In a previous paper we introduced the concept of semiseparable functor. Here we continue our study of these functors in connection with idempotent (Cauchy) completion. To this aim, we introduce and investigate the notions of (co)reflection and bireflection up to retracts. We show that the (co)comparison functor attached to an adjunction whose associated (co)monad is separable is a coreflection (reflection) up to retracts. This fact allows us to prove that a right (left) adjoint functor is semiseparable if and only if the associated (co)monad is separable and the (co)comparison functor is a bireflection up to retracts, extending a characterization pursued by X.-W. Chen in the separable case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in the framework of pre-triangulated categories.

Semiseparable Functors and Conditions up to Retracts / Ardizzoni, Alessandro; Bottegoni, Lucrezia. - In: APPLIED CATEGORICAL STRUCTURES. - ISSN 0927-2852. - ELETTRONICO. - 32:5(2024), pp. 1-33. [10.1007/s10485-024-09782-z]

Semiseparable Functors and Conditions up to Retracts

Bottegoni, Lucrezia
2024

Abstract

In a previous paper we introduced the concept of semiseparable functor. Here we continue our study of these functors in connection with idempotent (Cauchy) completion. To this aim, we introduce and investigate the notions of (co)reflection and bireflection up to retracts. We show that the (co)comparison functor attached to an adjunction whose associated (co)monad is separable is a coreflection (reflection) up to retracts. This fact allows us to prove that a right (left) adjoint functor is semiseparable if and only if the associated (co)monad is separable and the (co)comparison functor is a bireflection up to retracts, extending a characterization pursued by X.-W. Chen in the separable case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in the framework of pre-triangulated categories.
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Descrizione: Semiseparable functors and conditions up to retracts
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2992262