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

An abelian k-linear semisimple category having a finite number of simple objects, and endowed with a ribbon structure, is called premodular. It is modular (in the sense of Turaev) if the so-called S-matrix is invertible. A modular category defines invariants of 3-manifolds and a TQFT ([T]). When is it possible to construct a modularisation of a given premodular category, i.e. a functor to a modular category preserving the structures and 'dominant' in a certain sense? It turns out (2.3) that this amounts essentially to making 'transparent' objects trivial. We give a full answer to this problem in the case when k is a field of char. 0 (as well as partial answers in char. p) : under a few obvious hypotheses, a premodular category admits a modularisation, which is unique (th. 3.1, and cor. 3.5 in char. 0) The proof relies on two main ingredients : a new and very simple criterion for the S-matrix to be invertible (1.1) and Deligne's internal characterization of tannakian categories in char. 0 [D]. When simple transparent objects are invertible, the criterion is simpler (4.2) and the modularisation can be described more explicitly (prop. 4.4). We conclude with two examples : the premodular categories associated with quantum SL N and PGL N at roots of unity; in the first case, we obtain modular categories which were built independently by C. Blanchet [B]; in the second case, we obtain modularizations in all the cases where Y. Yokota [Y] found Reshetikhin-Turaev invariants of 3-manifolds, thereby improving as well as explaining Yokota's results. (1991): 18D10, 57M25, 57N10, 81R50, 81T10 Mathematics Subject Classification IntroductionMotivation. On sait que l'on peut associerà certains types de catégories monoïdales (voireà des n-catégories) des invariants topologiques en petite dimension.Ainsi, V. G. Turaev a introduit dans [T] la notion de catégorie modulaire, et montré comment l'on peut associerà une telle catégorie une TQFT (Topological Quantum Field Theory), et en particulier les invariants de 3-variétés de Une catégorie prémodulaire C sur un corps k est une 'catégorie rubanée', ou 'tortil', admettant une structure abélienne k-linéaire semi-simple avec un nombre fini d'objets 'simples'; elle est dite modulaire si une certaine matrice associée,

We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of tensor categories generalize strictly exact sequences of Hopf algebras as defined by Schneider, and in particular, exact sequences of (finite) groups. We classify exact sequences of tensor categories C ′ → C → C ′′ (such that C ′ is finite) in terms of normal faithful Hopf monads on C ′′ and also, in terms of self-trivializing commutative algebras in the center of C. More generally, we show that, given any dominant tensor functor C → D admitting an exact (right or left) adjoint there exists a canonical commutative algebra (A, σ) in the center of C such that F is tensor equivalent to the free module functor C → mod C (A, σ), where mod C (A, σ) denotes the category of A-modules in C endowed with a monoidal structure defined using σ. We re-interpret equivariantization under a finite group action on a tensor category and, in particular, the modularization construction, in terms of exact sequences, Hopf monads and commutative central algebras. As an application, we prove that a braided fusion category whose dimension is odd and square-free is equivalent, as a fusion category, to the representation category of a group.Definition 5.2. The commutative central algebra (A, σ) associated with a tensor functor F admitting a right adjoint is called the induced central algebra of F .