Abstract
Generalizations of right adjointness (i.e. having a left adjoint) of a functor G:A→X are studied. G is called weakly right adjoint, if for any X ∈ ObX there exists an AX ∈ ObA and an arrow eX:X→ GAX, such that for any f:X→GB there is a (not necessarily unique) morphism f′:AX→B inA with (Gf′)ex=f. As weakly right adjoint functors do not have so many interesting properties, it is useful to consider weakly right adjoint functors with a certain uniqueness condition. There are three ways for doing this, first by assuming uniqueness only for special f' s, second by assuming uniquness only “up to automorphisms”, and third by assuming a “canonical choice” of f′. A different way of generalizing right adjointness are the “locally adjunctable functors” of Kaput [5]. These weaker notions of adjointness are compared, their continuity properties are studied and the problem, when they imply right adjointness is discussed.
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