为什么left adjoint的存在性和comma category有关？ 第1页

首先我们知道adjunction有两种等价的定义：

Definition 1: Suppose and are categories and and are functors. Then F is left adjoint to G and G is right adjoint to F, notated , iff naturally in , . We say that F and G together with the associated isomorphism between the relevant com-set form an adjunction
Definition 2: Suppose and are categories and and are functors. Then F is left adjoint to G and G is right adjoint to F, notated , contain two natural transformations satisfies two triangle identities.

既然这两种定义等价，那么我们可以问一个自然的问题：能不能把adjunction的两种定义揉合在一起，各取一半，变成一个新的adjunction的定义？答案明显是可以的。我们有如下的theorem：

Suppose and are categories and and are functors. Then iff (i) there is a natural transformation , for which (ii) for any in , there is a unique in such that .

可以看出(i)就是definition 2的“一半”, (ii)就是defintion 1的“一半”。同时我们可以看出这个定义是一种universal property, 而universal property 暗示某种合适的dervied category 存在initial or terminal object。 这种dervied category 就是某种特殊的 comma category！

Suppose is a functor. If the derived comma category has an initial object for every , then G has a left adjoint.