Adjoint Pair In Triangulated Categories

This paper systematically reviews some theories of adjoint pair that disperse in various documents.This article starts from the concept of adjoint pair,discusses some basic properties.In addition category,we concludes with the equivalent descriptions of adjoint pair,and talk about the relationships between the existence of left and right adjoint functors with representable functor.Further,in abelian category,a left adjoint is right exact functor and a right adjoint is left exact functor;when the additive functor has left(right)adjoint,then it keeps direct sums and products.On this basis,it concludes that the equivalence functor preserves direct sums and products.Then it introduces the concept of compact objects,discusses the relationships between the functor preserving direct sums and preserving compact objects,and then we concluded that the equivalence functor preserves compact objects.We then discuss some basic properties of triangulated categories,by introducing perfectly generation,symmetrically generation,Brown representability theorem and Brown representability for the dual,we need an important conclusion in the triangulated categories,namely,one of functor in adjoint pair is a triangulated functor if and only if the other is also a triangulated functor.Applying this conclusion,we get the main conclusion of this article:a functor preserves direct sums if and only if it has a right(triangulated)adjoint;a functor preserves direct products if and only if it has a left(triangulated)adjoint.