Representable Functors for Corings

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<ul><li><p>This article was downloaded by: [University of Toronto Libraries]On: 05 November 2014, At: 08:47Publisher: Taylor &amp; FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK</p><p>Communications in AlgebraPublication details, including instructions for authors and subscription information:</p><p>Representable Functors for CoringsG. Militaru aa Faculty of Mathematics and Computer Science , University of Bucharest , Bucharest ,RomaniaPublished online: 15 May 2012.</p><p>To cite this article: G. Militaru (2012) Representable Functors for Corings, Communications in Algebra, 40:5, 1766-1796, DOI:10.1080/00927872.2011.556694</p><p>To link to this article:</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. However, Taylor &amp; Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor &amp; Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.</p><p>This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms &amp; Conditions of access and use can be found at</p><p></p></li><li><p>Communications in Algebra, 40: 17661796, 2012Copyright Taylor &amp; Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2011.556694</p><p>REPRESENTABLE FUNCTORS FOR CORINGS</p><p>G. MilitaruFaculty of Mathematics and Computer Science, University of Bucharest,Bucharest, Romania</p><p>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</p><p>CR S is</p><p>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.</p><p>Key Words: Corings; Representable; Separable and Frobenius functors.</p><p>2010 Mathematics Subject Classification: 16T15; 18A22.</p><p>INTRODUCTION</p><p>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 [19].</p><p>Received September 28, 2010; Revised January 12, 2011. Communicated by E. Puczylowski.Address correspondence to Prof. G. Militaru, Faculty of Mathematics and Computer</p><p>Science, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania; and</p><p>1766</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f T</p><p>oron</p><p>to L</p><p>ibra</p><p>ries</p><p>] at</p><p> 08:</p><p>47 0</p><p>5 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>REPRESENTABLE FUNCTORS FOR CORINGS 1767</p><p>The classical definition of representable functors corresponding to et-valuated functors was generalized [1] 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.</p><p>Problem A: Describe, whenever possible, the category Rep of allrepresentable functors F .</p><p>Problem B: Give a necessary and sufficient condition for a given functor F to be representable (possibly predefining the object of representability).</p><p>Problem C: When is a composition of two representable functors a representablefunctor?</p><p>Problem D: Give a necessary and sufficient condition for a representable functorF and for its left adjoint1 to be separable or Frobenius.</p><p>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 [1] 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 [1], 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 [16]. A crucial step relatedto problem A was made by Freyd in his seminal article [13]: 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 [2] and in Section 1 we shallgive more examples and motivations for each of the problems above.</p><p>The study of corings and their corepresentations has generated an explosionof interest and has become a distinctive research area after [5] 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</p><p>1A representable functor F has a left adjoint if the object of representability of F has copowers.</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f T</p><p>oron</p><p>to L</p><p>ibra</p><p>ries</p><p>] at</p><p> 08:</p><p>47 0</p><p>5 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>1768 MILITARU</p><p>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 [6].</p><p>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. Let R, S be rings and C an R-coring.We are focusing on the large categories Functors</p><p>(S</p><p>CR</p><p>)and Functors</p><p>(CR S</p><p>)of all covariant functors that connect the category of comodules over C and thecategory of modules over S. Three Yoneda type embeddings are constructed. Theclasses of all natural transformations between an induction functor and the identityfunctor on the category CR are explicitly computed. Section 3 contains the mainresults of the article. Theorem 3.1 gives an answer for Problem A: the categoryRepCR S is equivalent to the opposite of the category</p><p>CRS . Corollary 3.4 offers</p><p>an answer for Problem C.Let U be a S R-bimodule and the induction functor U R CR </p><p>S. Theorem 3.9 gives necessary and sufficient conditions for U R to be arepresentable functor, i.e., an answer for Problem B. It generalizes and unifiestwo theorems that at first glance have nothing in common: [18, Theorem 2.1] isrecovered for the trivial coring C = R, while [5, Theorem 4.1] is obtained as aparticular case for U = S = R if in addition to that we impose and predefine Cto be the object of representability of RR . Example 3.5 and Corollary 3.10explain that various theorems [10, Theorem 2.4], [5, Theorem 4.1], etc. givingnecessary and sufficient conditions for a forgetful functor to be Frobenius areparticular cases of representability. As a bonus of our approach, Theorem 3.6 givesnecessary and sufficient conditions for U R to be an equivalence of categories.The Morita equivalence between two categories of modules is recovered as a specialcase corresponding to the trivial coring C = R. Finally, Corollaries 3.11 and 3.15give necessary and sufficient conditions for two types of induction functors to beseparable functors in the case that there exists what we have called a comodule dualbasis of first (or second) kind. Moreover, (2) of Corollary 3.11 is in fact an answerfor Problem D, concerning the separability of a representable functor.</p><p>1. PRELIMINARIES</p><p>We denote by et the category of sets and b the category of abelian groups.All functors in this article will be covariant functors. op will be the opposite ofa category . Except the category below all categories are locally small, thatis, HomCD is a set for any objects C, D . We denote by NatFG theclass of all natural transformations between two functors F , G and by</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f T</p><p>oron</p><p>to L</p><p>ibra</p><p>ries</p><p>] at</p><p> 08:</p><p>47 0</p><p>5 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>REPRESENTABLE FUNCTORS FOR CORINGS 1769</p><p>Functors = the large category of all functors F . The morphismsbetween two functors F , G are all natural transformations F G.</p><p>Let R, S be two rings. We denote by R, S , RS the categories ofleft R-modules, right S-modules, R S-bimodules. RHomMN, HomSMN,RHomSMN will be the morphisms in the respective categories. For two R S-bimodules P, Q RS , we have that P S QS (natural isomorphism) if andonly if P Q (isomorphism of R S-bimodules). For an R-bimodule M , we denoteby MR = m M rm = mrr R the set of R-centralized elements.</p><p>Representable Functors</p><p>A covariant functor F et is called representable if there exists C ,called the representing (universal or generic) object of F , such that F HomCin et . Repet will be the full subcategory of Functorset of allrepresentable functors. The Yoneda lemma states that for any functor F etand C the map</p><p> NatHomC F FC = CIdC (1)</p><p>is a bijection between sets with the inverse given by</p><p>1xDf = Ffx (2)</p><p>for all x FC, D , and f HomCD. As a consequence, the functor calledYoneda embedding,</p><p>Y op Functorset YC = HomC Yf = Homf(3)</p><p>for all C, D and f HomCD is faithful and full. Thus, using thecharacterization of equivalences of categories [17], we obtain that the Yonedaembedding gives an equivalence of categories</p><p>op Repet C HomC</p><p>The above definition of representable functors is the classical one: it can begeneralized replacing the category et to a variety of algebras in the sense ofuniversal algebra [1] (for example, can be the category of groups, abelian groups,rings, k-algebras, R, etc.).</p><p>Definition 1.1. Let be a variety of algebras and et the forgetfulfunctor. A functor F is called representable if F et isrepresentable in the classical sense.</p><p>Let be a category and a variety of algebras. We shall give some examplesrelated to Problems A, B, C from the Introduction. The Yoneda lemma gives a firstanswer for Problem A in the trivial case = et. In the following, we shall give asan example one of the fundamental theorems of [1]. Let R be a ring; an R-ring is a</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f T</p><p>oron</p><p>to L</p><p>ibra</p><p>ries</p><p>] at</p><p> 08:</p><p>47 0</p><p>5 N</p><p>ovem</p><p>ber </p><p>2014</p></li><li><p>1770 MILITARU</p><p>semigroup in the monoidal category RRR of R-bimodules. Let Ringsbe the category of R-rings with R-bimodule maps that respect the multiplications onR-rings as morphisms. Then the functor</p><p> RRop RepRingsb M = RHomRM</p><p>is an equivalence of categories ([1, Theorem 13.15]).Concerning Problem B, besides the universal constructions in mathematics</p><p>that give answers for it in the trivial case = et, we shall indicate two moreexamples. The first one was proven by Morita in [18, Theorem 2.1]: for a S R-bimodule V the induction functor V R R S is representable if and onlyif V is finitely generated projective as a right R-module. As explained in theIntroduction, the property of a functor to be Frobenius can be restated moreelegantly as a representability problem, predefining the object of representability.For instance, [10, Theorem 4.2] can be res...</p></li></ul>