Eilenberg-Moore Constructions Research Articles

On the differential graded Eilenberg–Moore construction

The Eilenberg–Moore construction for modules over a differential graded monad is used to study a question of Balmer regarding existence of an exact adjoint pair representing an exact monad. A Bousfield-like localization for differential graded categories is realized as a special case of this construction using Drinfeld quotients. As applications, we study some example coming from G-equivariant triangulated categories and twisted derived categories.

Unifying structured recursion schemes

AbstractFolds and unfolds have been understood as fundamental building blocks for total programming, and have been extended to form an entire zoo of specialised structured recursion schemes. A great number of these schemes were unified by the introduction of adjoint folds, but more exotic beasts such as recursion schemes from comonads proved to be elusive. In this paper, we show how the two canonical derivations of adjunctions from (co)monads yield recursion schemes of significant computational importance: monadic catamorphisms come from the Kleisli construction, and more astonishingly, the elusive recursion schemes from comonads come from the Eilenberg–Moore construction. Thus, we demonstrate that adjoint folds are more unifying than previously believed.

On the 2-Categories of Weak Distributive Laws

A weak mixed distributive law (also called weak entwining structure [8]) in a 2-category consists of a monad and a comonad, together with a 2-cell relating them in a way which generalizes a mixed distributive law due to Beck. We show that a weak mixed distributive law can be described as a compatible pair of a monad and a comonad, in 2-categories extending, respectively, the 2-category of comonads and the 2-category of monads in [13]. Based on this observation, we define a 2-category whose 0-cells are weak mixed distributive laws. In a 2-category 𝒦 which admits Eilenberg–Moore constructions both for monads and comonads, and in which idempotent 2-cells split, we construct a fully faithful 2-functor from this 2-category of weak mixed distributive laws to 𝒦2×2.

The weak theory of monads

We construct a ‘weak’ version EM w ( K ) of Lack and Street's 2-category of monads in a 2-category K , by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between monads in EM w ( K ) and composite pre-monads in K is discussed. If K admits Eilenberg–Moore constructions for monads, we define two symmetrical notions of ‘weak liftings’ for monads in K . If moreover idempotent 2-cells in K split, we describe both kinds of weak lifting via an appropriate pseudo-functor EM w ( K ) → K . Weak entwining structures and partial entwining structures are shown to realize weak liftings of a comonad for a monad in these respective senses. Weak bialgebras are characterized as algebras and coalgebras, such that the corresponding monads weakly lift for the corresponding comonads and also the comonads weakly lift for the monads.

The Eilenberg-Moore Category and a Beck-type Theorem for a Morita Context

The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a Morita context comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a double adjunction to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the moritability of a pair of functors with a common domain) are derived.

The monoidal Eilenberg–Moore construction and bialgebroids

Strong monoidal functors U : C→ M with left adjoints determine, in a universal way, monoids T in the category of opmonoidal endofunctors on M . Treating such opmonoidal monads as abstract “quantum groupoids” we derive Tannaka duality between right adjoint strong monoidal functors and opmonoidal monads. Bialgebroids, i.e., Takeuchi's × R -bialgebras, appear as the special case when T has also a right adjoint. Street's 2-category of monads then leads to a natural definition of the 2-category of bialgebroids.

The formal theory of monads II

We give an explicit description of the free completion EM( K) of a 2-category K under the Eilenberg–Moore construction, and show that this has the same underlying category as the 2-category Mnd( K) of monads in K . We then demonstrate that much of the formal theory of monads can be deduced using only the universal property of this completion, provided that one is willing to work with EM( K) as the 2-category of monads rather than Mnd( K) . We also introduce the wreaths in K ; these are the objects of EM( EM( K)) , and are to be thought of as generalized distributive laws. We study these wreaths, and give examples to show how they arise in a variety of contexts.

Monads on Composition Graphs

Collections of objects and morphisms that fail to form categories, inasmuch as the expected composites of two morphisms need not always be defined, were introduced in [14,15] under the name composition graphs. In [14,16], notions of adjunction and weak adjunction for composition graphs are proposed. Building on these definitions, we now introduce a concept of monads for composition graphs and show that the usual correspondence between adjunctions and monads remains correct, i.e. that (weak) adjunctions give rise to monads and that all monads are induced by adjunctions. Monads are described in terms of natural transforms as well as in terms of Kleisli triples, which seem to be better suited in the absence of associativity. The realization of a monad by an adjunction relies on a generalization of the Kleisli construction to composition graphs; on the other hand, the Eilenberg–Moore construction produces only a weak adjunction and admits comparison functors from weak adjunctions. As a first nontrivial example, we determine the algebras of the word monad on the composition graph of full morphisms between sets with a distinguished subset.