Semiseparable functors and conditions up to retracts (Q6607369)

The authors [\textit{A. Ardizzoni} and \textit{L. Bottegoni}, J. Algebra 638, 862--917 (2024; Zbl 1531.18004)] introduced the notion of a semiseparable functor, where the properties of semiseparable functors with an adjoint are investigated. Any separable functors in the sense of \textit{C. Năstăsescu} et al. [J. Algebra 123, No. 2, 397--413 (1989; Zbl 0673.16026)] is semiseparable.\N\NThis paper continues the above study in connections with idempotent (Cauchy) completion, introducing and invesigating the notions of (co)reflection and bireflection up to retracts.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 1] recalls the known results on semiseparable functors.\N\item[\S 2] is concerned with results involving the idempotent completion. How the notions of faithful, full, fully faithful, semiseparable, separable or naaturally \ full functor behave with respect to idempotent completion is investigated. (Co)reflections up to retracts and bireflections up to retracts are introduced and invesigated. Semifunctors and semiadjunctions [\textit{S. Hayashi}, Theor. Comput. Sci. 41, 95--104 (1985; Zbl 0592.03010)] are considered as a tool to provide a characterization of (co)reflections up to retracts. It is shown that a (co)reflection up to retracts comes out to be always surjective up to retracts, and sufficient conditions guaranteeing that a functor is a (co)reflection up to retracts are given.\N\item[\S 3] collects the fall-outs of the results achieved so far. It is established that the quotient functor onto the coidentifier category is a coreflection up to retracts and that so is the comparison functor attached to an adjunction whose associated monad is separable. A dual result is obtained for the cocomparison functor in case that the associated comonad is coseparable. These results allow of characterizing a semiseparable right (left) adjoint in terms of (co)separability of the associated (co)monad and the requirement that the (co)comparison functor is a bireflection up to retracts. It is shown that two canonical factorizations attached to a semiseparable right adjoint functor are the same up to an equivalence up to retracts. The idempotent completions of the Kleisli category and Eilenberg-Moore category attached to a separable monad are related, while, in case this monad is induced by an adjunction with semiseparable right adjoint, the idempotent completion of the coidentifier category is added to the picture. An analogue for semiseparable functors of a result obtained by \textit{P. Balmer} [Adv. Math. 226, No. 5, 4352--4372 (2011; Zbl 1236.18013), Theorem 4.1] in the framework of pre-triangulated categories is established. \N\end{itemize}