Applied Categorical Structures

, Volume 21, Issue 3, pp 249–290 | Cite as

On Rational Pairings of Functors

  • Bachuki MesablishviliEmail author
  • Robert Wisbauer


In the theory of coalgebras C over a ring R, the rational functor relates the category \(_{C^*}{\mathbb{M}}\) of modules over the algebra C * (with convolution product) with the category \(^C{\mathbb{M}}\) of comodules over C. This is based on the pairing of the algebra C * with the coalgebra C provided by the evaluation map \({\rm ev}:C^*\otimes_R C\to R\). The (rationality) condition under consideration ensures that \(^C{\mathbb{M}}\) becomes a coreflective full subcategory of \(_{C^*}{\mathbb{M}}\). We generalise this situation by defining a pairing between endofunctors T and G on any category \({\mathbb{A}}\) as a map, natural in \(a,b\in {\mathbb{A}}\),
$$ \beta_{a,b}:{\mathbb{A}}(a, G(b)) \to {\mathbb{A}}(T(a),b), $$
and we call it rational if these all are injective. In case T = (T, m T , e T ) is a monad and G = (G, δ G , ε G ) is a comonad on \({\mathbb{A}}\), 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 \({\mathbb{A}}_{T}\) which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.


Rational module Rational functor (Idempotent) comonad 

Mathematics Subject Classifications (2010)

18C5 16T15 18C20 18D10 


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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Razmadze Mathematical InstituteTbilisiRepublic of Georgia
  2. 2.Tbilisi Centre for Mathematical SciencesTbilisiRepublic of Georgia
  3. 3.Department of Mathematics of HHUDüsseldorfGermany

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