On Rational Pairings of Functors

arXiv:1003.3221v1 [math.CT] 16 Mar 2010 ON RATIONAL PAIRINGS OF FUNCTORS BACHUKI MESABLISHVILI AND ROBERT WISBAUER Abstract. In the theory of coalgebras C over a ring R, the rational functor relates the category of modules over the algebra C∗(with convolution product) with the category of comodules over C. It is based on the pairing of the algebra C∗with the coalgebra C provided by the evaluation map ev : C∗⊗R C →R. We generalise this situation by defining a pairing between endofunctors T and G on any category A as a map, natural in a, b ∈A, βa,b : A(a, G(b)) →A(T(a), b), and we call it rational if these all are injective. In case T = (T, mT , eT ) is a monad and G = (G, δG, εG) is a comonad on A, additional compatibility conditions are imposed on a pairing between T and G. If such a pairing is given and is rational, and T has a right adjoint monad T⋄, we construct a rational functor as the functor-part of an idempotent comonad on the T-modules AT which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories. Contents 1. Introduction 1 2. Preliminaries 3 3. Pairings of functors 5 4. Rational functors 12 5. Pairings in monoidal categories 15 6. Entwinings in monoidal categories 19 References 31 1. Introduction The pairing of a k-vector space V with its dual space V ∗= Hom(V, k) provided by the evaluation map V ∗⊗V →k can be extended from base fields k to arbitrary base rings A. Then it can be applied to the study of A-corings C to obtain a faithful functor from the category of C-comodules to the category of C∗-modules. The purpose of this paper it to extend these results to (endo)functors on arbitrary categories. We begin by recalling some facts from module theory. 1.1. Pairing of modules. Let C be a bimodule over a ring A and C∗= HomA(C, A) the right dual. Then C ⊗A −and C∗⊗A −are endofunctors on the category AM of left A-modules and the evaluation ev : C∗⊗A C →A, f ⊗c 7→f(c), induces a pairing between these functors. For left A-modules X, Y , the map αY : C ⊗A Y →AHom(C∗, Y ), c ⊗y 7→[f 7→f(c)y], induces the map βX,Y : AHom(X, C ⊗A Y ) −→ AHom(X, AHom(C∗, Y )), X f→C ⊗A Y 7−→ X f→C ⊗A Y αY −→AHom(C∗, Y ). 1 2 BACHUKI MESABLISHVILI AND ROBERT WISBAUER Clearly βX,Y is injective for all left A-modules X, Y if and only if αY is a monomorphism (injective) for any left A-module Y , that is, CA is locally projective (see [1], [8, 42.10]). Now consider the situation above with some additional structure. 1.2. Pairings for corings. Let C = (C, ∆, ε) be a coring over the ring A, that is, C is an A-bimodule with bimodule morphisms coproduct ∆: C →C ⊗A C and counit ε : C →A. Then the right dual C∗= HomA(C, A) has a ring structure by the convolution product for f, g ∈C∗, f ∗g = f ◦(g ⊗A IC) ◦∆(convention opposite to [8, 17.8]) with unit ε, and we have a pairing between the comonad C ⊗A −and the monad C∗⊗A −on AM. In this case, HomA(C∗, −) is a comonad on AM and αY considered in 1.1 induces a comonad morphism α : C ⊗−→HomA(C∗, −). We have the commutative diagrams (1.1) C∗⊗A C∗⊗A C I⊗I⊗∆/ ∗⊗I  C∗⊗A C∗⊗A C ⊗A C I⊗ev⊗I/ C∗⊗A C ev  C ε⊗I o ε zvvvvvvvvvv C∗⊗A C ev / A . The Eilenberg-Moore category MHom(C∗,−) of Hom(C∗, −)-comodules is equivalent to the category C∗M of left C∗-modules (e.g. [5, Section 3]) and thus α induces a functor CM →MHom(C∗,−) ≃C∗M which is fully faithful if and only if the pairing (C∗, C, ev) is rational, that is αY is monomorph for all Y ∈AM (see [8, 19.2 and 19.3]). Moreover, α is an isomorphism if and only if the categories CM and C∗M are equivalent and this is tantamount to CA being finitely generated and projective. In Section 2 we recall the notions and some basic facts on natural transformations between endofuctors needed for our investigations. Weakening the conditions for an adjoint pair of functors, a pairing of two functors T : A →B and G : B →A is defined as a map βa,b : A(a, G(b)) →A(T (a), b), natural in a ∈A, b ∈B of two functors between arbitrary categories is defined in Section 3 (see 3.1) and it is called rational if all the βa,b are injective maps. For pairing of monads T with comonads G on a category A, additional conditions are imposed on the defining natural transformations (see 3.2). These imply the existence of a functor ΦP : AG →AT from the G-comodules to the T-modules (see 3.5), which is full and faithful provided the pairing is rational (see 3.7). Of special interest is the situation that the monad T has a right adjoint T⋄and the last part of Section 3 is dealing with this case. Referring to these results, a rational functor RatP : AT →AT is associated with any rational pairing in Section 4. This leads to the definition of rational T -modules and under some additional conditions they form a coreflective subcatgeory of AT (see 4.8). The application of the general notions of pairings to monoidal categories is outlined in Section 5. The resulting formalism is very close to the module case considered in

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