Recollements, comma categories and morphic enhancements

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$ , there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$ . Examples related to inflation categories and weighted projective lines are discussed.

Objective Triangle Functors in Adjoint Pairs

An additive functor [Formula: see text] → [Formula: see text] between additive categories is objective if any morphism f in [Formula: see text] with F(f) = 0 factors through an object K with F(K) = 0. We consider when a triangle functor in an adjoint pair is objective. We show that a triangle functor is objective provided that its adjoint (whatever left adjoint or right adjoint) is full or dense. We also give an example to show that the adjoint of a faithful triangle functor is not necessarily objective. In particular, the adjoint of an objective triangle functor is not necessarily objective. This is in contrast to the well-known fact that the adjoint of a triangle functor is always a triangle functor. Also, for an arbitrary adjoint pair (F, G) between categories which are not necessarily additive, we give a sufficient and necessary condition such that F (resp., G) is full or faithful.