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July 30th, 2019, 05:01 AM  #1 
Newbie Joined: Jul 2019 From: India Posts: 1 Thanks: 0  Category Theory  Monad
Hello, I am a programmer and encountered Monad as functional programming pattern. I want to deeply learn the concept so I learned some points from category theory which can explain me Monad from math's perspective. I learned composition, functor, natural transformation and adjunction. I learned that adjunction creates monadcomonad pair. Suppose there is category C and D. Each x from C maps to Gx via functor $\displaystyle G: C \rightarrow D$ in category D, and each y in D maps to Fy in C via functor $\displaystyle F: D \rightarrow C$ and there is $\displaystyle f : Fy \rightarrow x$ and $\displaystyle g : y \rightarrow Gx$ We have adjunction with $\displaystyle {Hom}_C(Fy, x) \cong {Hom}_D(y, Gx)$. Here, we get $\displaystyle {\eta}_y : y \rightarrow GFy$, $\displaystyle {\eta}_{Gx} : Gx \rightarrow GFGx$, $\displaystyle {\eta}_{GFy} : GFy \rightarrow GFGFy$, $\displaystyle {\eta}_{GFGx} : GFGx \rightarrow GFGFGx$ and $\displaystyle {\epsilon}_x : FGx \rightarrow x$, $\displaystyle {\epsilon}_{Fy} : FGFy \rightarrow Fy$, $\displaystyle {\epsilon}_{FGx} : FGFGx \rightarrow FGx$, $\displaystyle {\epsilon}_{FGFy} : FGFGFy \rightarrow FGFy$ in adjunction such that above homset satisfied. Now, $\displaystyle G \circ F$ is a monad and $\displaystyle F \circ G$ is a comonad Monad has two natural transformations $\displaystyle \eta : I \rightarrow GF$ and $\displaystyle \mu : GFGF \rightarrow GF$ and Comonad has below two natural transformations $\displaystyle \epsilon: FG \rightarrow I$ $\displaystyle \delta : FG \rightarrow FGFG$ I come to know that natural transformation $\displaystyle \eta : I \rightarrow GF$ has naturality square like $\displaystyle {\eta}_{Gx} \circ g = GF(g) \circ {\eta}_y $ and $\displaystyle \epsilon : FG \rightarrow I$ has $\displaystyle f \circ {\epsilon}_{Fy} = {\epsilon}_x \circ FG(f)$ That's how I can get first laws for monad and comonad, but I cannot wrap my head around second laws for both. $\displaystyle \mu : GFGF \rightarrow GF$ $\displaystyle \delta: FG\rightarrow FGFG$ How do this above laws emerge from adjunction? Which naturality squares exist for above two laws? It will be a great help as I am stuck only on the last point. Thanks 

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