Examples of Para-cocyclic Objects Induced by Bd-laws

In a recent paper [BŞ], we gave a general construction of a para-cocyclic structure on a cosimplex, associated to a so called admissible septuple – consisting of two categories, three functors and two natural transformations, subject to compatibility relations. The main examples of such admissible septuples were induced by algebra homomorphisms. In this note we provide more general examples coming from appropriate (‘locally braided’) morphisms of monads. 2000 Mathematics Subject Classification: 16E40 (primary), 16W30 (secondary) Introduction History of cyclic homology started in the early eighties of the last century. The seminal works of the pioneers A Connes, B Tsygan, D Quillen and J-L Loday were motivated by looking for non-commutative generalizations of de Rham cohomology on one hand, and Lie algebra homology of matrices on the other hand. In the subsequent decades cyclic homology has been extensively studied and became an important tool in diverse areas of mathematics, such as homological algebra, algebraic topology, Lie algebras, algebraic K-theory and so non-commutative differential geometry. Thus by most various motivations, lots of examples have been constructed. In order to study general features of the examples, and also to be able to construct new ones, it was desirable to find a unifying general description. A fundamental first step in this direction was made by A Kaygun in [Kay], who gave a construction of para-(co)cyclic objects in symmetric monoidal categories in terms of (co)monoids. In particular, in this way he managed to describe in a universal form all examples arising from Hopf cyclic theory (upto cyclic duality, cf. [KR]). Motivated by a generalization to bialgebroids (over non-commutative rings, in which case the underlying bimodule categories are not symmetric), in [BŞ] we made a further step of generalization and constructed para-(co)cyclic objects in arbitrary categories, in terms of (co)monads. Kaygun’s construction can be recovered as a particular case when the (co)monads in question are induced by (co)monoids. Mean examples of Kaygun’s construction are induced by algebras over a commutative ring. By analogy, in this paper we show that appropriate monad morphisms (which are ‘locally braided’ in a sense to be described) induce examples of para-cocyclic objects in [BŞ]. The paper is organized as follows. In Section 1 we recall some facts about monads and BD-laws that are used in the paper. In Section 2 we introduce the notion of a locally braid preserving morphism of monads, generalizing a homomorphism of algebras, and we investigate their basic properties that are needed to state and prove our main result. In Section 3 we show that any such morphism determines an ‘admissible septuple’ in the sense of [BŞ], hence can be used to construct para-cocyclic objects. Here we also illustrate how this construction works in the example of a morphism of algebras in a braided monoidal category (i.e. when there is a global braiding). Particular examples will be provided by appropriate homomorphisms of (co)module algebras of a (co)quasitriangular Hopf algebra. Acknowledgment. It is our pleasure to thank the organizers of the conference ‘Non-commutative Rings and Geometry’, Almeŕıa, Spain, 18-22 September 2007, held in the honour of the 60th birthday of Freddy Van Oystaeyen. The first author was partially supported by the Hungarian Scientific Research Fund OTKA K 68195 and the Bolyai János Research Scholarship. The second author was supported by Contract 2-CEx06-11-20 of the Romanian Ministry of Education and Research. 1 2 GABRIELLA BÖHM AND DRAGOŞ ŞTEFAN 1. Monads and the category of their (bi)modules Throughout the paper, we use the notations introduced in [BŞ]. That is, in the 2-category CAT horizontal composition (of functors) is denoted by juxtaposition, while ◦ is used for vertical composition (of natural transformations). For example, for two functors F : C → C, G : C → C and an object X in C, instead of G(F (X)) we write GFX . For two natural transformations μ : F → F ′ and ν : G → G we write GμX ◦ νFX : GFX → GF X instead of G(μX) ◦ νF (X). In equalities of natural transformations we shall omit the object X in our formulae. We shall also use a graphical representation of morphisms in a category. For functors F1, . . . , Fn, G1, . . . , Gm, which can be composed to F1F2 . . . Fn : D1 → C and G1G2 . . . Gm : D2 → C, and objects X in D1 and Y in D2, a morphism f : F1F2 . . . FnX → G1G2 . . . GmY will be represented vertically, with the domain up, as in Figure 1(a). Furthermore, for a functor T : C → C, the morphism Tf will be drawn as in (b). Keeping the notation from the first paragraph of this section, the picture representing μGX is shown in diagram (c). The composition g ◦ f of the morphisms f : X → Y and g : Y → Z will be represented as in diagram (d). For the multiplication t and the unit τ of a monad T on C (see Definition 1.1), and an object X in C, to draw tX and τX we shall use the diagrams (e) and (f), while for a distributive law l : RT → TR (see Definition 1.9) lX will be drawn as in the picture (g). If l is invertible, the representation of lX is shown in diagram (h). t T T R R F 1 F 1 FG