Pluri-adjoints and preservation of finite limits

Abstract

If G is a functor between two categories A and X , Freyd's Adjoint Functor Theorem provides, under suitable conditions, the connection between the existence of a (left) adjoint F for G and the preservation by G of all small limits (that are assumed to exist in A ). The present paper deals with the situation when the left adjoint for G fails to exist, yet some conditions are present that come close to its existence. Thus, for each object X of X , a set { F τ ( X) τ∈ T( X)} of objects, with certain properties, is given rather than a single object F( X). These data and properties define a pluri-adjoint for G. the existence of a pluri-adjoint for G is shown to be equivalent to the fact that G preserves finite limits rather than arbitrary small limits. Several examples are provided. In particular, it is shown that the distributive property in a lattice is equivalent to the existence of some pluri-adjoint.