adjunction

[dannypsnl]

Compare to equivalence of categories, adjunction is more weak. Instead of naturally isomorphic to identity functor, adjunction with two functors $R : C \to D$ and $L : C \to D$ only needs the followings exist

1. A natural transformation $\eta : I_D \to R \circ L$
2. and a natural transformation $\epsilon : L \circ R \to I_C$

The notation for adjunction is:

$$ L \dashv R $$

Naming

$L$ is called the left adjoint to the functor $R$

$R$ is called the right adjoint to the functor $L$

$\eta$ is called the unit of the adjunction

$\epsilon$ is called the counit of the adjunction

Fact

Adjunction forms monad, every monad can be break to an adjunction

[dannypsnl]

Adjunction 是兩個 functors $R : C \to D, L : D \to C$ 的一種關係，存在下列兩個 natural transformation 時

1. $\eta : I_D \to R \circ L$
2. $\epsilon : L \circ R \to I_C$

這時稱他們有 adjunction 關係，記成

$$L \dashv R$$

[dannypsnl]

其實好像也不用特別寫，就直接參考 nlab