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July 30th, 2019, 05:01 AM   #1
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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 monad-comonad 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 hom-set 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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