1,361 research outputs found
Lie monads and dualities
We study dualities between Lie algebras and Lie coalgebras, and their
respective (co)representations. To allow a study of dualities in an
infinite-dimensional setting, we introduce the notions of Lie monads and Lie
comonads, as special cases of YB-Lie algebras and YB-Lie coalgebras in additive
monoidal categories. We show that (strong) dualities between Lie algebras and
Lie coalgebras are closely related to (iso)morphisms between associated Lie
monads and Lie comonads. In the case of a duality between two Hopf algebras -in
the sense of Takeuchi- we recover a duality between a Lie algebra and a Lie
coalgebra -in the sense defined in this note- by computing the primitive and
the indecomposables elements, respectively.Comment: 27 pages, v2: some examples added and minor change
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
Lifting Coalgebra Modalities and Model Structure to Eilenberg-Moore Categories
A categorical model of the multiplicative and exponential fragments of
intuitionistic linear logic (), known as a \emph{linear
category}, is a symmetric monoidal closed category with a monoidal coalgebra
modality (also known as a linear exponential comonad). Inspired by Blute and
Scott's work on categories of modules of Hopf algebras as models of linear
logic, we study categories of algebras of monads (also known as Eilenberg-Moore
categories) as models of . We define a lifting
monad on a linear category as a Hopf monad -- in the Brugui{\`e}res, Lack, and
Virelizier sense -- with a special kind of mixed distributive law over the
monoidal coalgebra modality. As our main result, we show that the linear
category structure lifts to the category of algebras of lifting
monads. We explain how groups in the category of coalgebras of the monoidal
coalgebra modality induce lifting monads and provide a source
for such groups from enrichment over abelian groups. Along the way we also
define mixed distributive laws of symmetric comonoidal monads over symmetric
monoidal comonads and lifting differential category structure.Comment: An extend abstract version of this paper appears in the conference
proceedings of the 3rd International Conference on Formal Structures for
Computation and Deduction (FSCD 2018), under the title "Lifting Coalgebra
Modalities and Model Structure to Eilenberg-Moore Categories
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