The General Theory of Diads

A diad is a generalisation of a monad and a comonad. The idea is that we ignore the unit or counit, and consider only the natural transformations between T and T ². It turns out that almost all the constructions that we form for a monad or comonad can also be constructed from a related diad. Diads were introduced in Kenney (Appl. Categ. Structures, 2008), where they give a generalisation of the results that the category of coalgebras for a finite-limit preserving comonad on a topos is another topos, and that the category of algebras for a finite-limit preserving idempotent monad on a topos is another topos. In that paper, we were only interested in a special class of diads called codistributive diads, and we considered only the part of the theory of diads necessary to prove the result about finite-limit preserving diads in topoi. Here, we will study general diads in greater detail. We will develop the general theory with constructions that extend the standard constructions for monads and comonads.     

Algebras Versus Coalgebras

Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 1970s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between the base category and the category of the associated bimodules. For a set G, the endofunctor G× – on the category of sets shares these properties if and only if G admits a group structure. Finally, we report about the possibility of subsuming algebras and coalgebras in the notion of ( F , G )-dimodules associated to two functors between different categories. This observation, due to Tatsuya Hagino, was an outcome from the theory of categorical data types and may also be of use in classical algebra.      

Scalar Extension of Bicoalgebroids

After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter–Drinfel’d modules over a bicoalgebroid. It is proved that the Yetter–Drinfel’d category is monoidal and pre-braided just as in the case of bialgebroids, and is embedded into the one-sided center of the comodule category. We proceed to define braided cocommutative coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of Brzeziński and Militaru (J Algebra 251:279–294, 2002) and Bálint and Szlachányi (J Algebra 296:520–560, 2006), originally applied to bialgebras and bialgebroids, to bicoalgebroids. A few classical examples of this construction are given. Identifying the comodule category over a bicoalgebroid with the category of coalgebras of the associated comonad, we obtain a comonadic (weakened) version of Schauenburg’s theorem. Finally, we take a look at the scalar extension and braided cocommutative coalgebras from a (co-)monadic point of view.      

Hopf monads on monoidal categories

We define Hopf monads on an arbitrary monoidal category, extending the definition given in Bruguières and Virelizier (2007) [5] 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).     

Weak bimonoids in duoidal categories

Weak bimonoids in duoidal categories are introduced. They provide a common generalization of bimonoids in duoidal categories and of weak bimonoids in braided monoidal categories. Under the assumption that idempotent morphisms in the base category split, they are shown to induce weak bimonads (in four symmetric ways). As a consequence, they have four separable Frobenius base (co)monoids, two in each of the underlying monoidal categories. Hopf modules over weak bimonoids are defined by weakly lifting the induced comonad to the Eilenberg–Moore category of the induced monad. Making appropriate assumptions on the duoidal category in question, the fundamental theorem of Hopf modules is proven which says that the category of modules over one of the base monoids is equivalent to the category of Hopf modules if and only if a Galois-type comonad morphism is an isomorphism.     

The weak theory of monads

We construct a ‘weak’ version EMw(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 EMw(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 EMw(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.     

Pre-torsors and Galois Comodules Over Mixed Distributive Laws

We study comodule functors for comonads arising from mixed distributive laws. Their Galois property is reformulated in terms of a (so-called) regular arrow in Street’s bicategory of comonads. Between categories possessing equalizers, we introduce the notion of a regular adjunction. An equivalence is proven between the category of pre-torsors over two regular adjunctions (N _A ,R _A ) and (N _B ,R _B ) on one hand, and the category of regular comonad arrows (R _A ,ξ) from some equalizer preserving comonad \mathbb C{\mathbb C} to N _B R _B on the other. This generalizes a known relationship between pre-torsors over equal commutative rings and Galois objects of coalgebras. Developing a bi-Galois theory of comonads, we show that a pre-torsor over regular adjunctions determines also a second (equalizer preserving) comonad \mathbb D{\mathbb D} and a co-regular comonad arrow from \mathbb D{\mathbb D} to N _A R _A , such that the comodule categories of \mathbb C{\mathbb C} and \mathbb D{\mathbb D} are equivalent.     

Monads and distributive laws for Rota–Baxter and differential algebras

In recent years, algebraic studies of the differential calculus and integral calculus in the forms of differential algebra and Rota–Baxter algebra have been merged together to reflect the close relationship between the two calculi through the First Fundamental Theorem of Calculus. In this paper we study this relationship from a categorical point of view in the context of distributive laws which can be tracked back to the distributive law of multiplication over addition. The monad giving Rota–Baxter algebras and the comonad giving differential algebras are constructed. Then we obtain monads and comonads giving the composite structures of differential and Rota–Baxter algebras. As a consequence, a mixed distributive law of the monad giving Rota–Baxter algebras over the comonad giving differential algebras is established.     

Cartesian Differential Categories as Skew Enriched Categories

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

     

A logic of implications in algebra and coalgebra

Implications in a category can be presented as epimorphisms: an object satisfies the implication iff it is injective with respect to that epimorphism. G. Ro?u formulated a logic for deriving an implication from other implications. We present two versions of implicational logics: a general one and a finitary one (for epimorphisms with finitely presentable domains and codomains). In categories Alg Σ of algebras on a given signature our logic specializes to the implicational logic of R. Quackenbush. In categories Coalg H of coalgebras for a given accessible endofunctor H of sets we derive a logic for implications in the sense of P. Gumm.     

Chiral Koszul duality

We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld (American Mathematical Society Colloquium Publications, 51. American Mathematical Society, Providence, RI, 2004), to higher-dimensional varieties. This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen’s homotopy theory of differential graded Lie algebras. We prove the equivalence of higher-dimensional chiral and factorization algebras by embedding factorization algebras into a larger category of chiral commutative coalgebras, then realizing this interrelation as a chiral form of Koszul duality. We apply these techniques to rederive some fundamental results of Beilinson and Drinfeld (American Mathematical Society Colloquium Publications, 51. American Mathematical Society, Providence, RI, 2004) on chiral enveloping algebras of *{\star} -Lie algebras.     

Functors on the category of quasi-fibrations

We consider the following questions: when can we extend a continuous endofunctor on Top the category of topological spaces to a fibrewise continuous endofunctor on Top(2) the category of continuous maps? If this is true, does such fibrewise continuous endofunctor preserve fibrations? In this paper, we define Fib the topological category of cell-wise trivial fibre spaces over polyhedra and show that any continuous endofunctor on Top induces a fibrewise continuous endofunctor on Fib preserving the class of quasi-fibrations.     

Convexities Generated by <Emphasis Type="Italic">L</Emphasis>-Monads

Let \mathbbF\mathbb{F} be a monad in the category Comp. We build for each \mathbbF\mathbb{F}-algebra a convexity in general sense (see van de Vel 1993). We investigate properties of such convexities and apply them to prove that the multiplication map of the order-preserving functional monad is soft.     

Unicity of Enrichment over Cat or Gpd

Enrichment of ordinary monads over Cat or Gpd is fundamental to Max Kelly’s unified theory of coherence for categories with structure. So here, we investigate existence and unicity of enrichments of ordinary functors, natural transformations, and hence also monads, over Cat and Gpd. We show that every ordinary natural transformation between 2-functors whose domain 2-category has either tensors or cotensors with the arrow category is 2-natural. We use that to prove that an ordinary monad, or endofunctor, on such a 2-category has at most one enrichment over Cat or Gpd. We also describe a monad on Cat that has no enrichment. So enrichment over Cat is a non-trivial property of a monad rather than a structure that is additional to it. Finally, we present an example, due to Kelly, of V other than Cat or Gpd and an ordinary monad for which more than one enrichment over V exists, showing that our main theorem is specific to Cat and Gpd.     

A note on triangulated monads and categories of module spectra

Consider a monad on an idempotent complete triangulated category with the property that its Eilenberg–Moore category of modules inherits a triangulation. We show that any other triangulated adjunction realizing this monad is ‘essentially monadic’, i.e. becomes monadic after performing the two evident necessary operations of taking the Verdier quotient by the kernel of the right adjoint and idempotent completion. In this sense, the monad itself is ‘intrinsically monadic’. It follows that for any highly structured ring spectrum, its category of homotopy (aka naïve) modules is triangulated if and only if it is equivalent to its category of highly structured (aka strict) modules.     

Yoneda Structures from 2-toposes

A 2-categorical generalisation of the notion of elementary topos is provided, and some of the properties of the Yoneda structure (Street and Walters, J. Algebra, 50:350–379, 1978) it generates are explored. Results enabling one to exhibit objects as cocomplete in the sense definable within a Yoneda structure are presented. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2-category (Street, Lecture Notes in Math., 420:104–133, 1974) and provides a self-contained development of the necessary background material on Yoneda structures.      

INSTANCES AND RAMIFICATIONS OF THE SEMI-ADJOINT SITUATION I. PRESTABLE REFLECTION AND SYMMETRIC UNADS

Abstract

It is well known that there is a one to one correspondence between idempotent monads in a category and reflective subcategories. In this paper it is examined what replaces the reflective subcategory if the idempotent monad is replaced (a) by a monad and (b) by a symmetric unad. It is shown that in case (a) one obtains the weakly reflective subcategory of objects injective relative to the functor part of the monad. In case (b) one obtains a proto-reflection and it is shown that (for complete categories) the associated orthogonal subcategory is reflective if and only if there exists a free monad associated to the unad.     

CATEGORICAL ASPECTS OF POWER ALGEBRAS

Abstract

The well-known power algebra construction is investigated from a categorical point of view. We establish basic categorical properties, from which we deduce the Homomorphism and Isomorphism Theorems for power algebras. We show that the power algebra construction induces a monad, called the power algebra monad, and determine the associated Eilenberg-Moore category as well as the associated Kleisli category.