2010/03/16 by Mesablishvili, Bachuki, Wisbauer, Robert · 1 citation
#Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1003.3221
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^*\otR C→ R. We generalise this situation by defining a \em 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 \em rational if these all are injective. In case \bT=(T,mT,eT) is a monad and \bG=(G,δG,\veG) is a comonad on \A, additional compatibility conditions are imposed on a pairing between \bT and \bG. If such a pairing is given and is rational, and \bT has a right adjoint monad \bT^\di, we construct a \em rational functor as the functor-part of an idempotent comonad on the \bT-modules \A\rT which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.