92 research outputs found
On the center of fusion categories
M\"uger proved in 2003 that the center of a spherical fusion category C of
non-zero dimension over an algebraically closed field is a modular fusion
category whose dimension is the square of that of C. We generalize this theorem
to a pivotal fusion category C over an arbitrary commutative ring K, without
any condition on the dimension of the category. (In this generalized setting,
modularity is understood as 2-modularity in the sense of Lyubashenko.) Our
proof is based on an explicit description of the Hopf algebra structure of the
coend of the center of C. Moreover we show that the dimension of C is
invertible in K if and only if any object of the center of C is a retract of a
`free' half-braiding. As a consequence, if K is a field, then the center of C
is semisimple (as an abelian category) if and only if the dimension of C is
non-zero. If in addition K is algebraically closed, then this condition implies
that the center is a fusion category, so that we recover M\"uger's result
Hopf monads on monoidal categories
We define Hopf monads on an arbitrary monoidal category, extending the
definition given previously for monoidal categories with duals. A Hopf monad is
a bimonad (or opmonoidal monad) whose fusion operators are invertible. This
definition can be formulated in terms of Hopf adjunctions, which are comonoidal
adjunctions with an invertibility condition. On a monoidal category with
internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode.
Hopf monads generalize Hopf algebras to the non-braided setting. They also
generalize Hopf algebroids (which are linear Hopf monads on a category of
bimodules admitting a right adjoint). We show that any finite tensor category
is the category of finite-dimensional modules over a Hopf algebroid. Any Hopf
algebra in the center of a monoidal category C gives rise to a Hopf monad on C.
The Hopf monads so obtained are exactly the augmented Hopf monads. More
generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross
product of T by a Hopf algebra of the center of the category of T-modules
(generalizing the Radford-Majid bosonization of Hopf algebras). We show that
the comonoidal comonad of a Hopf adjunction is canonically represented by a
cocommutative central coalgebra. As a corollary, we obtain an extension of
Sweedler's Hopf module decomposition theorem to Hopf monads (in fact to the
weaker notion of pre-Hopf monad).Comment: 45 page
Quasi-tame substitudes and the Grothendieck construction
This paper continues the study of the homotopy theory of algebras over
polynomial monads initiated by the first author and Clemens Berger. We
introduce the notion of a quasi-tame polynomial monad (generalizing tame ones)
and produce transferred model structures (left proper in many settings) on
algebras over such a monad. Our motivating application is to produce model
structures on Grothendieck categories, which are used in a companion paper to
give a unified approach to the study of operads, their algebras, and their
modules. We prove a general result regarding when a Grothendieck construction
can be realized as a category of algebras over a polynomial monad, examples
illustrating that quasi-tameness is necessary as well as sufficient for
admissibility, and an extension of classifier methods to a non-polynomial
situation, namely the case of commutative monoids.Comment: Comments welcome. This paper has a companion paper, "Model structures
on operads and algebras from a global perspective
Higher operads, higher categories
Higher-dimensional category theory is the study of n-categories, operads,
braided monoidal categories, and other such exotic structures. It draws its
inspiration from areas as diverse as topology, quantum algebra, mathematical
physics, logic, and theoretical computer science. This is the first book on the
subject and lays its foundations.
Many examples are given throughout. There is also an introductory chapter
motivating the subject for topologists.Comment: Book, 410 page
Model structures on operads and algebras from a global perspective
This paper studies the homotopy theory of the Grothendieck construction using
model categories and semi-model categories, provides a unifying framework for
the homotopy theory of operads and their algebras and modules, and uses this
framework to produce model structures, rectification results, and properness
results in new settings. In contrast to previous authors, we begin with a
global (semi-)model structure on the Grothendieck and induce (semi-)model
structures on the base and fibers. In a companion paper, we show how to produce
such global model structures in general settings. Applications include numerous
flavors of operads encoded by polynomial monads and substitudes (symmetric,
non-symmetric, cyclic, modular, higher operads, dioperads, properads, and
PROPs), (commutative) monoids and their modules, and twisted modular operads.
We also prove a general result for upgrading a semi-model structure to a full
model structure.Comment: Comments welcome. This paper has a companion paper "Quasi-tame
substitudes and the Grothendieck construction
- …