * -operation is called hermitian (unitary). A unitary category with finite-dimensional hom-spaces and splitting idempotents is semisimple (since finite dimensional algebras with positive * -operation are semisimple, thus multi-matrix algebras).

Monoidal Categories Enriched in Braided Monoidal Categories

“…As mentioned earlier, we have seen recent interest in monoidal categories enriched in V = sVec. Brundan and Ellis de ned a super tensor category in [BE16] (see also [Müg13,§6]), and Usher worked out many basic properties in [Ush16]. Usher also indicated some interesting examples (his Example 6.9) which were earlier announced by Walker in the language of spin planar algebras.…”

Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter

“…M (f ⊗ g) = 0. Since B is a unitary fusion category, M (f ⊗ g) = 0 if and only if the following trace is equal to 0: (note here that the pictures are read bottom-up) (3.2) = 0.By definition of an (A, A)-bimodule, the condition (2) in Definition 3.2 is equivalent to the following two conditions[55]:…”

Braided Categories of Endomorphisms as Invariants for Local Quantum Field Theories

“…37] that the category of finitely reducible DHR representations of the net, denoted by DHR{A}, has the abstract structure of a unitary modular tensor category (UMTC). Referring to [DHR71], [FRS92], [BKLR15], [Müg12], [EGNO15] for the relevant definitions and further details, we just recall that DHR representations of a local quantum field theory satisfying Haag duality can be described in terms of DHR endomorphisms of the quasilocal algebra A, which enjoy covariance, localizability and transportability properties. They are the objects of the C * tensor category DHR{A}, and their intertwiners are the morphisms.…”