representable functors for corings
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Representable Functors for CoringsG. Militaru aa Faculty of Mathematics and Computer Science , University of Bucharest , Bucharest ,RomaniaPublished online: 15 May 2012.
To cite this article: G. Militaru (2012) Representable Functors for Corings, Communications in Algebra, 40:5, 1766-1796, DOI:10.1080/00927872.2011.556694
To link to this article: http://dx.doi.org/10.1080/00927872.2011.556694
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Communications in Algebra, 40: 17661796, 2012Copyright Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2011.556694
REPRESENTABLE FUNCTORS FOR CORINGS
G. MilitaruFaculty of Mathematics and Computer Science, University of Bucharest,Bucharest, Romania
We address four problems regarding representable functors and give answers to themfor functors connecting the category of comodules over a coring to the categoryof modules over a ring. A functors property of being Frobenius is restated as aparticular case of its representability by imposing the predefinition of the object ofrepresentability. Let R, S be two rings, C an R-coring and CR the category of leftC-comodules. The category RepCR S of all representable functors
CR S is
shown to be equivalent to the opposite of the category CRS . For U an SR-bimodulewe give necessary and sufficient conditions for the induction functor U R CR S to be: a representable functor, an equivalence of categories, a separable or aFrobenius functor. The latter results generalize and unify the classical theorems ofMorita for categories of modules over rings and the more recent theorems obtained byBrezinski, Caenepeel et al. for categories of comodules over corings.
Key Words: Corings; Representable; Separable and Frobenius functors.
2010 Mathematics Subject Classification: 16T15; 18A22.
The representable functor is a central concept in category theory for atleast two reasons. On one hand, fundamental mathematical constructions likefree groups, free modules, tensor products of modules, tensor algebras, symmetricalgebras, and algebras of noncommutative differential forms of an algebra are betterexplained using the language of representable functors. They are all answers to thesame question: Is a given functor F representable? On the other hand, representablefunctors are bridges from classical geometry to noncommutative geometry andquantum groups. Based on the fact that an affine scheme is a representable functoron the category of commutative k-algebras a quantum space is defined as a naturalgeneralization. More precisely, a quantum space is a representable functor on thecategory of k-algebras (not neccesarily commutative) while a quantum group is aquantum space for which the object of representability has a structure of Hopfalgebra .
Received September 28, 2010; Revised January 12, 2011. Communicated by E. Puczylowski.Address correspondence to Prof. G. Militaru, Faculty of Mathematics and Computer
Science, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania; E-mail:firstname.lastname@example.org and email@example.com
REPRESENTABLE FUNCTORS FOR CORINGS 1767
The classical definition of representable functors corresponding to et-valuated functors was generalized  replacing the category of sets with a variety ofalgebras in the sense of universal algebras (for example, can be the category ofsemigroups, monoids, groups, abelian groups, rings, algebras over commutative ringor modules over a ring). At this level, a functor F is called representableif F et is representable in the classical sense, where et is theforgetful functor. At first sight, this generalisation seems trivial. In reality, however,it poses great difficulties in the theory of representable functors. Let us take acategory and a variety of algebras. There are four general problems concerningrepresentable functors.
Problem A: Describe, whenever possible, the category Rep of allrepresentable functors F .
Problem B: Give a necessary and sufficient condition for a given functor F to be representable (possibly predefining the object of representability).
Problem C: When is a composition of two representable functors a representablefunctor?
Problem D: Give a necessary and sufficient condition for a representable functorF and for its left adjoint1 to be separable or Frobenius.
All universal constructions mentioned in the beginning are in fact answers toProblem B in the trivial case = et. An excellent book dedicated exclusively toProblem A is  where the category Rep is described for different categoriesof varieties of algebras and . In the case that has finite coproducts theProblem A is equivalent to description of the opposite of the category of so calledco- objects of [1, Corollary 8.13]. In general, the problem is difficult and theresults can be very interesting and surprising: moreover, as explained in , ProblemA has relevant applications in the study of fundamental groups of topologicalspaces, operator algebras, connected graded algebras, probability distributions,Hopf algebras, etc. The pioneer of studying Problem A was Kan, who described allrepresentable functors from semigroups to semigroups . A crucial step relatedto problem A was made by Freyd in his seminal article : if is a cocompletecategory and a variety of algebras then a functor F is representableif and only if F is a right adjoint ([1, Theorem 8.14]). Abstract properties of thecategory Rep are proven in the recent article  and in Section 1 we shallgive more examples and motivations for each of the problems above.
The study of corings and their corepresentations has generated an explosionof interest and has become a distinctive research area after  was published, for atleast two reasons. On one hand, the category of comodules over a coring is a verygeneral one. Different types of categories like modules over a ring, comodules overa coalgebra, descent data associated to a ring extension, Hopf modules and relativeHopf modules, DoiKoppinen modules, entwined modules, or YetterDrinfeldmodules are all special cases of comodules over various corings. Thus any resultproven for the category of comodules over a coring is also applicable for all
1A representable functor F has a left adjoint if the object of representability of F has copowers.
these particular types of categories. On the other hand, working with comodulesover corings is simple and most proofs are natural and transparent. For moredetails about the importance of coring theory and its applications we refer to themonograph .
In this article, we shall give answers to all the above problems in case = CR,the category of left C-comodules over an R-coring C and = S, the categoryof left S-modules over a ring S. The article is organized as follows. In Section 1,we recall the basic concepts that will be used throughout the article and givemore examples and motivations of the above problems. Theorem 1.5 is a structuretheorem that gives a first answer to Problem D: it describes all functors that areFrobenius and separable from an abelian category to the category of abelian groups.Representable functors having an object of representability with special propertiesplay the key role in this picture. Section 2 collects all technical results that we shalluse to prove the main theorems of the article.