### Citations

102 |
Adjoint functors and triples
- Eilenberg, Moore
- 1965
(Show Context)
Citation Context ...s. 1. Categorical setting For convenience we recall the basic definitions. The observations on the interplay between adjoint pairs of functors, monads and comonads go back to the EilenbergMoore paper =-=[9]-=- from 1965 but for a long time were hardly exploited by people working on rings and modules. Parallel to the new appreciation of these techniques in classical algebra they also turn out to be of consi... |

77 |
The predual theorem to the Jacobson-Bourbaki Theorem
- Sweedler
- 1975
(Show Context)
Citation Context ...gs are Frobenius algebras. In this case the counit ε : P ∗ ⊗R P → R is just the trace of the matrix. A further application of 4.12 for not necessarily commutative rings yields 4.14. Sweedler’s coring =-=[14]-=-. Let (A, m, 1A) be any ring and h : R → A a ring homomorphism. Then the (A, R)-bimodule A is finitely generated and projective as left A-module and A ⊗R A becomes an A-coring with coproduct and couni... |

68 |
Frobenius and separable functors for generalized module categories and non-linear equations
- Caenepeel, Militaru, et al.
- 2002
(Show Context)
Citation Context ...The question arises if there are other such linear maps τ making A ⊗R B an R-algebra. This was intensively studied for algebras and led to various notions of smash products. The reader is referred to =-=[7]-=- (and the references given there) for investigations in this direction.�� � �� COALGEBRAIC STRUCTURES 19 Considering the related functors A ⊗R − and B ⊗R −, the lifting conditions from 2.1 are concer... |

50 |
Categories for the Working
- MacLane
- 1971
(Show Context)
Citation Context ...cation of categorical results by van Osdol from 1971. The section concludes with a remark on the role of the Yang-Baxter equation for entwinings. For notions from category theory we refer to Mac Lane =-=[10]-=- and for facts from module theory the reader may consult [16] or any other introductory texts on these fields. 1. Categorical setting For convenience we recall the basic definitions. The observations ... |

46 |
Distributive laws, [in:] Seminar on Triples and Categorical Homology Theory
- Beck
- 1969
(Show Context)
Citation Context ... questions can be related with lifting properties of endofunctors to the category of the corresponding modules or comodules, respectively, which can be described by distributive laws as considered in =-=[3]-=- (also called entwinings). For the category of vector spaces these liftings come in to define suitable structures on the tensor product of algebras or coalgebras. For details we refer to Section 5. 2.... |

38 | Frobenius monads and pseudomonoids
- Street
(Show Context)
Citation Context ...onds to the condition that F is isomorphic to its right adjoint and, by 1.16, this means that F allows for a comonad structure and leads to the following characterisation of Frobenius monads (compare =-=[13]-=-). 1.17. Proposition. For a monad (F, m, η) on A, the following are equivalent: (a) F is a Frobenius monad;�� �� �� � � �� �� �� �� �� � � �� �� �� 8 R. WISBAUER (b) F has a comonad structure (F, δ, ... |

24 | Monads and comonads on module categories
- Böhm, Brzeziński, et al.
(Show Context)
Citation Context ...omodules by the functors AF → A G , F (A) A G → AF , A ρ h ηA �� A ↦−→ A �� GF (A) G(h) F (ρ) �� G(A) ↦−→ F (A) �� F G(A) �� G(A) , εA �� A . More about these structures may be found, for example, in =-=[4, 17]-=-. For any k-vector space V , (V ⊗k −, Homk(V, −)) forms an adjoint pair of endofunctors. Thus a monad structure on V ⊗k − (algebra) implies a comonad structure on Homk(V, −). Then, if V has finite dim... |

21 | Modules, comodules and cotensor products over Frobenius algebras - Abrams - 1999 |

19 |
Bimonads and Hopf monads on categories
- Mesablishvili, Wisbauer
(Show Context)
Citation Context ...a right adjoint G, then, by 1.16, G allows for a monad and a comonad structure and it is easy to check that the compatibility conditions for B are transferred to G, that is G is again a bimonad (e.g. =-=[11, 4]-=-). 2.5. Hopf monads. Given a bimonad B = (B, m, e, δ, ε), one defines Hopf modules as objects A in A allowing for module and comodule structure maps, ϱA : B(A) → A and ϱ A : A → B(A), with commutative... |

17 | Algebras versus coalgebras
- Wisbauer
- 2008
(Show Context)
Citation Context ...omodules by the functors AF → A G , F (A) A G → AF , A ρ h ηA �� A ↦−→ A �� GF (A) G(h) F (ρ) �� G(A) ↦−→ F (A) �� F G(A) �� G(A) , εA �� A . More about these structures may be found, for example, in =-=[4, 17]-=-. For any k-vector space V , (V ⊗k −, Homk(V, −)) forms an adjoint pair of endofunctors. Thus a monad structure on V ⊗k − (algebra) implies a comonad structure on Homk(V, −). Then, if V has finite dim... |

14 |
Foundations of Module and Ring
- Wisbauer
- 1991
(Show Context)
Citation Context ...tion concludes with a remark on the role of the Yang-Baxter equation for entwinings. For notions from category theory we refer to Mac Lane [10] and for facts from module theory the reader may consult =-=[16]-=- or any other introductory texts on these fields. 1. Categorical setting For convenience we recall the basic definitions. The observations on the interplay between adjoint pairs of functors, monads an... |

6 | Cotensor products of modules
- Abrams, Weibel
(Show Context)
Citation Context ... Of course, equality holds on the right side if and only if RC is finitely generated and projective. Clearly, if R is a field, C satisfies the α-condition and - as pointed out by Abrams and Weibel in =-=[2]-=- - C ∗ can be seen as a pro-object in the category of finite-dimensional algebras. In this case the isomorphism given coincides with [2, Theorem 4.3]. The constructions of this section can also be for... |

3 | Idempotent monads and ?-functors
- Clark, Wisbauer
(Show Context)
Citation Context ...enini-Orsatti). A ∗-module P that is in addition a subgenerator in RM (generates all injectives) is known as a tilting module. The equivalence noticed above implies the BrennerButler equivalence (see =-=[8]-=-). 4. Module and comodule categories In the following we apply our general results to describe algebras and coalgebras and their (co-) module categories and this will throw new light on familiar notio... |

3 | Lifting theorems for tensor functors on module categories - Wisbauer |

1 | On Rational Pairings of Functors
- Mesablishvili, Wisbauer
- 2011
(Show Context)
Citation Context ...nsional algebras. In this case the isomorphism given coincides with [2, Theorem 4.3]. The constructions of this section can also be formulated in a general categorical context and are investigated in =-=[12]-=-. Recall that a Frobenius algebra over a field k is defined as a finite dimensional kalgebra A for which A ≃ A ∗ = Homk(A, k) as (left) A-modules. This notion can be readily extended to algebras over ... |

1 |
Sheaves in regular categories, in Exact categories and categories of sheaves
- Osdol
- 1971
(Show Context)
Citation Context ...closely related to comonads (corings). In categorical language this is described by lifting monads to comodule categories as outlined in 2.3 (mixed entwinings) and this goes back to van Osdol’s paper =-=[15]-=- from 1971. Consider an R-algebra (A, m, e) and an R-coalgebra (C, δ, ε). An R-linear map λ : A ⊗R C → C ⊗R A is called a mixed entwining provided all the diagrams in 2.3 are commutative for T = A ⊗R ... |